Cf. "Type Inference in Stack-Based Programming Languages" by Rob Kleffner, 2017-03-10.
+"Typing Tools for Typeless Stack Languages" by Jaanus Pöial
+"Typing Tools for Typeless Stack Languages" by Jaanus Pöial
@INPROCEEDINGS{Pöial06typingtools,
author = {Jaanus Pöial},
@@ -11826,7 +11827,7 @@ div#notebook {
The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be unified or a type conflict declared.
+The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be unified or a type conflict declared.
(a -- b t[i])∘(c u[j] -- d) t <= u (t is subtype of u)
-------------------------------
@@ -11844,7 +11845,7 @@ div#notebook {
Let's work through some examples by hand to develop an intuition for the algorithm.
+Let's work through some examples by hand to develop an intuition for the algorithm.
Here's that trick to represent functions like rest and cons that manipulate lists. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works.
Here's that trick to represent functions like rest and cons that manipulate stacks. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works.
Compare with the stack effect comment and you can see it works fine:
-([4 5 ...] 2 3 1 -- [3 2 ...])([4 5 ...] 2 3 1 -- [3 2 ...])
+ 3 4 5 1 2 0 2 1 5def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -12951,7 +12953,7 @@ cons ( 1 [...] -- [1 ...] )
So far we have dealt with types of functions, those dealing with simple stack manipulation. Let's extend our machinery to deal with types of arguments.
+So far we have dealt with types of functions, those dealing with simple stack manipulation. Let's extend our machinery to deal with types of arguments.
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -14659,7 +14661,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
Consider the stack function which grabs the whole stack, quotes it, and puts it on itself:
Consider the stack function which grabs the whole stack, quotes it, and puts it on itself:
stack (... -- ... [...] )
stack (... a -- ... a [a ...] )
@@ -14752,7 +14754,7 @@ uncons = ((a, Z), S) -- (Z, (a, S))compose() version 2¶This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification.
+compose() version 2¶This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification. Further, the first and second of Pöial's rules are now handled automatically by the unification algorithm.
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- fg_in, fg_out = f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- fg_in, fg_out = stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if s == False: # s can also be the empty dict, which is ok.
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- fg_in, fg_out = update(s, (f_in, g_out))
-
- return fg_in, fg_out
-
-
-stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+ s = unify(g_in, f_out)
+ if s == False: # s can also be the empty dict, which is ok.
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ return update(s, (f_in, g_out))
concat¶How to deal with concat?
concat ([.0.] [.1.] -- [.0. .1.])
-
-We would like to represent this in Python somehow...
- -concat = (S[0], S[1]), ((S[0], S[1]),)
-But this is actually cons with the first argument restricted to be a stack:
([.0.] [.1.] -- [[.0.] .1.])
-
-What we have implemented so far would actually only permit:
- -([.0.] [.1.] -- [.2.])concat = (S[0], S[1]), (S[2],)
-Which works but can lose information. Consider cons concat, this is how much information we could retain:
(1 [.0.] [.1.] -- [1 .0. .1.])
-
-As opposed to just:
- -(1 [.0.] [.1.] -- [.2.])We can invent a new type of type variable, a "sequence type" (I think this is what they mean in the literature by that term...) or "Kleene Star" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more AnyJoyType variables would be:
We can borrow a trick from Brzozowski's Derivatives of Regular Expressions to invent a new type of type variable, a "sequence type" (I think this is what they mean in the literature by that term...) or "Kleene Star" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more AnyJoyType variables would be:
A*class KleeneStar(object):
@@ -15796,7 +15709,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
def unify(u, v, s=None):
@@ -15845,7 +15758,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
sn.update(s)
return t
- ses = unify(u[0], v[0])
+ ses = unify(u[0], v[0], s)
results = ()
for sn in ses:
results += unify(u[1], v[1], sn)
@@ -15877,7 +15790,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
a = (As[1], S[1])
@@ -15894,7 +15807,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Out[93]:
+Out[91]:
@@ -15911,7 +15824,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [94]:
+In [92]:
b = (A[1], S[2])
@@ -15928,7 +15841,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Out[94]:
+Out[92]:
@@ -15945,7 +15858,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [95]:
+In [93]:
for result in unify(b, a):
@@ -15978,7 +15891,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [96]:
+In [94]:
for result in unify(a, b):
@@ -16025,7 +15938,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [97]:
+In [95]:
sum_ = ((Ns[1], S[1]), S[0]), (N[0], S[0])
@@ -16058,7 +15971,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [98]:
+In [96]:
f = (N[1], (N[2], (N[3], S[1]))), S[0]
@@ -16091,7 +16004,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [99]:
+In [97]:
for result in unify(sum_[0], f):
@@ -16128,35 +16041,23 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [100]:
+In [98]:
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- yield f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- yield stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if not s:
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- for result in s:
- yield update(result, (f_in, g_out))
+ s = unify(g_in, f_out)
+ if not s:
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ for result in s:
+ yield update(result, (f_in, g_out))
@@ -16166,7 +16067,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [101]:
+In [99]:
def meta_compose(F, G):
@@ -16191,7 +16092,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [102]:
+In [100]:
for f in MC([dup], muls):
@@ -16212,8 +16113,8 @@ uncons = ([a1 .1.] -- a1 [.1.])
@@ -16224,7 +16125,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [103]:
+In [101]:
for f in MC([dup], [sum_]):
@@ -16256,7 +16157,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [104]:
+In [102]:
for f in MC([cons], [sum_]):
@@ -16289,7 +16190,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [105]:
+In [103]:
sum_ = (((N[1], (Ns[1], S[1])), S[0]), (N[0], S[0]))
@@ -16325,7 +16226,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [106]:
+In [104]:
a = (A[4], (As[1], (A[3], S[1])))
@@ -16342,7 +16243,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Out[106]:
+Out[104]:
@@ -16359,7 +16260,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [107]:
+In [105]:
b = (A[1], (A[2], S[2]))
@@ -16376,7 +16277,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Out[107]:
+Out[105]:
@@ -16393,7 +16294,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [108]:
+In [106]:
for result in unify(b, a):
@@ -16426,7 +16327,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [109]:
+In [107]:
for result in unify(a, b):
@@ -16456,6 +16357,798 @@ uncons = ([a1 .1.] -- a1 [.1.])
+
+
+
+
+
+Part VII: Typing Combinators¶
TBD
+This is an open subject.
+The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:
+
+dip (a [F] --)--(-- F a)
+
+
+One thing that might help is...
+
+
+
+
+
+
+
+
+Consider the type of:
+
+[cons] dip
+
+
+Obviously it would be:
+
+(a1 [..1] a2 -- [a1 ..1] a2)
+
+
+dip itself could have:
+
+(a1 [..1] -- ... then what?
+
+
+
+
+
+
+In [108]:
+
+
+class SymbolJoyType(AnyJoyType): prefix = 'F'
+
+W = map(SymbolJoyType, _R)
+
+k = S[0], ((W[1], S[2]), S[0])
+Symbol('cons')
+print doc_from_stack_effect(*k)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [109]:
+
+
+dip_a = ((W[1], S[2]), (A[1], S[0]))
+
+
+
+
+
+
+
+
+
+In [110]:
+
+
+d = relabel(S[0], dip_a)
+print doc_from_stack_effect(*d)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [111]:
+
+
+s = list(unify(d[1], k[1]))[0]
+s
+
+
+
+
+
+
+
+
+
+
+
+
+Out[111]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [112]:
+
+
+j = update(s, k)
+
+
+
+
+
+
+
+
+
+In [113]:
+
+
+print doc_from_stack_effect(*j)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [114]:
+
+
+j
+
+
+
+
+
+
+
+
+
+
+
+
+Out[114]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [115]:
+
+
+cons
+
+
+
+
+
+
+
+
+
+
+
+
+Out[115]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [116]:
+
+
+for f in MC([k], [dup]):
+ print doc_from_stack_effect(*f)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [117]:
+
+
+l = S[0], ((cons, S[2]), (A[1], S[0]))
+
+
+
+
+
+
+
+
+
+In [118]:
+
+
+print doc_from_stack_effect(*l)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [119]:
+
+
+def dip_t(F):
+ (quote, (a1, sec)) = F[1]
+ G = F[0], sec
+ P = S[3], (a1, S[3])
+ a = [P]
+ while isinstance(quote, tuple):
+ term, quote = quote
+ a.append(term)
+ a.append(G)
+ return a[::-1]
+
+
+
+
+
+
+
+
+
+In [120]:
+
+
+from joy.utils.stack import iter_stack
+
+
+
+
+
+
+
+
+
+In [121]:
+
+
+a, b, c = dip_t(l)
+
+
+
+
+
+
+
+
+
+In [122]:
+
+
+a
+
+
+
+
+
+
+
+
+
+
+
+
+Out[122]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [123]:
+
+
+b
+
+
+
+
+
+
+
+
+
+
+
+
+Out[123]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [124]:
+
+
+c
+
+
+
+
+
+
+
+
+
+
+
+
+Out[124]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [125]:
+
+
+MC([a], [b])
+
+
+
+
+
+
+
+
+
+
+
+
+Out[125]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [126]:
+
+
+kjs = MC(MC([a], [b]), [c])
+kjs
+
+
+
+
+
+
+
+
+
+
+
+
+Out[126]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+In [127]:
+
+
+print doc_from_stack_effect(*kjs[0])
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(a0 [.0.] -- [a0 .0.] a1)
+
+ a0 [.0.] a1 [cons] dip
+----------------------------
+ [a0 .0.] a1
+
+
+
+
+
+
+In [128]:
+
+
+stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+
+
+
+
+
+
+
+
+
+In [129]:
+
+
+class SymbolJoyType(AnyJoyType):
+ prefix = 'F'
+
+ def __init__(self, name, sec, number):
+ self.name = name
+ self.stack_effects = sec
+ self.number = number
+
+class CombinatorJoyType(SymbolJoyType): prefix = 'C'
+
+def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+CONS = SymbolJoyType('cons', [cons], 23)
+DIP = CombinatorJoyType('dip', [dip_t], 44)
+
+
+def kav(F, e):
+ #i, stack = F
+ if not e:
+ return [(F, e)]
+ n, e = e
+ if isinstance(n, SymbolJoyType):
+ Fs = []
+ for sec in n.stack_effects:
+ Fs.extend(MC([F], sec))
+ return [kav(Fn, e) for Fn in Fs]
+ if isinstance(n, CombinatorJoyType):
+ res = []
+ for f in n.stack_effects:
+ s, e = f(F[1], e)
+ new_F = F[0], s
+ res.extend(kav(new_F, e))
+ return res
+ lit = S[0], (n, S[0])
+ return [kav(Fn, e) for Fn in MC([F], [lit])]
+
+
+
+
+
+
+
+
+
+
+
+compare, and be amazed:
+
+
+
+
+
+
+In [130]:
+
+
+def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+
+
+
+
+
+
+
+
+In [131]:
+
+
+def dip(stack, expression, dictionary):
+ (quote, (x, stack)) = stack
+ expression = (x, expression)
+ return stack, concat(quote, expression), dictionary
+
+
+
+
+
+
+
+
+
+
+
+And that brings us to current Work-In-Progress. I'm pretty hopeful that the mixed-mode inferencer/interpreter kav() function along with the ability to specify multiple implementations for the combinators will permit modelling of the stack effects of e.g. ifte. If I can keep up the pace I should be able to verify that conjecture by the end of June.
+
+
+
+
+
+
+
+
+The rest of this stuff is junk and/or unfinished material.
+
+
+
+
+
+
+
+
+concat¶
How to deal with concat?
+
+concat ([.0.] [.1.] -- [.0. .1.])
+
+
+We would like to represent this in Python somehow...
+
+
+
+
+
+
+In [132]:
+
+
+concat = (S[0], S[1]), ((S[0], S[1]),)
+
+
+
+
+
+
+
+
+
+
+
+But this is actually cons with the first argument restricted to be a stack:
+
+([.0.] [.1.] -- [[.0.] .1.])
+
+
+What we have implemented so far would actually only permit:
+
+([.0.] [.1.] -- [.2.])
+
+
+
+
+
+
+In [133]:
+
+
+concat = (S[0], S[1]), (S[2],)
+
+
+
+
+
+
+
+
+
+
+
+Which works but can lose information. Consider cons concat, this is how much information we could retain:
+
+(1 [.0.] [.1.] -- [1 .0. .1.])
+
+
+As opposed to just:
+
+(1 [.0.] [.1.] -- [.2.])
+
+
+
@@ -16502,7 +17195,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [110]:
+In [134]:
class Astar(object):
@@ -16529,7 +17222,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [111]:
+In [135]:
a, b = (A[1], S[0]), (A[2], S[1])
@@ -16542,7 +17235,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-In [112]:
+In [136]:
concat(a, b)
@@ -16558,7 +17251,7 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Out[112]:
+Out[136]:
@@ -16577,14 +17270,14 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Joy in the Logical Paradigm¶
For this to work the type label classes have to be modified to let T >= t succeed, where e.g. T is IntJoyType and t is int
+Appendix: Joy in the Logical Paradigm¶
For this to work the type label classes have to be modified to let T >= t succeed, where e.g. T is IntJoyType and t is int
-In [113]:
+In [137]:
F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
@@ -16609,75 +17302,75 @@ uncons = ([a1 .1.] -- a1 [.1.])
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
-<ipython-input-113-4b4cb6ff86e5> in <module>()
+<ipython-input-137-4b4cb6ff86e5> in <module>()
1 F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
2
----> 3 print doc_from_stack_effect(*F)
-<ipython-input-101-ddee30dbb1a6> in C(f, g)
+<ipython-input-99-ddee30dbb1a6> in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
-<ipython-input-100-4237a6bb159d> in compose(f, g)
+<ipython-input-98-5eb7ac5ad2c2> in compose(f, g)
1 def compose(f, g):
- 2
-----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
-<ipython-input-101-ddee30dbb1a6> in C(f, g)
+<ipython-input-99-ddee30dbb1a6> in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
-<ipython-input-100-4237a6bb159d> in compose(f, g)
+<ipython-input-98-5eb7ac5ad2c2> in compose(f, g)
1 def compose(f, g):
- 2
-----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
-<ipython-input-101-ddee30dbb1a6> in C(f, g)
+<ipython-input-99-ddee30dbb1a6> in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
-<ipython-input-100-4237a6bb159d> in compose(f, g)
+<ipython-input-98-5eb7ac5ad2c2> in compose(f, g)
1 def compose(f, g):
- 2
-----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
-<ipython-input-101-ddee30dbb1a6> in C(f, g)
+<ipython-input-99-ddee30dbb1a6> in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
-<ipython-input-100-4237a6bb159d> in compose(f, g)
+<ipython-input-98-5eb7ac5ad2c2> in compose(f, g)
1 def compose(f, g):
- 2
-----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
-<ipython-input-101-ddee30dbb1a6> in C(f, g)
+<ipython-input-99-ddee30dbb1a6> in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
-<ipython-input-100-4237a6bb159d> in compose(f, g)
+<ipython-input-98-5eb7ac5ad2c2> in compose(f, g)
1 def compose(f, g):
- 2
-----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
ValueError: need more than 1 value to unpack
@@ -16771,42 +17464,8 @@ uncons = ([a1 .1.] -- a1 [.1.])
-Typing Combinators¶
TBD
-This is an open subject.
-The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:
-
-dip (a [F] --)--(-- F a)
-
-
-One thing that might help is...
-
-
-
-
-
-
-
-
-Abstract Interpretation¶
-
-
-
-
-
-
-
-Something else...¶
-[4 5 ...] 2 1 0 pop∘swap∘roll<∘rest∘rest∘cons∘cons
-[4 5 ...] 2 1 swap∘roll<∘rest∘rest∘cons∘cons
-[4 5 ...] 1 2 roll<∘rest∘rest∘cons∘cons
-1 2 [4 5 ...] rest∘rest∘cons∘cons
-1 2 [5 ...] rest∘cons∘cons
-1 2 [...] cons∘cons
-1 [2 ...] cons
- [1 2 ...]
-
-
-Eh?
+Abstract Interpretation¶
I think this might be sorta what I'm doing above with the kav() function...
+ In any event "mixed-mode" interpreters that include values and type variables and can track constraints, etc. will be, uh, super-useful. And Abstract Interpretation should be a rich source of ideas.
diff --git a/docs/Types.ipynb b/docs/Types.ipynb
index ed16c7f..2f9ddf1 100644
--- a/docs/Types.ipynb
+++ b/docs/Types.ipynb
@@ -4,14 +4,16 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "# Type Inference"
+ "# Type Inference\n",
+ "\n",
+ "Cf. [\"Type Inference in Stack-Based Programming Languages\"](http://prl.ccs.neu.edu/blog/2017/03/10/type-inference-in-stack-based-programming-languages/) by Rob Kleffner, 2017-03-10."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Pöial's Rules\n",
+ "## Part I: Pöial's Rules\n",
"\n",
"[\"Typing Tools for Typeless Stack Languages\" by Jaanus Pöial\n",
"](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.212.6026)\n",
@@ -55,7 +57,7 @@
"metadata": {},
"source": [
"### Third Rule\n",
- "The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be *unified* or a type conflict declared.\n",
+ "The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be [*unified*](https://en.wikipedia.org/wiki/Robinson's_unification_algorithm) or a type conflict declared.\n",
"\n",
" (a -- b t[i])∘(c u[j] -- d) t <= u (t is subtype of u)\n",
" -------------------------------\n",
@@ -71,7 +73,6 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Examples\n",
"Let's work through some examples by hand to develop an intuition for the algorithm."
]
},
@@ -394,7 +395,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Implementation"
+ "## Part II: Implementation"
]
},
{
@@ -733,8 +734,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "### List Functions\n",
- "Here's that trick to represent functions like `rest` and `cons` that manipulate lists. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works. "
+ "### Stack Functions\n",
+ "Here's that trick to represent functions like `rest` and `cons` that manipulate stacks. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works. "
]
},
{
@@ -816,7 +817,8 @@
"source": [
"Compare with the stack effect comment and you can see it works fine:\n",
"\n",
- " ([4 5 ...] 2 3 1 -- [3 2 ...])"
+ " ([4 5 ...] 2 3 1 -- [3 2 ...])\n",
+ " 3 4 5 1 2 0 2 1 5"
]
},
{
@@ -873,7 +875,7 @@
"def unify(u, v, s=None):\n",
" if s is None:\n",
" s = {}\n",
- " else:\n",
+ " elif s:\n",
" u = update(s, u)\n",
" v = update(s, v)\n",
"\n",
@@ -924,7 +926,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Compiling\n",
+ "## Part III: Compiling Stack Functions\n",
"Now consider the Python function we would like to derive:"
]
},
@@ -1379,7 +1381,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Types and Subtypes of Arguments\n",
+ "## Part IV: Types and Subtypes of Arguments\n",
"So far we have dealt with types of functions, those dealing with simple stack manipulation. Let's extend our machinery to deal with types of arguments."
]
},
@@ -1753,7 +1755,7 @@
"def unify(u, v, s=None):\n",
" if s is None:\n",
" s = {}\n",
- " else:\n",
+ " elif s:\n",
" u = update(s, u)\n",
" v = update(s, v)\n",
"\n",
@@ -2208,7 +2210,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Functions that use the Stack\n",
+ "## Part V: Functions that use the Stack\n",
"\n",
"Consider the `stack` function which grabs the whole stack, quotes it, and puts it on itself:\n",
"\n",
@@ -2291,7 +2293,7 @@
"metadata": {},
"source": [
"#### `compose()` version 2\n",
- "This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification."
+ "This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification. Further, the first and second of Pöial's rules are now handled automatically by the unification algorithm."
]
},
{
@@ -2301,28 +2303,11 @@
"outputs": [],
"source": [
"def compose(f, g):\n",
- "\n",
" (f_in, f_out), (g_in, g_out) = f, g\n",
- "\n",
- " if not g_in:\n",
- " fg_in, fg_out = f_in, stack_concat(g_out, f_out)\n",
- "\n",
- " elif not f_out:\n",
- " fg_in, fg_out = stack_concat(f_in, g_in), g_out\n",
- "\n",
- " else: # Unify and update.\n",
- "\n",
- " s = unify(g_in, f_out)\n",
- "\n",
- " if s == False: # s can also be the empty dict, which is ok.\n",
- " raise TypeError('Cannot unify %r and %r.' % (fo, gi))\n",
- "\n",
- " fg_in, fg_out = update(s, (f_in, g_out))\n",
- "\n",
- " return fg_in, fg_out\n",
- "\n",
- "\n",
- "stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e"
+ " s = unify(g_in, f_out)\n",
+ " if s == False: # s can also be the empty dict, which is ok.\n",
+ " raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))\n",
+ " return update(s, (f_in, g_out))"
]
},
{
@@ -2766,7 +2751,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Multiple Stack Effects\n",
+ "## Part VI: Multiple Stack Effects\n",
"..."
]
},
@@ -2952,67 +2937,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## `concat`\n",
+ "### Representing an Unbounded Sequence of Types\n",
"\n",
- "How to deal with `concat`?\n",
- "\n",
- " concat ([.0.] [.1.] -- [.0. .1.])\n",
- " \n",
- "We would like to represent this in Python somehow..."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 89,
- "metadata": {},
- "outputs": [],
- "source": [
- "concat = (S[0], S[1]), ((S[0], S[1]),)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "But this is actually `cons` with the first argument restricted to be a stack:\n",
- "\n",
- " ([.0.] [.1.] -- [[.0.] .1.])\n",
- "\n",
- "What we have implemented so far would actually only permit:\n",
- "\n",
- " ([.0.] [.1.] -- [.2.])"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 90,
- "metadata": {},
- "outputs": [],
- "source": [
- "concat = (S[0], S[1]), (S[2],)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- " \n",
- "Which works but can lose information. Consider `cons concat`, this is how much information we *could* retain:\n",
- "\n",
- " (1 [.0.] [.1.] -- [1 .0. .1.])\n",
- "\n",
- "As opposed to just:\n",
- "\n",
- " (1 [.0.] [.1.] -- [.2.])"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Brzo...'s Derivitives of Regular Expressions\n",
- "\n",
- "We can invent a new type of type variable, a \"sequence type\" (I think this is what they mean in the literature by that term...) or \"Kleene Star\" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more `AnyJoyType` variables would be:\n",
+ "We can borrow a trick from [Brzozowski's Derivatives of Regular Expressions](https://en.wikipedia.org/wiki/Brzozowski_derivative) to invent a new type of type variable, a \"sequence type\" (I think this is what they mean in the literature by that term...) or \"[Kleene Star](https://en.wikipedia.org/wiki/Kleene_star)\" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more `AnyJoyType` variables would be:\n",
"\n",
" A*"
]
@@ -3060,7 +2987,7 @@
" w/ {d: e}\n",
" [ A* b .0.] U [ .1.]\n",
" w/ {.1.: A* b .0.}\n",
- " [ A* b .0.] U [ .1.]"
+ " [ A* b .0.] U [ A* b .0.]"
]
},
{
@@ -3075,7 +3002,7 @@
},
{
"cell_type": "code",
- "execution_count": 91,
+ "execution_count": 89,
"metadata": {},
"outputs": [],
"source": [
@@ -3140,7 +3067,7 @@
},
{
"cell_type": "code",
- "execution_count": 92,
+ "execution_count": 90,
"metadata": {},
"outputs": [],
"source": [
@@ -3190,7 +3117,7 @@
" sn.update(s)\n",
" return t\n",
"\n",
- " ses = unify(u[0], v[0])\n",
+ " ses = unify(u[0], v[0], s)\n",
" results = ()\n",
" for sn in ses:\n",
" results += unify(u[1], v[1], sn)\n",
@@ -3217,7 +3144,7 @@
},
{
"cell_type": "code",
- "execution_count": 93,
+ "execution_count": 91,
"metadata": {},
"outputs": [
{
@@ -3226,7 +3153,7 @@
"(a1*, s1)"
]
},
- "execution_count": 93,
+ "execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
@@ -3238,7 +3165,7 @@
},
{
"cell_type": "code",
- "execution_count": 94,
+ "execution_count": 92,
"metadata": {},
"outputs": [
{
@@ -3247,7 +3174,7 @@
"(a1, s2)"
]
},
- "execution_count": 94,
+ "execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
@@ -3259,7 +3186,7 @@
},
{
"cell_type": "code",
- "execution_count": 95,
+ "execution_count": 93,
"metadata": {},
"outputs": [
{
@@ -3278,7 +3205,7 @@
},
{
"cell_type": "code",
- "execution_count": 96,
+ "execution_count": 94,
"metadata": {},
"outputs": [
{
@@ -3309,7 +3236,7 @@
},
{
"cell_type": "code",
- "execution_count": 97,
+ "execution_count": 95,
"metadata": {},
"outputs": [
{
@@ -3328,7 +3255,7 @@
},
{
"cell_type": "code",
- "execution_count": 98,
+ "execution_count": 96,
"metadata": {},
"outputs": [
{
@@ -3347,7 +3274,7 @@
},
{
"cell_type": "code",
- "execution_count": 99,
+ "execution_count": 97,
"metadata": {},
"outputs": [
{
@@ -3385,34 +3312,22 @@
"metadata": {},
"source": [
"#### `compose()` version 3\n",
- "This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification."
+ "This function has to be modified to yield multiple results."
]
},
{
"cell_type": "code",
- "execution_count": 100,
+ "execution_count": 98,
"metadata": {},
"outputs": [],
"source": [
"def compose(f, g):\n",
- "\n",
" (f_in, f_out), (g_in, g_out) = f, g\n",
- "\n",
- " if not g_in:\n",
- " yield f_in, stack_concat(g_out, f_out)\n",
- "\n",
- " elif not f_out:\n",
- " yield stack_concat(f_in, g_in), g_out\n",
- "\n",
- " else: # Unify and update.\n",
- "\n",
- " s = unify(g_in, f_out)\n",
- "\n",
- " if not s:\n",
- " raise TypeError('Cannot unify %r and %r.' % (fo, gi))\n",
- "\n",
- " for result in s:\n",
- " yield update(result, (f_in, g_out))\n"
+ " s = unify(g_in, f_out)\n",
+ " if not s:\n",
+ " raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))\n",
+ " for result in s:\n",
+ " yield update(result, (f_in, g_out))\n"
]
},
{
@@ -3424,7 +3339,7 @@
},
{
"cell_type": "code",
- "execution_count": 101,
+ "execution_count": 99,
"metadata": {},
"outputs": [],
"source": [
@@ -3445,15 +3360,15 @@
},
{
"cell_type": "code",
- "execution_count": 102,
+ "execution_count": 100,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "(a0 -- f0)\n",
- "(a0 -- i0)\n"
+ "(f0 -- f1)\n",
+ "(i0 -- i1)\n"
]
}
],
@@ -3464,7 +3379,7 @@
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 101,
"metadata": {},
"outputs": [
{
@@ -3484,7 +3399,7 @@
},
{
"cell_type": "code",
- "execution_count": 104,
+ "execution_count": 102,
"metadata": {},
"outputs": [
{
@@ -3505,7 +3420,7 @@
},
{
"cell_type": "code",
- "execution_count": 105,
+ "execution_count": 103,
"metadata": {},
"outputs": [
{
@@ -3548,7 +3463,7 @@
},
{
"cell_type": "code",
- "execution_count": 106,
+ "execution_count": 104,
"metadata": {},
"outputs": [
{
@@ -3557,7 +3472,7 @@
"(a4, (a1*, (a3, s1)))"
]
},
- "execution_count": 106,
+ "execution_count": 104,
"metadata": {},
"output_type": "execute_result"
}
@@ -3569,7 +3484,7 @@
},
{
"cell_type": "code",
- "execution_count": 107,
+ "execution_count": 105,
"metadata": {},
"outputs": [
{
@@ -3578,7 +3493,7 @@
"(a1, (a2, s2))"
]
},
- "execution_count": 107,
+ "execution_count": 105,
"metadata": {},
"output_type": "execute_result"
}
@@ -3590,7 +3505,7 @@
},
{
"cell_type": "code",
- "execution_count": 108,
+ "execution_count": 106,
"metadata": {},
"outputs": [
{
@@ -3609,7 +3524,7 @@
},
{
"cell_type": "code",
- "execution_count": 109,
+ "execution_count": 107,
"metadata": {},
"outputs": [
{
@@ -3654,6 +3569,615 @@
"outputs": [],
"source": []
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Part VII: Typing Combinators\n",
+ "\n",
+ "TBD\n",
+ "\n",
+ "This is an open subject.\n",
+ "\n",
+ "The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:\n",
+ "\n",
+ " dip (a [F] --)--(-- F a)\n",
+ "\n",
+ "One thing that might help is..."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Consider the type of:\n",
+ "\n",
+ " [cons] dip\n",
+ "\n",
+ "Obviously it would be:\n",
+ "\n",
+ " (a1 [..1] a2 -- [a1 ..1] a2)\n",
+ "\n",
+ "`dip` itself could have:\n",
+ "\n",
+ " (a1 [..1] -- ... then what?\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 108,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(-- [F1 .2.])\n"
+ ]
+ }
+ ],
+ "source": [
+ "class SymbolJoyType(AnyJoyType): prefix = 'F'\n",
+ "\n",
+ "W = map(SymbolJoyType, _R)\n",
+ "\n",
+ "k = S[0], ((W[1], S[2]), S[0])\n",
+ "Symbol('cons')\n",
+ "print doc_from_stack_effect(*k)\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 109,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "dip_a = ((W[1], S[2]), (A[1], S[0]))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 110,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(-- a1001 [F1001 .1002.])\n"
+ ]
+ }
+ ],
+ "source": [
+ "d = relabel(S[0], dip_a)\n",
+ "print doc_from_stack_effect(*d)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 111,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{s0: (a1001, s1000), s1002: s2, F1001: F1}"
+ ]
+ },
+ "execution_count": 111,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "s = list(unify(d[1], k[1]))[0]\n",
+ "s"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 112,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [],
+ "source": [
+ "j = update(s, k)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 113,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(a1001 -- a1001 [F1 .2.])\n"
+ ]
+ }
+ ],
+ "source": [
+ "print doc_from_stack_effect(*j)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 114,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "((a1001, s1000), ((F1, s2), (a1001, s1000)))"
+ ]
+ },
+ "execution_count": 114,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "j"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 115,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "((s1, (a1, s23)), ((a1, s1), s23))"
+ ]
+ },
+ "execution_count": 115,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "cons"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 116,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(-- [F0 .1.] [F0 .1.])\n"
+ ]
+ }
+ ],
+ "source": [
+ "for f in MC([k], [dup]):\n",
+ " print doc_from_stack_effect(*f)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 117,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "l = S[0], ((cons, S[2]), (A[1], S[0]))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 118,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(-- a1 [[[[.1.] a1 .23.] [a1 .1.] .23.] .2.])\n"
+ ]
+ }
+ ],
+ "source": [
+ "print doc_from_stack_effect(*l)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 119,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "\n",
+ "def dip_t(F):\n",
+ " (quote, (a1, sec)) = F[1]\n",
+ " G = F[0], sec\n",
+ " P = S[3], (a1, S[3])\n",
+ " a = [P]\n",
+ " while isinstance(quote, tuple):\n",
+ " term, quote = quote\n",
+ " a.append(term)\n",
+ " a.append(G)\n",
+ " return a[::-1]\n",
+ "\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 120,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from joy.utils.stack import iter_stack"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 121,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "a, b, c = dip_t(l)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 122,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(s0, s0)"
+ ]
+ },
+ "execution_count": 122,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "a"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 123,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "((s1, (a1, s23)), ((a1, s1), s23))"
+ ]
+ },
+ "execution_count": 123,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "b"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 124,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(s3, (a1, s3))"
+ ]
+ },
+ "execution_count": 124,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "c"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 125,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[((s0, (a0, s1)), ((a0, s0), s1))]"
+ ]
+ },
+ "execution_count": 125,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "MC([a], [b])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 126,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[((s0, (a0, s1)), (a1, ((a0, s0), s1)))]"
+ ]
+ },
+ "execution_count": 126,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "kjs = MC(MC([a], [b]), [c])\n",
+ "kjs"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 127,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "(a0 [.0.] -- [a0 .0.] a1)\n"
+ ]
+ }
+ ],
+ "source": [
+ "print doc_from_stack_effect(*kjs[0])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " (a0 [.0.] -- [a0 .0.] a1)\n",
+ " \n",
+ " a0 [.0.] a1 [cons] dip\n",
+ " ----------------------------\n",
+ " [a0 .0.] a1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 128,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 129,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "class SymbolJoyType(AnyJoyType):\n",
+ " prefix = 'F'\n",
+ "\n",
+ " def __init__(self, name, sec, number):\n",
+ " self.name = name\n",
+ " self.stack_effects = sec\n",
+ " self.number = number\n",
+ "\n",
+ "class CombinatorJoyType(SymbolJoyType): prefix = 'C'\n",
+ "\n",
+ "def dip_t(stack, expression):\n",
+ " (quote, (a1, stack)) = stack\n",
+ " expression = stack_concat(quote, (a1, expression))\n",
+ " return stack, expression\n",
+ "\n",
+ "CONS = SymbolJoyType('cons', [cons], 23)\n",
+ "DIP = CombinatorJoyType('dip', [dip_t], 44)\n",
+ "\n",
+ "\n",
+ "def kav(F, e):\n",
+ " #i, stack = F\n",
+ " if not e:\n",
+ " return [(F, e)]\n",
+ " n, e = e\n",
+ " if isinstance(n, SymbolJoyType):\n",
+ " Fs = []\n",
+ " for sec in n.stack_effects:\n",
+ " Fs.extend(MC([F], sec))\n",
+ " return [kav(Fn, e) for Fn in Fs]\n",
+ " if isinstance(n, CombinatorJoyType):\n",
+ " res = []\n",
+ " for f in n.stack_effects:\n",
+ " s, e = f(F[1], e)\n",
+ " new_F = F[0], s\n",
+ " res.extend(kav(new_F, e))\n",
+ " return res\n",
+ " lit = S[0], (n, S[0])\n",
+ " return [kav(Fn, e) for Fn in MC([F], [lit])]\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "compare, and be amazed:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 130,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def dip_t(stack, expression):\n",
+ " (quote, (a1, stack)) = stack\n",
+ " expression = stack_concat(quote, (a1, expression))\n",
+ " return stack, expression"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 131,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def dip(stack, expression, dictionary):\n",
+ " (quote, (x, stack)) = stack\n",
+ " expression = (x, expression)\n",
+ " return stack, concat(quote, expression), dictionary"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "And that brings us to current Work-In-Progress. I'm pretty hopeful that the mixed-mode inferencer/interpreter `kav()` function along with the ability to specify multiple implementations for the combinators will permit modelling of the stack effects of e.g. `ifte`. If I can keep up the pace I should be able to verify that conjecture by the end of June."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The rest of this stuff is junk and/or unfinished material."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### `concat`\n",
+ "\n",
+ "How to deal with `concat`?\n",
+ "\n",
+ " concat ([.0.] [.1.] -- [.0. .1.])\n",
+ " \n",
+ "We would like to represent this in Python somehow..."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 132,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "concat = (S[0], S[1]), ((S[0], S[1]),)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "But this is actually `cons` with the first argument restricted to be a stack:\n",
+ "\n",
+ " ([.0.] [.1.] -- [[.0.] .1.])\n",
+ "\n",
+ "What we have implemented so far would actually only permit:\n",
+ "\n",
+ " ([.0.] [.1.] -- [.2.])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 133,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "concat = (S[0], S[1]), (S[2],)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " \n",
+ "Which works but can lose information. Consider `cons concat`, this is how much information we *could* retain:\n",
+ "\n",
+ " (1 [.0.] [.1.] -- [1 .0. .1.])\n",
+ "\n",
+ "As opposed to just:\n",
+ "\n",
+ " (1 [.0.] [.1.] -- [.2.])"
+ ]
+ },
{
"cell_type": "markdown",
"metadata": {},
@@ -3696,7 +4220,7 @@
},
{
"cell_type": "code",
- "execution_count": 110,
+ "execution_count": 134,
"metadata": {},
"outputs": [],
"source": [
@@ -3719,7 +4243,7 @@
},
{
"cell_type": "code",
- "execution_count": 111,
+ "execution_count": 135,
"metadata": {},
"outputs": [],
"source": [
@@ -3728,7 +4252,7 @@
},
{
"cell_type": "code",
- "execution_count": 112,
+ "execution_count": 136,
"metadata": {},
"outputs": [
{
@@ -3737,7 +4261,7 @@
"(a1, (A*, (a2, s1)))"
]
},
- "execution_count": 112,
+ "execution_count": 136,
"metadata": {},
"output_type": "execute_result"
}
@@ -3769,13 +4293,13 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Joy in the Logical Paradigm\n",
+ "## Appendix: Joy in the Logical Paradigm\n",
"For this to work the type label classes have to be modified to let `T >= t` succeed, where e.g. `T` is `IntJoyType` and `t` is `int`"
]
},
{
"cell_type": "code",
- "execution_count": 113,
+ "execution_count": 137,
"metadata": {},
"outputs": [
{
@@ -3785,17 +4309,17 @@
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
- "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mF\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mswap\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mroll_down\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrest\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrest\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcons\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcons\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;32mprint\u001b[0m \u001b[0mdoc_from_stack_effect\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mF\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mswap\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mroll_down\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrest\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrest\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcons\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcons\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0;32mprint\u001b[0m \u001b[0mdoc_from_stack_effect\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0munify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Cannot unify %r and %r.'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf_out\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0munify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Cannot unify %r and %r.'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf_out\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0munify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Cannot unify %r and %r.'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf_out\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0munify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Cannot unify %r and %r.'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf_out\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mC\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0;32mfor\u001b[0m \u001b[0mfg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0;32myield\u001b[0m \u001b[0mdelabel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36mcompose\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcompose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;34m(\u001b[0m\u001b[0mf_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_out\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0ms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0munify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg_in\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_out\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Cannot unify %r and %r.'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf_out\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg_in\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mValueError\u001b[0m: need more than 1 value to unpack"
]
}
@@ -3880,41 +4404,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## Typing Combinators\n",
- "\n",
- "TBD\n",
- "\n",
- "This is an open subject.\n",
- "\n",
- "The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:\n",
- "\n",
- " dip (a [F] --)--(-- F a)\n",
- "\n",
- "One thing that might help is..."
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Abstract Interpretation"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Something else...\n",
- " [4 5 ...] 2 1 0 pop∘swap∘roll<∘rest∘rest∘cons∘cons\n",
- " [4 5 ...] 2 1 swap∘roll<∘rest∘rest∘cons∘cons\n",
- " [4 5 ...] 1 2 roll<∘rest∘rest∘cons∘cons\n",
- " 1 2 [4 5 ...] rest∘rest∘cons∘cons\n",
- " 1 2 [5 ...] rest∘cons∘cons\n",
- " 1 2 [...] cons∘cons\n",
- " 1 [2 ...] cons\n",
- " [1 2 ...]\n",
- "\n",
- "Eh?"
+ "## [Abstract Interpretation](https://en.wikipedia.org/wiki/Abstract_interpretation)\n",
+ "I *think* this might be sorta what I'm doing above with the `kav()` function...\n",
+ " In any event \"mixed-mode\" interpreters that include values and type variables and can track constraints, etc. will be, uh, super-useful. And Abstract Interpretation should be a rich source of ideas.\n"
]
},
{
diff --git a/docs/Types.md b/docs/Types.md
index 35be81b..b3f9f2a 100644
--- a/docs/Types.md
+++ b/docs/Types.md
@@ -1,7 +1,9 @@
# Type Inference
-## Pöial's Rules
+Cf. ["Type Inference in Stack-Based Programming Languages"](http://prl.ccs.neu.edu/blog/2017/03/10/type-inference-in-stack-based-programming-languages/) by Rob Kleffner, 2017-03-10.
+
+## Part I: Pöial's Rules
["Typing Tools for Typeless Stack Languages" by Jaanus Pöial
](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.212.6026)
@@ -30,7 +32,7 @@ This rule deals with functions that consume items from the stack `(a --)`:
(c a -- d)
### Third Rule
-The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be *unified* or a type conflict declared.
+The third rule is actually two rules. These two rules deal with composing functions when the second one will consume one of items the first one produces. The two types must be [*unified*](https://en.wikipedia.org/wiki/Robinson's_unification_algorithm) or a type conflict declared.
(a -- b t[i])∘(c u[j] -- d) t <= u (t is subtype of u)
-------------------------------
@@ -41,7 +43,6 @@ The third rule is actually two rules. These two rules deal with composing funct
-------------------------------
(a -- b )∘(c -- d) t[i] == u[k] == u[j]
-## Examples
Let's work through some examples by hand to develop an intuition for the algorithm.
There's a function in one of the other notebooks.
@@ -267,7 +268,7 @@ def F(stack):
return (d, (c, S0)), stack
```
-## Implementation
+## Part II: Implementation
### Representing Stack Effect Comments in Python
@@ -485,8 +486,8 @@ poswrd
-### List Functions
-Here's that trick to represent functions like `rest` and `cons` that manipulate lists. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works.
+### Stack Functions
+Here's that trick to represent functions like `rest` and `cons` that manipulate stacks. We use a cons-list of tuples and give the tails their own numbers. Then everything above already works.
```python
@@ -539,6 +540,7 @@ F
Compare with the stack effect comment and you can see it works fine:
([4 5 ...] 2 3 1 -- [3 2 ...])
+ 3 4 5 1 2 0 2 1 5
### Dealing with `cons` and `uncons`
However, if we try to compose e.g. `cons` and `uncons` it won't work:
@@ -567,7 +569,7 @@ The problem is that the `unify()` function as written doesn't handle the case wh
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -606,7 +608,7 @@ C(cons, uncons)
-## Compiling
+## Part III: Compiling Stack Functions
Now consider the Python function we would like to derive:
@@ -904,7 +906,7 @@ for name, stack_effect_comment in sorted(defs().items()):
-## Types and Subtypes of Arguments
+## Part IV: Types and Subtypes of Arguments
So far we have dealt with types of functions, those dealing with simple stack manipulation. Let's extend our machinery to deal with types of arguments.
### "Number" Type
@@ -1158,7 +1160,7 @@ delabel(foo)
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -1460,7 +1462,7 @@ for name, stack_effect_comment in sorted(defs().items()):
uncons = ([a1 .1.] -- a1 [.1.])
-## Functions that use the Stack
+## Part V: Functions that use the Stack
Consider the `stack` function which grabs the whole stack, quotes it, and puts it on itself:
@@ -1518,33 +1520,16 @@ Let's try `stack∘uncons∘uncons`:
It works.
#### `compose()` version 2
-This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification.
+This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification. Further, the first and second of Pöial's rules are now handled automatically by the unification algorithm.
```python
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- fg_in, fg_out = f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- fg_in, fg_out = stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if s == False: # s can also be the empty dict, which is ok.
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- fg_in, fg_out = update(s, (f_in, g_out))
-
- return fg_in, fg_out
-
-
-stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+ s = unify(g_in, f_out)
+ if s == False: # s can also be the empty dict, which is ok.
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ return update(s, (f_in, g_out))
```
I don't want to rewrite all the defs myself, so I'll write a little conversion function instead. This is programmer's laziness.
@@ -1796,7 +1781,7 @@ C(cons, unstack)
-## Multiple Stack Effects
+## Part VI: Multiple Stack Effects
...
@@ -1880,44 +1865,9 @@ for f in MC([dup], [mul]):
(n0 -- n1)
-## `concat`
+### Representing an Unbounded Sequence of Types
-How to deal with `concat`?
-
- concat ([.0.] [.1.] -- [.0. .1.])
-
-We would like to represent this in Python somehow...
-
-
-```python
-concat = (S[0], S[1]), ((S[0], S[1]),)
-```
-
-But this is actually `cons` with the first argument restricted to be a stack:
-
- ([.0.] [.1.] -- [[.0.] .1.])
-
-What we have implemented so far would actually only permit:
-
- ([.0.] [.1.] -- [.2.])
-
-
-```python
-concat = (S[0], S[1]), (S[2],)
-```
-
-
-Which works but can lose information. Consider `cons concat`, this is how much information we *could* retain:
-
- (1 [.0.] [.1.] -- [1 .0. .1.])
-
-As opposed to just:
-
- (1 [.0.] [.1.] -- [.2.])
-
-### Brzo...'s Derivitives of Regular Expressions
-
-We can invent a new type of type variable, a "sequence type" (I think this is what they mean in the literature by that term...) or "Kleene Star" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more `AnyJoyType` variables would be:
+We can borrow a trick from [Brzozowski's Derivatives of Regular Expressions](https://en.wikipedia.org/wiki/Brzozowski_derivative) to invent a new type of type variable, a "sequence type" (I think this is what they mean in the literature by that term...) or "[Kleene Star](https://en.wikipedia.org/wiki/Kleene_star)" type. I'm going to represent it as a type letter and the asterix, so a sequence of zero or more `AnyJoyType` variables would be:
A*
@@ -2049,7 +1999,7 @@ def unify(u, v, s=None):
sn.update(s)
return t
- ses = unify(u[0], v[0])
+ ses = unify(u[0], v[0], s)
results = ()
for sn in ses:
results += unify(u[1], v[1], sn)
@@ -2160,29 +2110,17 @@ for result in unify(sum_[0], f):
#### `compose()` version 3
-This function has to be modified to use the new datastructures and it is no longer recursive, instead recursion happens as part of unification.
+This function has to be modified to yield multiple results.
```python
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- yield f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- yield stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if not s:
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- for result in s:
- yield update(result, (f_in, g_out))
+ s = unify(g_in, f_out)
+ if not s:
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ for result in s:
+ yield update(result, (f_in, g_out))
```
@@ -2209,8 +2147,8 @@ for f in MC([dup], muls):
print doc_from_stack_effect(*f)
```
- (a0 -- f0)
- (a0 -- i0)
+ (f0 -- f1)
+ (i0 -- i1)
@@ -2295,6 +2233,341 @@ for result in unify(a, b):
{s2: (a1*, (a3, s1)), a2: a10004, a4: a1}
+## Part VII: Typing Combinators
+
+TBD
+
+This is an open subject.
+
+The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:
+
+ dip (a [F] --)--(-- F a)
+
+One thing that might help is...
+
+Consider the type of:
+
+ [cons] dip
+
+Obviously it would be:
+
+ (a1 [..1] a2 -- [a1 ..1] a2)
+
+`dip` itself could have:
+
+ (a1 [..1] -- ... then what?
+
+
+
+```python
+class SymbolJoyType(AnyJoyType): prefix = 'F'
+
+W = map(SymbolJoyType, _R)
+
+k = S[0], ((W[1], S[2]), S[0])
+Symbol('cons')
+print doc_from_stack_effect(*k)
+
+```
+
+ (-- [F1 .2.])
+
+
+
+```python
+dip_a = ((W[1], S[2]), (A[1], S[0]))
+```
+
+
+```python
+d = relabel(S[0], dip_a)
+print doc_from_stack_effect(*d)
+```
+
+ (-- a1001 [F1001 .1002.])
+
+
+
+```python
+s = list(unify(d[1], k[1]))[0]
+s
+```
+
+
+
+
+ {s0: (a1001, s1000), s1002: s2, F1001: F1}
+
+
+
+
+```python
+j = update(s, k)
+```
+
+
+```python
+print doc_from_stack_effect(*j)
+```
+
+ (a1001 -- a1001 [F1 .2.])
+
+
+
+```python
+j
+```
+
+
+
+
+ ((a1001, s1000), ((F1, s2), (a1001, s1000)))
+
+
+
+
+```python
+cons
+```
+
+
+
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+
+```python
+for f in MC([k], [dup]):
+ print doc_from_stack_effect(*f)
+```
+
+ (-- [F0 .1.] [F0 .1.])
+
+
+
+```python
+l = S[0], ((cons, S[2]), (A[1], S[0]))
+```
+
+
+```python
+print doc_from_stack_effect(*l)
+```
+
+ (-- a1 [[[[.1.] a1 .23.] [a1 .1.] .23.] .2.])
+
+
+
+```python
+
+def dip_t(F):
+ (quote, (a1, sec)) = F[1]
+ G = F[0], sec
+ P = S[3], (a1, S[3])
+ a = [P]
+ while isinstance(quote, tuple):
+ term, quote = quote
+ a.append(term)
+ a.append(G)
+ return a[::-1]
+
+
+
+
+```
+
+
+```python
+from joy.utils.stack import iter_stack
+```
+
+
+```python
+a, b, c = dip_t(l)
+```
+
+
+```python
+a
+```
+
+
+
+
+ (s0, s0)
+
+
+
+
+```python
+b
+```
+
+
+
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+
+```python
+c
+```
+
+
+
+
+ (s3, (a1, s3))
+
+
+
+
+```python
+MC([a], [b])
+```
+
+
+
+
+ [((s0, (a0, s1)), ((a0, s0), s1))]
+
+
+
+
+```python
+kjs = MC(MC([a], [b]), [c])
+kjs
+```
+
+
+
+
+ [((s0, (a0, s1)), (a1, ((a0, s0), s1)))]
+
+
+
+
+```python
+print doc_from_stack_effect(*kjs[0])
+```
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+ a0 [.0.] a1 [cons] dip
+ ----------------------------
+ [a0 .0.] a1
+
+
+```python
+stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+```
+
+
+```python
+class SymbolJoyType(AnyJoyType):
+ prefix = 'F'
+
+ def __init__(self, name, sec, number):
+ self.name = name
+ self.stack_effects = sec
+ self.number = number
+
+class CombinatorJoyType(SymbolJoyType): prefix = 'C'
+
+def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+CONS = SymbolJoyType('cons', [cons], 23)
+DIP = CombinatorJoyType('dip', [dip_t], 44)
+
+
+def kav(F, e):
+ #i, stack = F
+ if not e:
+ return [(F, e)]
+ n, e = e
+ if isinstance(n, SymbolJoyType):
+ Fs = []
+ for sec in n.stack_effects:
+ Fs.extend(MC([F], sec))
+ return [kav(Fn, e) for Fn in Fs]
+ if isinstance(n, CombinatorJoyType):
+ res = []
+ for f in n.stack_effects:
+ s, e = f(F[1], e)
+ new_F = F[0], s
+ res.extend(kav(new_F, e))
+ return res
+ lit = S[0], (n, S[0])
+ return [kav(Fn, e) for Fn in MC([F], [lit])]
+
+```
+
+compare, and be amazed:
+
+
+```python
+def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+```
+
+
+```python
+def dip(stack, expression, dictionary):
+ (quote, (x, stack)) = stack
+ expression = (x, expression)
+ return stack, concat(quote, expression), dictionary
+```
+
+And that brings us to current Work-In-Progress. I'm pretty hopeful that the mixed-mode inferencer/interpreter `kav()` function along with the ability to specify multiple implementations for the combinators will permit modelling of the stack effects of e.g. `ifte`. If I can keep up the pace I should be able to verify that conjecture by the end of June.
+
+The rest of this stuff is junk and/or unfinished material.
+
+### `concat`
+
+How to deal with `concat`?
+
+ concat ([.0.] [.1.] -- [.0. .1.])
+
+We would like to represent this in Python somehow...
+
+
+```python
+concat = (S[0], S[1]), ((S[0], S[1]),)
+```
+
+But this is actually `cons` with the first argument restricted to be a stack:
+
+ ([.0.] [.1.] -- [[.0.] .1.])
+
+What we have implemented so far would actually only permit:
+
+ ([.0.] [.1.] -- [.2.])
+
+
+```python
+concat = (S[0], S[1]), (S[2],)
+```
+
+
+Which works but can lose information. Consider `cons concat`, this is how much information we *could* retain:
+
+ (1 [.0.] [.1.] -- [1 .0. .1.])
+
+As opposed to just:
+
+ (1 [.0.] [.1.] -- [.2.])
+
### represent `concat`
([.0.] [.1.] -- [A*(.0.) .1.])
@@ -2356,7 +2629,7 @@ concat(a, b)
-## Joy in the Logical Paradigm
+## Appendix: Joy in the Logical Paradigm
For this to work the type label classes have to be modified to let `T >= t` succeed, where e.g. `T` is `IntJoyType` and `t` is `int`
@@ -2371,85 +2644,85 @@ print doc_from_stack_effect(*F)
ValueError Traceback (most recent call last)
- in ()
+ in ()
1 F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
2
----> 3 print doc_from_stack_effect(*F)
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
ValueError: need more than 1 value to unpack
@@ -2487,28 +2760,7 @@ F[1][0]
s[0]
```
-## Typing Combinators
+## [Abstract Interpretation](https://en.wikipedia.org/wiki/Abstract_interpretation)
+I *think* this might be sorta what I'm doing above with the `kav()` function...
+ In any event "mixed-mode" interpreters that include values and type variables and can track constraints, etc. will be, uh, super-useful. And Abstract Interpretation should be a rich source of ideas.
-TBD
-
-This is an open subject.
-
-The obvious thing is that you now need two pairs of tuples to describe the combinators' effects, a stack effect comment and an expression effect comment:
-
- dip (a [F] --)--(-- F a)
-
-One thing that might help is...
-
-## Abstract Interpretation
-
-## Something else...
- [4 5 ...] 2 1 0 pop∘swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 2 1 swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 1 2 roll<∘rest∘rest∘cons∘cons
- 1 2 [4 5 ...] rest∘rest∘cons∘cons
- 1 2 [5 ...] rest∘cons∘cons
- 1 2 [...] cons∘cons
- 1 [2 ...] cons
- [1 2 ...]
-
-Eh?
diff --git a/docs/Types.rst b/docs/Types.rst
index c6ab68e..0385776 100644
--- a/docs/Types.rst
+++ b/docs/Types.rst
@@ -2,8 +2,12 @@
Type Inference
==============
-Pöial's Rules
--------------
+Cf. `"Type Inference in Stack-Based Programming
+Languages" `__
@@ -47,8 +51,9 @@ Third Rule
The third rule is actually two rules. These two rules deal with
composing functions when the second one will consume one of items the
-first one produces. The two types must be *unified* or a type conflict
-declared.
+first one produces. The two types must be
+`*unified* `__
+or a type conflict declared.
::
@@ -61,9 +66,6 @@ declared.
-------------------------------
(a -- b )∘(c -- d) t[i] == u[k] == u[j]
-Examples
---------
-
Let's work through some examples by hand to develop an intuition for the
algorithm.
@@ -387,8 +389,8 @@ following Python code:
(_, (d, (c, ((a, (b, S0)), stack)))) = stack
return (d, (c, S0)), stack
-Implementation
---------------
+Part II: Implementation
+-----------------------
Representing Stack Effect Comments in Python
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -618,11 +620,11 @@ Let's try it out.
-List Functions
-~~~~~~~~~~~~~~
+Stack Functions
+~~~~~~~~~~~~~~~
Here's that trick to represent functions like ``rest`` and ``cons`` that
-manipulate lists. We use a cons-list of tuples and give the tails their
+manipulate stacks. We use a cons-list of tuples and give the tails their
own numbers. Then everything above already works.
.. code:: ipython2
@@ -683,6 +685,7 @@ Compare with the stack effect comment and you can see it works fine:
::
([4 5 ...] 2 3 1 -- [3 2 ...])
+ 3 4 5 1 2 0 2 1 5
Dealing with ``cons`` and ``uncons``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -719,7 +722,7 @@ deal with this recursively:
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -758,8 +761,8 @@ deal with this recursively:
-Compiling
----------
+Part III: Compiling Stack Functions
+-----------------------------------
Now consider the Python function we would like to derive:
@@ -1094,8 +1097,8 @@ from their stack effect comments:
-Types and Subtypes of Arguments
--------------------------------
+Part IV: Types and Subtypes of Arguments
+----------------------------------------
So far we have dealt with types of functions, those dealing with simple
stack manipulation. Let's extend our machinery to deal with types of
@@ -1396,7 +1399,7 @@ of how many labels of each domain it has "seen".
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -1719,8 +1722,8 @@ comments. We can write a function to check that:
uncons = ([a1 .1.] -- a1 [.1.])
-Functions that use the Stack
-----------------------------
+Part V: Functions that use the Stack
+------------------------------------
Consider the ``stack`` function which grabs the whole stack, quotes it,
and puts it on itself:
@@ -1803,32 +1806,17 @@ It works.
This function has to be modified to use the new datastructures and it is
no longer recursive, instead recursion happens as part of unification.
+Further, the first and second of Pöial's rules are now handled
+automatically by the unification algorithm.
.. code:: ipython2
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- fg_in, fg_out = f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- fg_in, fg_out = stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if s == False: # s can also be the empty dict, which is ok.
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- fg_in, fg_out = update(s, (f_in, g_out))
-
- return fg_in, fg_out
-
-
- stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+ s = unify(g_in, f_out)
+ if s == False: # s can also be the empty dict, which is ok.
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ return update(s, (f_in, g_out))
I don't want to rewrite all the defs myself, so I'll write a little
conversion function instead. This is programmer's laziness.
@@ -2100,8 +2088,8 @@ comments are now already in the form needed for the Python code:
-Multiple Stack Effects
-----------------------
+Part VI: Multiple Stack Effects
+-------------------------------
...
@@ -2191,58 +2179,16 @@ Multiple Stack Effects
(n0 -- n1)
-``concat``
-----------
+Representing an Unbounded Sequence of Types
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-How to deal with ``concat``?
-
-::
-
- concat ([.0.] [.1.] -- [.0. .1.])
-
-We would like to represent this in Python somehow...
-
-.. code:: ipython2
-
- concat = (S[0], S[1]), ((S[0], S[1]),)
-
-But this is actually ``cons`` with the first argument restricted to be a
-stack:
-
-::
-
- ([.0.] [.1.] -- [[.0.] .1.])
-
-What we have implemented so far would actually only permit:
-
-::
-
- ([.0.] [.1.] -- [.2.])
-
-.. code:: ipython2
-
- concat = (S[0], S[1]), (S[2],)
-
-Which works but can lose information. Consider ``cons concat``, this is
-how much information we *could* retain:
-
-::
-
- (1 [.0.] [.1.] -- [1 .0. .1.])
-
-As opposed to just:
-
-::
-
- (1 [.0.] [.1.] -- [.2.])
-
-Brzo...'s Derivitives of Regular Expressions
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-We can invent a new type of type variable, a "sequence type" (I think
-this is what they mean in the literature by that term...) or "Kleene
-Star" type. I'm going to represent it as a type letter and the asterix,
-so a sequence of zero or more ``AnyJoyType`` variables would be:
+We can borrow a trick from `Brzozowski's Derivatives of Regular
+Expressions `__ to
+invent a new type of type variable, a "sequence type" (I think this is
+what they mean in the literature by that term...) or "`Kleene
+Star `__" type. I'm going to
+represent it as a type letter and the asterix, so a sequence of zero or
+more ``AnyJoyType`` variables would be:
::
@@ -2392,7 +2338,7 @@ Can now return multiple results...
sn.update(s)
return t
- ses = unify(u[0], v[0])
+ ses = unify(u[0], v[0], s)
results = ()
for sn in ses:
results += unify(u[1], v[1], sn)
@@ -2517,30 +2463,17 @@ Can now return multiple results...
``compose()`` version 3
^^^^^^^^^^^^^^^^^^^^^^^
-This function has to be modified to use the new datastructures and it is
-no longer recursive, instead recursion happens as part of unification.
+This function has to be modified to yield multiple results.
.. code:: ipython2
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- yield f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- yield stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if not s:
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- for result in s:
- yield update(result, (f_in, g_out))
+ s = unify(g_in, f_out)
+ if not s:
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ for result in s:
+ yield update(result, (f_in, g_out))
.. code:: ipython2
@@ -2567,8 +2500,8 @@ no longer recursive, instead recursion happens as part of unification.
.. parsed-literal::
- (a0 -- f0)
- (a0 -- i0)
+ (f0 -- f1)
+ (i0 -- i1)
.. code:: ipython2
@@ -2665,6 +2598,377 @@ no longer recursive, instead recursion happens as part of unification.
{s2: (a1*, (a3, s1)), a2: a10004, a4: a1}
+Part VII: Typing Combinators
+----------------------------
+
+TBD
+
+This is an open subject.
+
+The obvious thing is that you now need two pairs of tuples to describe
+the combinators' effects, a stack effect comment and an expression
+effect comment:
+
+::
+
+ dip (a [F] --)--(-- F a)
+
+One thing that might help is...
+
+Consider the type of:
+
+::
+
+ [cons] dip
+
+Obviously it would be:
+
+::
+
+ (a1 [..1] a2 -- [a1 ..1] a2)
+
+``dip`` itself could have:
+
+::
+
+ (a1 [..1] -- ... then what?
+
+.. code:: ipython2
+
+ class SymbolJoyType(AnyJoyType): prefix = 'F'
+
+ W = map(SymbolJoyType, _R)
+
+ k = S[0], ((W[1], S[2]), S[0])
+ Symbol('cons')
+ print doc_from_stack_effect(*k)
+
+
+
+.. parsed-literal::
+
+ (-- [F1 .2.])
+
+
+.. code:: ipython2
+
+ dip_a = ((W[1], S[2]), (A[1], S[0]))
+
+.. code:: ipython2
+
+ d = relabel(S[0], dip_a)
+ print doc_from_stack_effect(*d)
+
+
+.. parsed-literal::
+
+ (-- a1001 [F1001 .1002.])
+
+
+.. code:: ipython2
+
+ s = list(unify(d[1], k[1]))[0]
+ s
+
+
+
+
+.. parsed-literal::
+
+ {s0: (a1001, s1000), s1002: s2, F1001: F1}
+
+
+
+.. code:: ipython2
+
+ j = update(s, k)
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*j)
+
+
+.. parsed-literal::
+
+ (a1001 -- a1001 [F1 .2.])
+
+
+.. code:: ipython2
+
+ j
+
+
+
+
+.. parsed-literal::
+
+ ((a1001, s1000), ((F1, s2), (a1001, s1000)))
+
+
+
+.. code:: ipython2
+
+ cons
+
+
+
+
+.. parsed-literal::
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+.. code:: ipython2
+
+ for f in MC([k], [dup]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (-- [F0 .1.] [F0 .1.])
+
+
+.. code:: ipython2
+
+ l = S[0], ((cons, S[2]), (A[1], S[0]))
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*l)
+
+
+.. parsed-literal::
+
+ (-- a1 [[[[.1.] a1 .23.] [a1 .1.] .23.] .2.])
+
+
+.. code:: ipython2
+
+
+ def dip_t(F):
+ (quote, (a1, sec)) = F[1]
+ G = F[0], sec
+ P = S[3], (a1, S[3])
+ a = [P]
+ while isinstance(quote, tuple):
+ term, quote = quote
+ a.append(term)
+ a.append(G)
+ return a[::-1]
+
+
+
+
+
+.. code:: ipython2
+
+ from joy.utils.stack import iter_stack
+
+.. code:: ipython2
+
+ a, b, c = dip_t(l)
+
+.. code:: ipython2
+
+ a
+
+
+
+
+.. parsed-literal::
+
+ (s0, s0)
+
+
+
+.. code:: ipython2
+
+ b
+
+
+
+
+.. parsed-literal::
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+.. code:: ipython2
+
+ c
+
+
+
+
+.. parsed-literal::
+
+ (s3, (a1, s3))
+
+
+
+.. code:: ipython2
+
+ MC([a], [b])
+
+
+
+
+.. parsed-literal::
+
+ [((s0, (a0, s1)), ((a0, s0), s1))]
+
+
+
+.. code:: ipython2
+
+ kjs = MC(MC([a], [b]), [c])
+ kjs
+
+
+
+
+.. parsed-literal::
+
+ [((s0, (a0, s1)), (a1, ((a0, s0), s1)))]
+
+
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*kjs[0])
+
+
+.. parsed-literal::
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+
+::
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+ a0 [.0.] a1 [cons] dip
+ ----------------------------
+ [a0 .0.] a1
+
+.. code:: ipython2
+
+ stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+
+.. code:: ipython2
+
+ class SymbolJoyType(AnyJoyType):
+ prefix = 'F'
+
+ def __init__(self, name, sec, number):
+ self.name = name
+ self.stack_effects = sec
+ self.number = number
+
+ class CombinatorJoyType(SymbolJoyType): prefix = 'C'
+
+ def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+ CONS = SymbolJoyType('cons', [cons], 23)
+ DIP = CombinatorJoyType('dip', [dip_t], 44)
+
+
+ def kav(F, e):
+ #i, stack = F
+ if not e:
+ return [(F, e)]
+ n, e = e
+ if isinstance(n, SymbolJoyType):
+ Fs = []
+ for sec in n.stack_effects:
+ Fs.extend(MC([F], sec))
+ return [kav(Fn, e) for Fn in Fs]
+ if isinstance(n, CombinatorJoyType):
+ res = []
+ for f in n.stack_effects:
+ s, e = f(F[1], e)
+ new_F = F[0], s
+ res.extend(kav(new_F, e))
+ return res
+ lit = S[0], (n, S[0])
+ return [kav(Fn, e) for Fn in MC([F], [lit])]
+
+
+compare, and be amazed:
+
+.. code:: ipython2
+
+ def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+.. code:: ipython2
+
+ def dip(stack, expression, dictionary):
+ (quote, (x, stack)) = stack
+ expression = (x, expression)
+ return stack, concat(quote, expression), dictionary
+
+And that brings us to current Work-In-Progress. I'm pretty hopeful that
+the mixed-mode inferencer/interpreter ``kav()`` function along with the
+ability to specify multiple implementations for the combinators will
+permit modelling of the stack effects of e.g. ``ifte``. If I can keep up
+the pace I should be able to verify that conjecture by the end of June.
+
+The rest of this stuff is junk and/or unfinished material.
+
+``concat``
+~~~~~~~~~~
+
+How to deal with ``concat``?
+
+::
+
+ concat ([.0.] [.1.] -- [.0. .1.])
+
+We would like to represent this in Python somehow...
+
+.. code:: ipython2
+
+ concat = (S[0], S[1]), ((S[0], S[1]),)
+
+But this is actually ``cons`` with the first argument restricted to be a
+stack:
+
+::
+
+ ([.0.] [.1.] -- [[.0.] .1.])
+
+What we have implemented so far would actually only permit:
+
+::
+
+ ([.0.] [.1.] -- [.2.])
+
+.. code:: ipython2
+
+ concat = (S[0], S[1]), (S[2],)
+
+Which works but can lose information. Consider ``cons concat``, this is
+how much information we *could* retain:
+
+::
+
+ (1 [.0.] [.1.] -- [1 .0. .1.])
+
+As opposed to just:
+
+::
+
+ (1 [.0.] [.1.] -- [.2.])
+
represent ``concat``
~~~~~~~~~~~~~~~~~~~~
@@ -2731,8 +3035,8 @@ or...
-Joy in the Logical Paradigm
----------------------------
+Appendix: Joy in the Logical Paradigm
+-------------------------------------
For this to work the type label classes have to be modified to let
``T >= t`` succeed, where e.g. ``T`` is ``IntJoyType`` and ``t`` is
@@ -2752,85 +3056,85 @@ For this to work the type label classes have to be modified to let
ValueError Traceback (most recent call last)
- in ()
+ in ()
1 F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
2
----> 3 print doc_from_stack_effect(*F)
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
- in C(f, g)
+ in C(f, g)
10 def C(f, g):
11 f, g = relabel(f, g)
---> 12 for fg in compose(f, g):
13 yield delabel(fg)
- in compose(f, g)
+ in compose(f, g)
1 def compose(f, g):
- 2
- ----> 3 (f_in, f_out), (g_in, g_out) = f, g
- 4
- 5 if not g_in:
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
ValueError: need more than 1 value to unpack
@@ -2862,38 +3166,11 @@ For this to work the type label classes have to be modified to let
s[0]
-Typing Combinators
-------------------
+`Abstract Interpretation `__
+-----------------------------------------------------------------------------------
-TBD
-
-This is an open subject.
-
-The obvious thing is that you now need two pairs of tuples to describe
-the combinators' effects, a stack effect comment and an expression
-effect comment:
-
-::
-
- dip (a [F] --)--(-- F a)
-
-One thing that might help is...
-
-Abstract Interpretation
------------------------
-
-Something else...
------------------
-
-::
-
- [4 5 ...] 2 1 0 pop∘swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 2 1 swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 1 2 roll<∘rest∘rest∘cons∘cons
- 1 2 [4 5 ...] rest∘rest∘cons∘cons
- 1 2 [5 ...] rest∘cons∘cons
- 1 2 [...] cons∘cons
- 1 [2 ...] cons
- [1 2 ...]
-
-Eh?
+I *think* this might be sorta what I'm doing above with the ``kav()``
+function... In any event "mixed-mode" interpreters that include values
+and type variables and can track constraints, etc. will be, uh,
+super-useful. And Abstract Interpretation should be a rich source of
+ideas.
diff --git a/docs/sphinx_docs/_build/html/notebooks/index.html b/docs/sphinx_docs/_build/html/notebooks/index.html
index 4c74e30..4269353 100644
--- a/docs/sphinx_docs/_build/html/notebooks/index.html
+++ b/docs/sphinx_docs/_build/html/notebooks/index.html
@@ -119,17 +119,15 @@
Type Inference
-- Pöial’s Rules
-- Examples
-- Implementation
-- Compiling
-- Types and Subtypes of Arguments
-- Functions that use the Stack
-- Sets of Stack Effects
-concat- Typing Combinators
+- Part I: Pöial’s Rules
+- Part II: Implementation
+- Part III: Compiling Stack Functions
+- Part IV: Types and Subtypes of Arguments
+- Part V: Functions that use the Stack
+- Part VI: Multiple Stack Effects
+- Part VII: Typing Combinators
+- Appendix: Joy in the Logical Paradigm
- Abstract Interpretation
-- Something else…
No Updates
diff --git a/docs/sphinx_docs/_build/html/searchindex.js b/docs/sphinx_docs/_build/html/searchindex.js
index c2ac2b0..d3e2225 100644
--- a/docs/sphinx_docs/_build/html/searchindex.js
+++ b/docs/sphinx_docs/_build/html/searchindex.js
@@ -1 +1 @@
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0.2.0 Documentation","Joy Interpreter","Functions Grouped by, er, Function with Examples","Function Reference","Categorical Programming","Developing a Program in Joy","Using x to Generate Values","Thun: Joy in Python","Newton\u2019s method","No Updates","Treating Trees I: Ordered Binary Trees","Quadratic formula","Recursive Combinators","Replacing Functions in the Dictionary","Treating Trees II: treestep","Type Inference","Traversing Datastructures with Zippers","Essays about Programming in Joy","Parsing Text into Joy Expressions","Tracing Joy Execution","Stack or Quote or Sequence or List\u2026"],titleterms:{"abstract":15,"case":[8,10],"function":[2,3,7,8,10,12,13,14,15],"long":13,"new":10,"p\u00f6ial":15,"void":2,"while":2,Adding:10,One:[6,10],The:[5,7,10,12,14],There:7,Using:6,With:14,about:17,add:[2,10],adding:10,address:16,altern:14,ana:12,analysi:5,anamorph:[2,12],app1:2,app2:2,app3:2,appendix:[10,12,15],approxim:8,argument:15,auto:3,averag:2,base:[8,10],binari:[2,10,14],both:10,branch:[2,10],can:10,cata:12,catamorph:12,categor:4,chatter:2,child:10,choic:2,clear:2,cleav:2,cmp:10,code:[7,10],combin:[2,10,12,15],comment:15,compar:10,comparison:2,compil:[6,15],compile_:15,compos:15,comput:8,con:[2,15],concat:[2,15],conclus:12,consecut:8,continu:7,current:10,datastructur:[7,10,16],deal:15,defin:[10,14],definit:11,delabel:15,delet:10,deriv:[11,12,14],design:12,determin:16,develop:5,dialect:0,dictionari:13,dip:[2,16],dipd:2,dipdd:2,direco:6,disenstacken:2,distinguish:15,div:2,doc_from_stack_effect:15,document:0,doe:10,down_to_zero:2,drop:2,dup:[2,15],dupd:2,dupdip:2,effect:15,els:10,empti:10,enstacken:2,equal:10,essai:17,euler:[5,6],eval:7,even:6,exampl:[2,7,10,12,14],execut:19,express:[7,18],extract:14,factori:12,fibonacci:6,filter:5,find:[8,10,12],first:[2,5,15],five:6,flatten:2,flexibl:14,floordiv:2,formula:11,found:10,four:12,fun:12,further:5,gcd:2,gener:[3,5,6,8],genrec:2,get:[10,14],getitem:2,given:14,greater:10,group:2,have:[10,14],help:2,highest:10,host:0,how:[5,6],hylo:12,hylomorph:12,identifi:15,ift:2,iii:15,implement:15,indic:0,infer:15,inferenc:15,inform:0,infra:[2,16],integ:5,interest:6,interlud:10,intern:18,interpret:[1,7,15],item:16,iter:[5,10],joi:[0,1,3,5,7,12,15,16,17,18,19,20],just:5,kei:10,languag:0,least_fract:2,left:10,less:10,let:5,librari:[3,7,15],like:10,list:[2,15,20],literari:7,littl:5,logic:[2,15],loop:[2,7],lower:10,lshift:2,make:[6,8],mani:5,map:2,math:2,method:8,min:2,miscellan:2,mod:2,modifi:15,modulu:2,more:10,most:10,mul:[2,15],multipl:[5,6,15],must:10,name:11,neg:2,newton:8,next:8,node:10,non:10,now:10,nullari:2,number:[12,15],one:7,onli:7,order:[10,14],osdn:0,our:10,over:2,pack:5,pam:2,para:12,paradigm:15,parameter:[10,14],pars:[2,18],parser:[7,18],part:15,pass:7,path:16,pattern:12,per:10,pop:[2,15],popd:2,popop:2,pow:2,power:6,pred:2,predic:[5,8,10,14],pretty_print:19,primit:12,primrec:2,print:7,problem:[5,6],process:10,product:2,program:[4,5,11,14,17],project:[0,5,6],pure:7,put:[10,11,14],python:[7,13,15],quadrat:11,quick:0,quot:[2,20],rang:[2,5,12],range_to_zero:2,read:7,recur:[8,10],recurs:[10,12,14],redefin:[10,14],refactor:[5,10],refer:3,regular:7,reimplement:14,relabel:15,rem:2,remaind:2,remov:2,render:5,repl:7,replac:[10,13],repres:15,reset:6,rest:[2,15],revers:2,right:[10,16],rightmost:10,roll:[2,15],rolldown:2,rollup:2,rshift:2,rule:15,run:[2,6],second:[2,15],select:2,sequenc:[6,15,20],set:[8,10],shorter:13,should:7,shunt:2,simplest:5,size:[2,13],someth:[],sourc:10,special:12,sqr:[2,15],sqrt:[2,11],stack:[2,7,15,20],start:0,step:[2,12,14],straightforward:11,structur:10,style:7,sub:[2,10],subtyp:15,succ:2,sum:[2,5],swaack:2,swap:[2,15],swon:2,swoncat:2,symbol:[7,12],tabl:0,tail:12,take:2,term:[5,6,14],ternari:2,text:18,than:10,them:11,thi:10,third:[2,15],three:6,thun:[0,7],time:[2,6],togeth:[10,11,14],token:7,toler:8,trace:[13,19],traceprint:7,travers:[10,14,16],treat:[10,14],tree:[10,14,16],treegrind:14,treestep:14,triangular:12,truediv:2,truthi:2,tuck:2,two:6,type:15,unari:2,unbound:15,uncon:[2,15],unifi:15,unit:2,unnecessari:5,unquot:2,unstack:2,updat:[9,15],use:15,util:[19,20],valu:[6,10],variabl:11,variat:6,version:[5,10,13,15],view:7,vii:15,within:8,word:2,write:11,xor:2,zero:6,zip:2,zipper:16}})
\ No newline at end of file
diff --git a/docs/sphinx_docs/notebooks/Types.rst b/docs/sphinx_docs/notebooks/Types.rst
index 971fdfc..0385776 100644
--- a/docs/sphinx_docs/notebooks/Types.rst
+++ b/docs/sphinx_docs/notebooks/Types.rst
@@ -2,8 +2,12 @@
Type Inference
==============
-Pöial's Rules
--------------
+Cf. `"Type Inference in Stack-Based Programming
+Languages" `__
@@ -47,8 +51,9 @@ Third Rule
The third rule is actually two rules. These two rules deal with
composing functions when the second one will consume one of items the
-first one produces. The two types must be *unified* or a type conflict
-declared.
+first one produces. The two types must be
+`*unified* `__
+or a type conflict declared.
::
@@ -61,9 +66,6 @@ declared.
-------------------------------
(a -- b )∘(c -- d) t[i] == u[k] == u[j]
-Examples
---------
-
Let's work through some examples by hand to develop an intuition for the
algorithm.
@@ -387,8 +389,8 @@ following Python code:
(_, (d, (c, ((a, (b, S0)), stack)))) = stack
return (d, (c, S0)), stack
-Implementation
---------------
+Part II: Implementation
+-----------------------
Representing Stack Effect Comments in Python
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -618,11 +620,11 @@ Let's try it out.
-List Functions
-~~~~~~~~~~~~~~
+Stack Functions
+~~~~~~~~~~~~~~~
Here's that trick to represent functions like ``rest`` and ``cons`` that
-manipulate lists. We use a cons-list of tuples and give the tails their
+manipulate stacks. We use a cons-list of tuples and give the tails their
own numbers. Then everything above already works.
.. code:: ipython2
@@ -683,6 +685,7 @@ Compare with the stack effect comment and you can see it works fine:
::
([4 5 ...] 2 3 1 -- [3 2 ...])
+ 3 4 5 1 2 0 2 1 5
Dealing with ``cons`` and ``uncons``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
@@ -719,7 +722,7 @@ deal with this recursively:
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -758,8 +761,8 @@ deal with this recursively:
-Compiling
----------
+Part III: Compiling Stack Functions
+-----------------------------------
Now consider the Python function we would like to derive:
@@ -1094,8 +1097,8 @@ from their stack effect comments:
-Types and Subtypes of Arguments
--------------------------------
+Part IV: Types and Subtypes of Arguments
+----------------------------------------
So far we have dealt with types of functions, those dealing with simple
stack manipulation. Let's extend our machinery to deal with types of
@@ -1396,7 +1399,7 @@ of how many labels of each domain it has "seen".
def unify(u, v, s=None):
if s is None:
s = {}
- else:
+ elif s:
u = update(s, u)
v = update(s, v)
@@ -1410,11 +1413,11 @@ of how many labels of each domain it has "seen".
if v >= u:
s[v] = u
return s
- raise ValueError('Cannot unify %r and %r.' % (u, v))
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
if isinstance(u, tuple) and isinstance(v, tuple):
if len(u) != len(v) != 2:
- raise ValueError(repr((u, v)))
+ raise TypeError(repr((u, v)))
for uu, vv in zip(u, v):
s = unify(uu, vv, s)
if s == False: # (instead of a substitution dict.)
@@ -1423,13 +1426,13 @@ of how many labels of each domain it has "seen".
if isinstance(v, tuple):
if not stacky(u):
- raise ValueError('Cannot unify %r and %r.' % (u, v))
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
s[u] = v
return s
if isinstance(u, tuple):
if not stacky(v):
- raise ValueError('Cannot unify %r and %r.' % (v, u))
+ raise TypeError('Cannot unify %r and %r.' % (v, u))
s[v] = u
return s
@@ -1719,8 +1722,8 @@ comments. We can write a function to check that:
uncons = ([a1 .1.] -- a1 [.1.])
-Functions that use the Stack
-----------------------------
+Part V: Functions that use the Stack
+------------------------------------
Consider the ``stack`` function which grabs the whole stack, quotes it,
and puts it on itself:
@@ -1803,32 +1806,17 @@ It works.
This function has to be modified to use the new datastructures and it is
no longer recursive, instead recursion happens as part of unification.
+Further, the first and second of Pöial's rules are now handled
+automatically by the unification algorithm.
.. code:: ipython2
def compose(f, g):
-
(f_in, f_out), (g_in, g_out) = f, g
-
- if not g_in:
- fg_in, fg_out = f_in, stack_concat(g_out, f_out)
-
- elif not f_out:
- fg_in, fg_out = stack_concat(f_in, g_in), g_out
-
- else: # Unify and update.
-
- s = unify(g_in, f_out)
-
- if s == False: # s can also be the empty dict, which is ok.
- raise TypeError('Cannot unify %r and %r.' % (fo, gi))
-
- fg_in, fg_out = update(s, (f_in, g_out))
-
- return fg_in, fg_out
-
-
- stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+ s = unify(g_in, f_out)
+ if s == False: # s can also be the empty dict, which is ok.
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ return update(s, (f_in, g_out))
I don't want to rewrite all the defs myself, so I'll write a little
conversion function instead. This is programmer's laziness.
@@ -2100,13 +2088,844 @@ comments are now already in the form needed for the Python code:
-Sets of Stack Effects
----------------------
+Part VI: Multiple Stack Effects
+-------------------------------
...
+.. code:: ipython2
+
+ class IntJoyType(NumberJoyType): prefix = 'i'
+
+
+ F = map(FloatJoyType, _R)
+ I = map(IntJoyType, _R)
+
+.. code:: ipython2
+
+ muls = [
+ ((I[2], (I[1], S[0])), (I[3], S[0])),
+ ((F[2], (I[1], S[0])), (F[3], S[0])),
+ ((I[2], (F[1], S[0])), (F[3], S[0])),
+ ((F[2], (F[1], S[0])), (F[3], S[0])),
+ ]
+
+.. code:: ipython2
+
+ for f in muls:
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (i1 i2 -- i3)
+ (i1 f2 -- f3)
+ (f1 i2 -- f3)
+ (f1 f2 -- f3)
+
+
+.. code:: ipython2
+
+ for f in muls:
+ try:
+ e = C(dup, f)
+ except TypeError:
+ continue
+ print doc_from_stack_effect(*dup), doc_from_stack_effect(*f), doc_from_stack_effect(*e)
+
+
+.. parsed-literal::
+
+ (a1 -- a1 a1) (i1 i2 -- i3) (i0 -- i1)
+ (a1 -- a1 a1) (f1 f2 -- f3) (f0 -- f1)
+
+
+.. code:: ipython2
+
+ from itertools import product
+
+
+ def meta_compose(F, G):
+ for f, g in product(F, G):
+ try:
+ yield C(f, g)
+ except TypeError:
+ pass
+
+
+ def MC(F, G):
+ return sorted(set(meta_compose(F, G)))
+
+.. code:: ipython2
+
+ for f in MC([dup], muls):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (f0 -- f1)
+ (i0 -- i1)
+
+
+.. code:: ipython2
+
+ for f in MC([dup], [mul]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (n0 -- n1)
+
+
+Representing an Unbounded Sequence of Types
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+We can borrow a trick from `Brzozowski's Derivatives of Regular
+Expressions `__ to
+invent a new type of type variable, a "sequence type" (I think this is
+what they mean in the literature by that term...) or "`Kleene
+Star `__" type. I'm going to
+represent it as a type letter and the asterix, so a sequence of zero or
+more ``AnyJoyType`` variables would be:
+
+::
+
+ A*
+
+The ``A*`` works by splitting the universe into two alternate histories:
+
+::
+
+ A* -> 0 | A A*
+
+The Kleene star variable disappears in one universe, and in the other it
+turns into an ``AnyJoyType`` variable followed by itself again. We have
+to return all universes (represented by their substitution dicts, the
+"unifiers") that don't lead to type conflicts.
+
+Consider unifying two stacks (the lowercase letters are any type
+variables of the kinds we have defined so far):
+
+::
+
+ [a A* b .0.] U [c d .1.]
+ w/ {c: a}
+ [ A* b .0.] U [ d .1.]
+
+Now we have to split universes to unify ``A*``. In the first universe it
+disappears:
+
+::
+
+ [b .0.] U [d .1.]
+ w/ {d: b, .1.: .0.}
+ [] U []
+
+While in the second it spawns an ``A``, which we will label ``e``:
+
+::
+
+ [e A* b .0.] U [d .1.]
+ w/ {d: e}
+ [ A* b .0.] U [ .1.]
+ w/ {.1.: A* b .0.}
+ [ A* b .0.] U [ .1.]
+
+Giving us two unifiers:
+
+::
+
+ {c: a, d: b, .1.: .0.}
+ {c: a, d: e, .1.: A* b .0.}
+
+.. code:: ipython2
+
+ class KleeneStar(object):
+
+ kind = AnyJoyType
+
+ def __init__(self, number):
+ self.number = number
+ self.count = 0
+ self.prefix = repr(self)
+
+ def __repr__(self):
+ return '%s%i*' % (self.kind.prefix, self.number)
+
+ def another(self):
+ self.count += 1
+ return self.kind(10000 * self.number + self.count)
+
+ def __eq__(self, other):
+ return (
+ isinstance(other, self.__class__)
+ and other.number == self.number
+ )
+
+ def __ge__(self, other):
+ return self.kind >= other.kind
+
+ def __add__(self, other):
+ return self.__class__(self.number + other)
+ __radd__ = __add__
+
+ def __hash__(self):
+ return hash(repr(self))
+
+ class AnyStarJoyType(KleeneStar): kind = AnyJoyType
+ class NumberStarJoyType(KleeneStar): kind = NumberJoyType
+ #class FloatStarJoyType(KleeneStar): kind = FloatJoyType
+ #class IntStarJoyType(KleeneStar): kind = IntJoyType
+ class StackStarJoyType(KleeneStar): kind = StackJoyType
+
+
+ As = map(AnyStarJoyType, _R)
+ Ns = map(NumberStarJoyType, _R)
+ Ss = map(StackStarJoyType, _R)
+
+``unify()`` version 4
+^^^^^^^^^^^^^^^^^^^^^
+
+Can now return multiple results...
+
+.. code:: ipython2
+
+ def unify(u, v, s=None):
+ if s is None:
+ s = {}
+ elif s:
+ u = update(s, u)
+ v = update(s, v)
+
+ if u == v:
+ return s,
+
+ if isinstance(u, AnyJoyType) and isinstance(v, AnyJoyType):
+ if u >= v:
+ s[u] = v
+ return s,
+ if v >= u:
+ s[v] = u
+ return s,
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
+
+ if isinstance(u, tuple) and isinstance(v, tuple):
+ if len(u) != len(v) != 2:
+ raise TypeError(repr((u, v)))
+
+ a, b = v
+ if isinstance(a, KleeneStar):
+ # Two universes, in one the Kleene star disappears and unification
+ # continues without it...
+ s0 = unify(u, b)
+
+ # In the other it spawns a new variable.
+ s1 = unify(u, (a.another(), v))
+
+ t = s0 + s1
+ for sn in t:
+ sn.update(s)
+ return t
+
+ a, b = u
+ if isinstance(a, KleeneStar):
+ s0 = unify(v, b)
+ s1 = unify(v, (a.another(), u))
+ t = s0 + s1
+ for sn in t:
+ sn.update(s)
+ return t
+
+ ses = unify(u[0], v[0], s)
+ results = ()
+ for sn in ses:
+ results += unify(u[1], v[1], sn)
+ return results
+
+ if isinstance(v, tuple):
+ if not stacky(u):
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
+ s[u] = v
+ return s,
+
+ if isinstance(u, tuple):
+ if not stacky(v):
+ raise TypeError('Cannot unify %r and %r.' % (v, u))
+ s[v] = u
+ return s,
+
+ return ()
+
+
+ def stacky(thing):
+ return thing.__class__ in {AnyJoyType, StackJoyType}
+
+.. code:: ipython2
+
+ a = (As[1], S[1])
+ a
+
+
+
+
+.. parsed-literal::
+
+ (a1*, s1)
+
+
+
+.. code:: ipython2
+
+ b = (A[1], S[2])
+ b
+
+
+
+
+.. parsed-literal::
+
+ (a1, s2)
+
+
+
+.. code:: ipython2
+
+ for result in unify(b, a):
+ print result, '->', update(result, a), update(result, b)
+
+
+.. parsed-literal::
+
+ {s1: (a1, s2)} -> (a1*, (a1, s2)) (a1, s2)
+ {a1: a10001, s2: (a1*, s1)} -> (a1*, s1) (a10001, (a1*, s1))
+
+
+.. code:: ipython2
+
+ for result in unify(a, b):
+ print result, '->', update(result, a), update(result, b)
+
+
+.. parsed-literal::
+
+ {s1: (a1, s2)} -> (a1*, (a1, s2)) (a1, s2)
+ {a1: a10002, s2: (a1*, s1)} -> (a1*, s1) (a10002, (a1*, s1))
+
+
+::
+
+ (a1*, s1) [a1*] (a1, s2) [a1]
+
+ (a1*, (a1, s2)) [a1* a1] (a1, s2) [a1]
+
+ (a1*, s1) [a1*] (a2, (a1*, s1)) [a2 a1*]
+
+.. code:: ipython2
+
+ sum_ = ((Ns[1], S[1]), S[0]), (N[0], S[0])
+
+ print doc_from_stack_effect(*sum_)
+
+
+.. parsed-literal::
+
+ ([n1* .1.] -- n0)
+
+
+.. code:: ipython2
+
+ f = (N[1], (N[2], (N[3], S[1]))), S[0]
+
+ print doc_from_stack_effect(S[0], f)
+
+
+.. parsed-literal::
+
+ (-- [n1 n2 n3 .1.])
+
+
+.. code:: ipython2
+
+ for result in unify(sum_[0], f):
+ print result, '->', update(result, sum_[1])
+
+
+.. parsed-literal::
+
+ {s1: (n1, (n2, (n3, s1)))} -> (n0, s0)
+ {n1: n10001, s1: (n2, (n3, s1))} -> (n0, s0)
+ {n1: n10001, s1: (n3, s1), n2: n10002} -> (n0, s0)
+ {n1: n10001, s1: (n1*, s1), n3: n10003, n2: n10002} -> (n0, s0)
+
+
+``compose()`` version 3
+^^^^^^^^^^^^^^^^^^^^^^^
+
+This function has to be modified to yield multiple results.
+
+.. code:: ipython2
+
+ def compose(f, g):
+ (f_in, f_out), (g_in, g_out) = f, g
+ s = unify(g_in, f_out)
+ if not s:
+ raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+ for result in s:
+ yield update(result, (f_in, g_out))
+
+
+.. code:: ipython2
+
+ def meta_compose(F, G):
+ for f, g in product(F, G):
+ try:
+ for result in C(f, g):
+ yield result
+ except TypeError:
+ pass
+
+
+ def C(f, g):
+ f, g = relabel(f, g)
+ for fg in compose(f, g):
+ yield delabel(fg)
+
+.. code:: ipython2
+
+ for f in MC([dup], muls):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (f0 -- f1)
+ (i0 -- i1)
+
+
+.. code:: ipython2
+
+
+
+ for f in MC([dup], [sum_]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ ([n0* .0.] -- [n0* .0.] n0)
+
+
+.. code:: ipython2
+
+
+
+ for f in MC([cons], [sum_]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (a0 [.0.] -- n0)
+ (n0 [n0* .0.] -- n1)
+
+
+.. code:: ipython2
+
+ sum_ = (((N[1], (Ns[1], S[1])), S[0]), (N[0], S[0]))
+ print doc_from_stack_effect(*cons),
+ print doc_from_stack_effect(*sum_),
+
+ for f in MC([cons], [sum_]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (a1 [.1.] -- [a1 .1.]) ([n1 n1* .1.] -- n0) (n0 [n0* .0.] -- n1)
+
+
+.. code:: ipython2
+
+ a = (A[4], (As[1], (A[3], S[1])))
+ a
+
+
+
+
+.. parsed-literal::
+
+ (a4, (a1*, (a3, s1)))
+
+
+
+.. code:: ipython2
+
+ b = (A[1], (A[2], S[2]))
+ b
+
+
+
+
+.. parsed-literal::
+
+ (a1, (a2, s2))
+
+
+
+.. code:: ipython2
+
+ for result in unify(b, a):
+ print result
+
+
+.. parsed-literal::
+
+ {a1: a4, s2: s1, a2: a3}
+ {a1: a4, s2: (a1*, (a3, s1)), a2: a10003}
+
+
+.. code:: ipython2
+
+ for result in unify(a, b):
+ print result
+
+
+.. parsed-literal::
+
+ {s2: s1, a2: a3, a4: a1}
+ {s2: (a1*, (a3, s1)), a2: a10004, a4: a1}
+
+
+Part VII: Typing Combinators
+----------------------------
+
+TBD
+
+This is an open subject.
+
+The obvious thing is that you now need two pairs of tuples to describe
+the combinators' effects, a stack effect comment and an expression
+effect comment:
+
+::
+
+ dip (a [F] --)--(-- F a)
+
+One thing that might help is...
+
+Consider the type of:
+
+::
+
+ [cons] dip
+
+Obviously it would be:
+
+::
+
+ (a1 [..1] a2 -- [a1 ..1] a2)
+
+``dip`` itself could have:
+
+::
+
+ (a1 [..1] -- ... then what?
+
+.. code:: ipython2
+
+ class SymbolJoyType(AnyJoyType): prefix = 'F'
+
+ W = map(SymbolJoyType, _R)
+
+ k = S[0], ((W[1], S[2]), S[0])
+ Symbol('cons')
+ print doc_from_stack_effect(*k)
+
+
+
+.. parsed-literal::
+
+ (-- [F1 .2.])
+
+
+.. code:: ipython2
+
+ dip_a = ((W[1], S[2]), (A[1], S[0]))
+
+.. code:: ipython2
+
+ d = relabel(S[0], dip_a)
+ print doc_from_stack_effect(*d)
+
+
+.. parsed-literal::
+
+ (-- a1001 [F1001 .1002.])
+
+
+.. code:: ipython2
+
+ s = list(unify(d[1], k[1]))[0]
+ s
+
+
+
+
+.. parsed-literal::
+
+ {s0: (a1001, s1000), s1002: s2, F1001: F1}
+
+
+
+.. code:: ipython2
+
+ j = update(s, k)
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*j)
+
+
+.. parsed-literal::
+
+ (a1001 -- a1001 [F1 .2.])
+
+
+.. code:: ipython2
+
+ j
+
+
+
+
+.. parsed-literal::
+
+ ((a1001, s1000), ((F1, s2), (a1001, s1000)))
+
+
+
+.. code:: ipython2
+
+ cons
+
+
+
+
+.. parsed-literal::
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+.. code:: ipython2
+
+ for f in MC([k], [dup]):
+ print doc_from_stack_effect(*f)
+
+
+.. parsed-literal::
+
+ (-- [F0 .1.] [F0 .1.])
+
+
+.. code:: ipython2
+
+ l = S[0], ((cons, S[2]), (A[1], S[0]))
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*l)
+
+
+.. parsed-literal::
+
+ (-- a1 [[[[.1.] a1 .23.] [a1 .1.] .23.] .2.])
+
+
+.. code:: ipython2
+
+
+ def dip_t(F):
+ (quote, (a1, sec)) = F[1]
+ G = F[0], sec
+ P = S[3], (a1, S[3])
+ a = [P]
+ while isinstance(quote, tuple):
+ term, quote = quote
+ a.append(term)
+ a.append(G)
+ return a[::-1]
+
+
+
+
+
+.. code:: ipython2
+
+ from joy.utils.stack import iter_stack
+
+.. code:: ipython2
+
+ a, b, c = dip_t(l)
+
+.. code:: ipython2
+
+ a
+
+
+
+
+.. parsed-literal::
+
+ (s0, s0)
+
+
+
+.. code:: ipython2
+
+ b
+
+
+
+
+.. parsed-literal::
+
+ ((s1, (a1, s23)), ((a1, s1), s23))
+
+
+
+.. code:: ipython2
+
+ c
+
+
+
+
+.. parsed-literal::
+
+ (s3, (a1, s3))
+
+
+
+.. code:: ipython2
+
+ MC([a], [b])
+
+
+
+
+.. parsed-literal::
+
+ [((s0, (a0, s1)), ((a0, s0), s1))]
+
+
+
+.. code:: ipython2
+
+ kjs = MC(MC([a], [b]), [c])
+ kjs
+
+
+
+
+.. parsed-literal::
+
+ [((s0, (a0, s1)), (a1, ((a0, s0), s1)))]
+
+
+
+.. code:: ipython2
+
+ print doc_from_stack_effect(*kjs[0])
+
+
+.. parsed-literal::
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+
+::
+
+ (a0 [.0.] -- [a0 .0.] a1)
+
+ a0 [.0.] a1 [cons] dip
+ ----------------------------
+ [a0 .0.] a1
+
+.. code:: ipython2
+
+ stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if q else e
+
+.. code:: ipython2
+
+ class SymbolJoyType(AnyJoyType):
+ prefix = 'F'
+
+ def __init__(self, name, sec, number):
+ self.name = name
+ self.stack_effects = sec
+ self.number = number
+
+ class CombinatorJoyType(SymbolJoyType): prefix = 'C'
+
+ def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+ CONS = SymbolJoyType('cons', [cons], 23)
+ DIP = CombinatorJoyType('dip', [dip_t], 44)
+
+
+ def kav(F, e):
+ #i, stack = F
+ if not e:
+ return [(F, e)]
+ n, e = e
+ if isinstance(n, SymbolJoyType):
+ Fs = []
+ for sec in n.stack_effects:
+ Fs.extend(MC([F], sec))
+ return [kav(Fn, e) for Fn in Fs]
+ if isinstance(n, CombinatorJoyType):
+ res = []
+ for f in n.stack_effects:
+ s, e = f(F[1], e)
+ new_F = F[0], s
+ res.extend(kav(new_F, e))
+ return res
+ lit = S[0], (n, S[0])
+ return [kav(Fn, e) for Fn in MC([F], [lit])]
+
+
+compare, and be amazed:
+
+.. code:: ipython2
+
+ def dip_t(stack, expression):
+ (quote, (a1, stack)) = stack
+ expression = stack_concat(quote, (a1, expression))
+ return stack, expression
+
+.. code:: ipython2
+
+ def dip(stack, expression, dictionary):
+ (quote, (x, stack)) = stack
+ expression = (x, expression)
+ return stack, concat(quote, expression), dictionary
+
+And that brings us to current Work-In-Progress. I'm pretty hopeful that
+the mixed-mode inferencer/interpreter ``kav()`` function along with the
+ability to specify multiple implementations for the combinators will
+permit modelling of the stack effects of e.g. ``ifte``. If I can keep up
+the pace I should be able to verify that conjecture by the end of June.
+
+The rest of this stuff is junk and/or unfinished material.
+
``concat``
-----------
+~~~~~~~~~~
How to deal with ``concat``?
@@ -2150,57 +2969,208 @@ As opposed to just:
(1 [.0.] [.1.] -- [.2.])
-Which works but can lose information. Consider ``cons concat``, this is
-how much information we *could* retain:
+represent ``concat``
+~~~~~~~~~~~~~~~~~~~~
::
- (1 [.0.] [.1.] -- [1 .0. .1.]) uncons uncons
+ ([.0.] [.1.] -- [A*(.0.) .1.])
- (1 [.0.] [.1.] -- 1 [.0. .1.]) uncons
- So far so good...
- (1 [2 .2.] [.1.] -- 1 2 [.2. .1.])
-
-
-
-
- (1 [.0.] [.1.] -- 1 [.0. .1.]) ([a1 .10.] -- a1 [.10.])
- w/ { [a1 .10.] : [ .0. .1.] }
- -or-
- w/ { [ .0. .1.] : [a1 .10. ] }
-
-Typing Combinators
-------------------
-
-TBD
-
-This is an open subject.
-
-The obvious thing is that you now need two pairs of tuples to describe
-the combinators' effects, a stack effect comment and an expression
-effect comment:
+Meaning that ``A*`` on the right-hand side should all the crap from
+``.0.``.
::
- dip (a [F] --)--(-- F a)
+ ([ .0.] [.1.] -- [ A* .1.])
+ ([a .0.] [.1.] -- [a A* .1.])
+ ([a b .0.] [.1.] -- [a b A* .1.])
+ ([a b c .0.] [.1.] -- [a b c A* .1.])
-One thing that might help is...
-
-Abstract Interpretation
------------------------
-
-Something else...
------------------
+or...
::
- [4 5 ...] 2 1 0 pop∘swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 2 1 swap∘roll<∘rest∘rest∘cons∘cons
- [4 5 ...] 1 2 roll<∘rest∘rest∘cons∘cons
- 1 2 [4 5 ...] rest∘rest∘cons∘cons
- 1 2 [5 ...] rest∘cons∘cons
- 1 2 [...] cons∘cons
- 1 [2 ...] cons
- [1 2 ...]
+ ([ .0.] [.1.] -- [ .1.])
+ ([a .0.] [.1.] -- [a .1.])
+ ([a b .0.] [.1.] -- [a b .1.])
+ ([a b c .0.] [.1.] -- [a b c .1.])
+ ([a A* c .0.] [.1.] -- [a A* c .1.])
-Eh?
+::
+
+ (a, (b, S0)) . S1 = (a, (b, (A*, S1)))
+
+.. code:: ipython2
+
+ class Astar(object):
+ def __repr__(self):
+ return 'A*'
+
+
+ def concat(s0, s1):
+ a = []
+ while isinstance(s0, tuple):
+ term, s0 = s0
+ a.append(term)
+ assert isinstance(s0, StackJoyType), repr(s0)
+ s1 = Astar(), s1
+ for term in reversed(a):
+ s1 = term, s1
+ return s1
+
+.. code:: ipython2
+
+ a, b = (A[1], S[0]), (A[2], S[1])
+
+.. code:: ipython2
+
+ concat(a, b)
+
+
+
+
+.. parsed-literal::
+
+ (a1, (A*, (a2, s1)))
+
+
+
+Appendix: Joy in the Logical Paradigm
+-------------------------------------
+
+For this to work the type label classes have to be modified to let
+``T >= t`` succeed, where e.g. ``T`` is ``IntJoyType`` and ``t`` is
+``int``
+
+.. code:: ipython2
+
+ F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
+
+ print doc_from_stack_effect(*F)
+
+
+::
+
+
+ ---------------------------------------------------------------------------
+
+ ValueError Traceback (most recent call last)
+
+ in ()
+ 1 F = reduce(C, (pop, swap, roll_down, rest, rest, cons, cons))
+ 2
+ ----> 3 print doc_from_stack_effect(*F)
+
+
+ in C(f, g)
+ 10 def C(f, g):
+ 11 f, g = relabel(f, g)
+ ---> 12 for fg in compose(f, g):
+ 13 yield delabel(fg)
+
+
+ in compose(f, g)
+ 1 def compose(f, g):
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+
+
+ in C(f, g)
+ 10 def C(f, g):
+ 11 f, g = relabel(f, g)
+ ---> 12 for fg in compose(f, g):
+ 13 yield delabel(fg)
+
+
+ in compose(f, g)
+ 1 def compose(f, g):
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+
+
+ in C(f, g)
+ 10 def C(f, g):
+ 11 f, g = relabel(f, g)
+ ---> 12 for fg in compose(f, g):
+ 13 yield delabel(fg)
+
+
+ in compose(f, g)
+ 1 def compose(f, g):
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+
+
+ in C(f, g)
+ 10 def C(f, g):
+ 11 f, g = relabel(f, g)
+ ---> 12 for fg in compose(f, g):
+ 13 yield delabel(fg)
+
+
+ in compose(f, g)
+ 1 def compose(f, g):
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+
+
+ in C(f, g)
+ 10 def C(f, g):
+ 11 f, g = relabel(f, g)
+ ---> 12 for fg in compose(f, g):
+ 13 yield delabel(fg)
+
+
+ in compose(f, g)
+ 1 def compose(f, g):
+ ----> 2 (f_in, f_out), (g_in, g_out) = f, g
+ 3 s = unify(g_in, f_out)
+ 4 if not s:
+ 5 raise TypeError('Cannot unify %r and %r.' % (f_out, g_in))
+
+
+ ValueError: need more than 1 value to unpack
+
+
+.. code:: ipython2
+
+ from joy.parser import text_to_expression
+
+.. code:: ipython2
+
+ s = text_to_expression('[3 4 ...] 2 1')
+ s
+
+.. code:: ipython2
+
+ L = unify(F[1], s)
+ L
+
+.. code:: ipython2
+
+ F[1]
+
+.. code:: ipython2
+
+ F[1][0]
+
+.. code:: ipython2
+
+ s[0]
+
+`Abstract Interpretation `__
+-----------------------------------------------------------------------------------
+
+I *think* this might be sorta what I'm doing above with the ``kav()``
+function... In any event "mixed-mode" interpreters that include values
+and type variables and can track constraints, etc. will be, uh,
+super-useful. And Abstract Interpretation should be a rich source of
+ideas.
diff --git a/joy/utils/polytypes.py b/joy/utils/polytypes.py
new file mode 100644
index 0000000..4171893
--- /dev/null
+++ b/joy/utils/polytypes.py
@@ -0,0 +1,331 @@
+'''
+
+Multiple Stack Effects
+
+By adjusting the machinery in types.py to handles lists of stackeffect comments
+we can capture more information about the type signatures of some functions,
+and we can introduce a kind of Kleene Star or sequence type that can stand for
+an unbounded sequence of other types.
+
+'''
+import sys
+sys.path.append('/home/sforman/Desktop/Joypy-hg')
+from itertools import product
+from joy.utils.types import (
+ AnyJoyType, A,
+ C,
+ DEFS,
+ doc_from_stack_effect,
+ FloatJoyType,
+ NumberJoyType, N,
+ StackJoyType, S,
+ stacky,
+ _R,
+ relabel, delabel,
+ update,
+ )
+
+
+# We no longer want FloatJoyType to accept IntJoyType.
+class IntJoyType(NumberJoyType): prefix = 'i'
+
+
+a0, a1, a2, a3, a4, a5, a6, a7, a8, a9 = A
+n0, n1, n2, n3, n4, n5, n6, n7, n8, n9 = N
+s0, s1, s2, s3, s4, s5, s6, s7, s8, s9 = S
+f0, f1, f2, f3, f4, f5, f6, f7, f8, f9 = F = map(FloatJoyType, _R)
+i0, i1, i2, i3, i4, i5, i6, i7, i8, i9 = I = map(IntJoyType, _R)
+
+
+class KleeneStar(object):
+ '''
+ A sequence of zero or more `AnyJoyType` variables would be:
+
+ A*
+
+ The `A*` works by splitting the universe into two alternate histories:
+
+ A* -> 0
+
+ A* -> A A*
+
+ The Kleene star variable disappears in one universe, and in the other
+ it turns into an `AnyJoyType` variable followed by itself again.
+
+ We have to return all universes (represented by their substitution
+ dicts, the "unifiers") that don't lead to type conflicts.
+ '''
+
+ kind = AnyJoyType
+
+ def __init__(self, number):
+ self.number = number
+ self.count = 0
+ self.prefix = repr(self)
+
+ def __repr__(self):
+ return '%s%i*' % (self.kind.prefix, self.number)
+
+ def another(self):
+ self.count += 1
+ return self.kind(10000 * self.number + self.count)
+
+ def __eq__(self, other):
+ return (
+ isinstance(other, self.__class__)
+ and other.number == self.number
+ )
+
+ def __ge__(self, other):
+ return self.kind >= other.kind
+
+ def __add__(self, other):
+ return self.__class__(self.number + other)
+ __radd__ = __add__
+
+ def __hash__(self):
+ return hash(repr(self))
+
+
+class AnyStarJoyType(KleeneStar): kind = AnyJoyType
+class NumberStarJoyType(KleeneStar): kind = NumberJoyType
+#class FloatStarJoyType(KleeneStar): kind = FloatJoyType
+#class IntStarJoyType(KleeneStar): kind = IntJoyType
+class StackStarJoyType(KleeneStar): kind = StackJoyType
+
+
+As = map(AnyStarJoyType, _R)
+Ns = map(NumberStarJoyType, _R)
+Ss = map(StackStarJoyType, _R)
+
+
+def _lil_uni(u, v, s):
+ if u >= v:
+ s[u] = v
+ return s,
+ if v >= u:
+ s[v] = u
+ return s,
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
+
+
+
+def unify(u, v, s=None):
+ if s is None:
+ s = {}
+ elif s:
+ u = update(s, u)
+ v = update(s, v)
+
+ if u == v:
+ return s,
+
+ if isinstance(u, AnyJoyType) and isinstance(v, AnyJoyType):
+ return _lil_uni(u, v, s)
+
+ if isinstance(u, tuple) and isinstance(v, tuple):
+ if len(u) != len(v) != 2:
+ raise TypeError(repr((u, v)))
+
+ a, b = v
+ if isinstance(a, KleeneStar):
+ if isinstance(b, KleeneStar):
+ return _lil_uni(a, b, s)
+
+ # Two universes, in one the Kleene star disappears and unification
+ # continues without it...
+ s0 = unify(u, b)
+
+ # In the other it spawns a new variable.
+ s1 = unify(u, (a.another(), v))
+
+ t = s0 + s1
+ for sn in t:
+ sn.update(s)
+ return t
+
+ a, b = u
+ if isinstance(a, KleeneStar):
+ s0 = unify(v, b)
+ s1 = unify(v, (a.another(), u))
+ t = s0 + s1
+ for sn in t:
+ sn.update(s)
+ return t
+
+ ses = unify(u[0], v[0], s)
+ results = ()
+ for sn in ses:
+ results += unify(u[1], v[1], sn)
+ return results
+
+ if isinstance(v, tuple):
+ if not stacky(u):
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
+ s[u] = v
+ return s,
+
+ if isinstance(u, tuple):
+ if not stacky(v):
+ raise TypeError('Cannot unify %r and %r.' % (v, u))
+ s[v] = u
+ return s,
+
+ return ()
+
+
+def compose(f, g):
+ (f_in, f_out), (g_in, g_out) = f, g
+ s = unify(g_in, f_out)
+ if not s:
+ raise TypeError('Cannot unify %r and %r.' % (fo, gi))
+ for result in s:
+ yield update(result, (f_in, g_out))
+
+
+def C(f, g):
+ f, g = relabel(f, g)
+ for fg in compose(f, g):
+ yield delabel(fg)
+
+
+def meta_compose(F, G):
+ for f, g in product(F, G):
+ try:
+ for result in C(f, g):
+ yield result
+ except TypeError:
+ pass
+
+
+def MC(F, G):
+ res = sorted(set(meta_compose(F, G)))
+ if not res:
+ raise TypeError('Cannot unify %r and %r.' % (F, G))
+ return res
+
+
+
+mul = [
+ ((i2, (i1, s0)), (i3, s0)),
+ ((f2, (i1, s0)), (f3, s0)),
+ ((i2, (f1, s0)), (f3, s0)),
+ ((f2, (f1, s0)), (f3, s0)),
+]
+
+
+dup = [DEFS['dup']]
+cons = [DEFS['cons']]
+
+
+sum_ = [(((Ns[1], s1), s0), (n0, s0))]
+f = [(s0, ((n1, (n2, (n3, s1))), s0))]
+
+
+print doc_from_stack_effect(*f[0]), doc_from_stack_effect(*sum_[0])
+print '.......................'
+##for result in unify(sum_[0][0], f[0][1]):
+## print result, '->', update(result, sum_[0][1])
+for g in MC(f, sum_):
+ print doc_from_stack_effect(*g)
+
+print
+print '.......................'
+print
+
+for g in MC(dup, mul):
+ print doc_from_stack_effect(*g)
+
+
+
+##stack_concat = lambda q, e: (q[0], stack_concat(q[1], e)) if isinstance(q, tuple) else e
+
+
+class FunctionJoyType(AnyJoyType):
+ def __init__(self, name, sec, number):
+ self.name = name
+ self.stack_effects = sec
+ self.number = number
+ def __add__(self, other):
+ return self
+ __radd__ = __add__
+
+
+class SymbolJoyType(FunctionJoyType): prefix = 'F'
+class CombinatorJoyType(FunctionJoyType): prefix = 'C'
+
+from joy.library import dip, dipd
+
+##def dip_t(stack, expression):
+## (quote, (a1, stack)) = stack
+## expression = stack_concat(quote, (a1, expression))
+## return stack, expression
+
+CONS = SymbolJoyType('cons', cons, 23)
+DIP = CombinatorJoyType('dip', [dip], 44)
+DIPD = CombinatorJoyType('dipd', [dipd], 45)
+
+from itertools import chain
+
+flatten = lambda g: list(chain.from_iterable(g))
+
+def kav(F, e):
+ if not e:
+ return [F]
+
+ n, e = e
+
+ if isinstance(n, SymbolJoyType):
+ Fs = MC([F], n.stack_effects)
+ res = flatten(kav(Fn, e) for Fn in Fs)
+
+ elif isinstance(n, CombinatorJoyType):
+ res = []
+ for combinator in n.stack_effects:
+ fi, fo = F
+ new_fo, ee, _ = combinator(fo, e, {})
+ new_F = fi, new_fo
+ res.extend(kav(new_F, ee))
+ else:
+ lit = s9, (n, s9)
+ res = flatten(kav(Fn, e) for Fn in MC([F], [lit]))
+
+ return res
+
+
+##l = [(s0, ((CONS, s2), (A[1], s0)))]
+##
+##e = (DIP, ())
+##
+##h = kav(l[0], e)
+##
+##for z in h:
+## print doc_from_stack_effect(*z)
+
+
+ID = s0, s0
+expression = (a1, ((CONS, s0), (DIP, ())))
+
+for sec in kav(ID, expression):
+ print doc_from_stack_effect(*sec)
+
+
+
+
+expression = (a1, (a3, ((CONS, s0), (DIPD, ()))))
+
+for sec in kav(ID, expression):
+ print doc_from_stack_effect(*sec)
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/joy/utils/types.py b/joy/utils/types.py
index e018d61..e70cb7c 100644
--- a/joy/utils/types.py
+++ b/joy/utils/types.py
@@ -34,7 +34,12 @@ class NumberJoyType(AnyJoyType): prefix = 'n'
class FloatJoyType(NumberJoyType): prefix = 'f'
class IntJoyType(FloatJoyType): prefix = 'i'
-class StackJoyType(AnyJoyType): prefix = 's'
+class StackJoyType(AnyJoyType):
+ prefix = 's'
+ def __nonzero__(self):
+ # Imitate () at the end of cons list.
+ return False
+
_R = range(10)
A = a0, a1, a2, a3, a4, a5, a6, a7, a8, a9 = map(AnyJoyType, _R)
@@ -69,8 +74,12 @@ def delabel(f, seen=None, c=None):
pass
if not isinstance(f, tuple):
- seen[f] = f.__class__(c[f.prefix])
- c[f.prefix] += 1
+ try:
+ seen[f] = f.__class__(c[f.prefix])
+ except TypeError: # FunctionJoyTypes break this.
+ seen[f] = f
+ else:
+ c[f.prefix] += 1
return seen[f]
return tuple(delabel(inner, seen, c) for inner in f)
@@ -97,7 +106,7 @@ def unify(u, v, s=None):
if v >= u:
s[v] = u
return s
- raise ValueError('Cannot unify %r and %r.' % (u, v))
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
if isinstance(u, tuple) and isinstance(v, tuple):
if len(u) != len(v) != 2:
@@ -110,13 +119,13 @@ def unify(u, v, s=None):
if isinstance(v, tuple):
if not stacky(u):
- raise ValueError('Cannot unify %r and %r.' % (u, v))
+ raise TypeError('Cannot unify %r and %r.' % (u, v))
s[u] = v
return s
if isinstance(u, tuple):
if not stacky(v):
- raise ValueError('Cannot unify %r and %r.' % (v, u))
+ raise TypeError('Cannot unify %r and %r.' % (v, u))
s[v] = u
return s