diff --git a/docs/Derivatives_of_Regular_Expressions.html b/docs/Derivatives_of_Regular_Expressions.html index 72a0866..294a4ee 100644 --- a/docs/Derivatives_of_Regular_Expressions.html +++ b/docs/Derivatives_of_Regular_Expressions.html @@ -12215,11 +12215,7 @@ expressions grow each derivation.

In [11]:
-
# This is the straightforward version with no "compaction".
-# It works fine, but does waaaay too much work because the
-# expressions grow each derivation.
-
-def D(symbol):
+
def D(symbol):
 
     def derv(R):
 
diff --git a/docs/Derivatives_of_Regular_Expressions.md b/docs/Derivatives_of_Regular_Expressions.md
index 9d2cd68..57cf09b 100644
--- a/docs/Derivatives_of_Regular_Expressions.md
+++ b/docs/Derivatives_of_Regular_Expressions.md
@@ -232,10 +232,6 @@ expressions grow each derivation.
 
 
 ```python
-# This is the straightforward version with no "compaction".
-# It works fine, but does waaaay too much work because the
-# expressions grow each derivation.
-
 def D(symbol):
 
     def derv(R):
diff --git a/docs/Derivatives_of_Regular_Expressions.rst b/docs/Derivatives_of_Regular_Expressions.rst
index 6f87ac6..48516f6 100644
--- a/docs/Derivatives_of_Regular_Expressions.rst
+++ b/docs/Derivatives_of_Regular_Expressions.rst
@@ -266,10 +266,6 @@ derivation.
 
 .. code:: ipython2
 
-    # This is the straightforward version with no "compaction".
-    # It works fine, but does waaaay too much work because the
-    # expressions grow each derivation.
-    
     def D(symbol):
     
         def derv(R):
diff --git a/docs/sphinx_docs/_build/html/_images/omg.svg b/docs/sphinx_docs/_build/html/_images/omg.svg
new file mode 100644
index 0000000..a3be69c
--- /dev/null
+++ b/docs/sphinx_docs/_build/html/_images/omg.svg
@@ -0,0 +1,185 @@
+
+
+
+
+
+
+finite_state_machine
+
+
+i
+
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+i
+
+
+i->i
+
+
+0
+
+
+j
+
+j
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+i->j
+
+
+1
+
+
+h
+
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+h
+
+
+h->i
+
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+0
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+
+h->h
+
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+1
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+a
+
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+b
+
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+
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+a->b
+
+
+0
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+c
+
+c
+
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+a->c
+
+
+1
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+b->b
+
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+0
+
+
+d
+
+d
+
+
+b->d
+
+
+1
+
+
+c->b
+
+
+0
+
+
+e
+
+e
+
+
+c->e
+
+
+1
+
+
+d->b
+
+
+0
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+f
+
+f
+
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+d->f
+
+
+1
+
+
+e->b
+
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+0
+
+
+g
+
+g
+
+
+e->g
+
+
+1
+
+
+f->h
+
+
+1
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+
+f->b
+
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+
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+g->i
+
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+0
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+g->g
+
+
+1
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+j->i
+
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+0
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+
+j->h
+
+
+1
+
+
+
diff --git a/docs/sphinx_docs/_build/html/genindex.html b/docs/sphinx_docs/_build/html/genindex.html
index 09d787a..37e4c73 100644
--- a/docs/sphinx_docs/_build/html/genindex.html
+++ b/docs/sphinx_docs/_build/html/genindex.html
@@ -473,6 +473,8 @@
       
  • TracePrinter (class in joy.utils.pretty_print)
 
    • 
       
  • tuck() (in module joy.utils.generated_library)
+
    • 
+      
  • type_check() (in module joy.utils.polytypes)
 
    • 
   
 
diff --git a/docs/sphinx_docs/_build/html/index.html b/docs/sphinx_docs/_build/html/index.html
index 4172f40..ce8501b 100644
--- a/docs/sphinx_docs/_build/html/index.html
+++ b/docs/sphinx_docs/_build/html/index.html
@@ -147,8 +147,11 @@ interesting aspects.  It’s quite a treasure trove.
    
 
  • Newton’s method
    • 
 
  • Traversing Datastructures with Zippers
    • 
 
  • The Blissful Elegance of Typing Joy
    • 
+
  • Type Checking
    • 
 
  • No Updates
    • 
 
  • Categorical Programming
    • 
+
  • The Four Fundamental Operations of Definite Action
    • 
+
  • ∂RE
    • 
 
 
 
diff --git a/docs/sphinx_docs/_build/html/notebooks/Derivatives_of_Regular_Expressions.html b/docs/sphinx_docs/_build/html/notebooks/Derivatives_of_Regular_Expressions.html
new file mode 100644
index 0000000..13f339d
--- /dev/null
+++ b/docs/sphinx_docs/_build/html/notebooks/Derivatives_of_Regular_Expressions.html
@@ -0,0 +1,923 @@
+
+
+
+
+  
+    
+    
+    ∂RE — Thun 0.2.0 documentation
+    
+    
+    
+    
+    
+    
+    
+    
+    
+   
+  
+  
+  
+  
+
+  
+  
+
+    
    
+      
    
+        
    
+          
    
+            
+  
    
+
    ∂RE
    
+
    
+
    Brzozowski’s Derivatives of Regular Expressions
    
+
    Legend:
    
+
    ∧ intersection
+∨ union
+∘ concatenation (see below)
+¬ complement
+ϕ empty set (aka ∅)
+λ singleton set containing just the empty string
+I set of all letters in alphabet
+
    
+
    
+
    Derivative of a set R of strings and a string a:
    
+
    ∂a(R)
+
+∂a(a) → λ
+∂a(λ) → ϕ
+∂a(ϕ) → ϕ
+∂a(¬a) → ϕ
+∂a(R*) → ∂a(R)∘R*
+∂a(¬R) → ¬∂a(R)
+∂a(R∘S) → ∂a(R)∘S ∨ δ(R)∘∂a(S)
+∂a(R ∧ S) → ∂a(R) ∧ ∂a(S)
+∂a(R ∨ S) → ∂a(R) ∨ ∂a(S)
+
+∂ab(R) = ∂b(∂a(R))
+
    
+
    
+
    Auxiliary predicate function δ (I call it nully) returns either
+λ if λ ⊆ R or ϕ otherwise:
    
+
    δ(a) → ϕ
+δ(λ) → λ
+δ(ϕ) → ϕ
+δ(R*) → λ
+δ(¬R) δ(R)≟ϕ → λ
+δ(¬R) δ(R)≟λ → ϕ
+δ(R∘S) → δ(R) ∧ δ(S)
+δ(R ∧ S) → δ(R) ∧ δ(S)
+δ(R ∨ S) → δ(R) ∨ δ(S)
+
    
+
    
+
    Some rules we will use later for “compaction”:
    
+
    R ∧ ϕ = ϕ ∧ R = ϕ
+
+R ∧ I = I ∧ R = R
+
+R ∨ ϕ = ϕ ∨ R = R
+
+R ∨ I = I ∨ R = I
+
+R∘ϕ = ϕ∘R = ϕ
+
+R∘λ = λ∘R = R
+
    
+
    
+
    Concatination of sets: for two sets A and B the set A∘B is defined as:
    
+
    {a∘b for a in A for b in B}
    
+
    E.g.:
    
+
    {‘a’, ‘b’}∘{‘c’, ‘d’} → {‘ac’, ‘ad’, ‘bc’, ‘bd’}
    
+
    
+
    
+
    Implementation
    
+
    from functools import partial as curry
+from itertools import product
+
    
+
    
+
    
+
    ϕ and λ
    
+
    The empty set and the set of just the empty string.
    
+
    phi = frozenset()   # ϕ
+y = frozenset({''}) # λ
+
    
+
    
+
    
+
    
+
    Two-letter Alphabet
    
+
    I’m only going to use two symbols (at first) becaase this is enough to
+illustrate the algorithm and because you can represent any other
+alphabet with two symbols (if you had to.)
    
+
    I chose the names O and l (uppercase “o” and lowercase “L”) to
+look like 0 and 1 (zero and one) respectively.
    
+
    syms = O, l = frozenset({'0'}), frozenset({'1'})
+
    
+
    
+
    
+
    
+
    Representing Regular Expressions
    
+
    To represent REs in Python I’m going to use tagged tuples. A regular
+expression is one of:
    
+
    O
+l
+(KSTAR, R)
+(NOT, R)
+(AND, R, S)
+(CONS, R, S)
+(OR, R, S)
+
    
+
    
+
    Where R and S stand for regular expressions.
    
+
    AND, CONS, KSTAR, NOT, OR = 'and cons * not or'.split()  # Tags are just strings.
+
    
+
    
+
    Because they are formed of frozenset, tuple and str objects
+only, these datastructures are immutable.
    
+
    
+
    
+
    String Representation of RE Datastructures
    
+
    def stringy(re):
+    '''
+    Return a nice string repr for a regular expression datastructure.
+    '''
+    if re == I: return '.'
+    if re in syms: return next(iter(re))
+    if re == y: return '^'
+    if re == phi: return 'X'
+
+    assert isinstance(re, tuple), repr(re)
+    tag = re[0]
+
+    if tag == KSTAR:
+        body = stringy(re[1])
+        if not body: return body
+        if len(body) > 1: return '(' + body + ")*"
+        return body + '*'
+
+    if tag == NOT:
+        body = stringy(re[1])
+        if not body: return body
+        if len(body) > 1: return '(' + body + ")'"
+        return body + "'"
+
+    r, s = stringy(re[1]), stringy(re[2])
+    if tag == CONS: return r + s
+    if tag == OR:   return '%s | %s' % (r, s)
+    if tag == AND:  return '(%s) & (%s)' % (r, s)
+
+    raise ValueError
+
    
+
    
+
    
+
    
+
    I
    
+
    Match anything. Often spelled “.”
    
+
    I = (0|1)*
+
    
+
    
+
    I = (KSTAR, (OR, O, l))
+
    
+
    
+
    print stringy(I)
+
    
+
    
+
    .
+
    
+
    
+
    
+
    
+
    (.111.) & (.01 + 11*)'
    
+
    The example expression from Brzozowski:
    
+
    (.111.) & (.01 + 11*)'
+   a    &  (b  +  c)'
+
    
+
    
+
    Note that it contains one of everything.
    
+
    a = (CONS, I, (CONS, l, (CONS, l, (CONS, l, I))))
+b = (CONS, I, (CONS, O, l))
+c = (CONS, l, (KSTAR, l))
+it = (AND, a, (NOT, (OR, b, c)))
+
    
+
    
+
    print stringy(it)
+
    
+
    
+
    (.111.) & ((.01 | 11*)')
+
    
+
    
+
    
+
    
+
    nully()
    
+
    Let’s get that auxiliary predicate function δ out of the way.
    
+
    def nully(R):
+    '''
+    δ - Return λ if λ ⊆ R otherwise ϕ.
+    '''
+
+    # δ(a) → ϕ
+    # δ(ϕ) → ϕ
+    if R in syms or R == phi:
+        return phi
+
+    # δ(λ) → λ
+    if R == y:
+        return y
+
+    tag = R[0]
+
+    # δ(R*) → λ
+    if tag == KSTAR:
+        return y
+
+    # δ(¬R) δ(R)≟ϕ → λ
+    # δ(¬R) δ(R)≟λ → ϕ
+    if tag == NOT:
+        return phi if nully(R[1]) else y
+
+    # δ(R∘S) → δ(R) ∧ δ(S)
+    # δ(R ∧ S) → δ(R) ∧ δ(S)
+    # δ(R ∨ S) → δ(R) ∨ δ(S)
+    r, s = nully(R[1]), nully(R[2])
+    return r & s if tag in {AND, CONS} else r | s
+
    
+
    
+
    
+
    
+
    No “Compaction”
    
+
    This is the straightforward version with no “compaction”. It works fine,
+but does waaaay too much work because the expressions grow each
+derivation.
    
+
    def D(symbol):
+
+    def derv(R):
+
+        # ∂a(a) → λ
+        if R == {symbol}:
+            return y
+
+        # ∂a(λ) → ϕ
+        # ∂a(ϕ) → ϕ
+        # ∂a(¬a) → ϕ
+        if R == y or R == phi or R in syms:
+            return phi
+
+        tag = R[0]
+
+        # ∂a(R*) → ∂a(R)∘R*
+        if tag == KSTAR:
+            return (CONS, derv(R[1]), R)
+
+        # ∂a(¬R) → ¬∂a(R)
+        if tag == NOT:
+            return (NOT, derv(R[1]))
+
+        r, s = R[1:]
+
+        # ∂a(R∘S) → ∂a(R)∘S ∨ δ(R)∘∂a(S)
+        if tag == CONS:
+            A = (CONS, derv(r), s)  # A = ∂a(R)∘S
+            # A ∨ δ(R) ∘ ∂a(S)
+            # A ∨  λ   ∘ ∂a(S) → A ∨ ∂a(S)
+            # A ∨  ϕ   ∘ ∂a(S) → A ∨ ϕ → A
+            return (OR, A, derv(s)) if nully(r) else A
+
+        # ∂a(R ∧ S) → ∂a(R) ∧ ∂a(S)
+        # ∂a(R ∨ S) → ∂a(R) ∨ ∂a(S)
+        return (tag, derv(r), derv(s))
+
+    return derv
+
    
+
    
+
    
+
    
+
    Compaction Rules
    
+
    def _compaction_rule(relation, one, zero, a, b):
+    return (
+        b if a == one else  # R*1 = 1*R = R
+        a if b == one else
+        zero if a == zero or b == zero else  # R*0 = 0*R = 0
+        (relation, a, b)
+        )
+
    
+
    
+
    An elegant symmetry.
    
+
    # R ∧ I = I ∧ R = R
+# R ∧ ϕ = ϕ ∧ R = ϕ
+_and = curry(_compaction_rule, AND, I, phi)
+
+# R ∨ ϕ = ϕ ∨ R = R
+# R ∨ I = I ∨ R = I
+_or = curry(_compaction_rule, OR, phi, I)
+
+# R∘λ = λ∘R = R
+# R∘ϕ = ϕ∘R = ϕ
+_cons = curry(_compaction_rule, CONS, y, phi)
+
    
+
    
+
    
+
    
+
    Memoizing
    
+
    We can save re-processing by remembering results we have already
+computed. RE datastructures are immutable and the derv() functions
+are pure so this is fine.
    
+
    class Memo(object):
+
+    def __init__(self, f):
+        self.f = f
+        self.calls = self.hits = 0
+        self.mem = {}
+
+    def __call__(self, key):
+        self.calls += 1
+        try:
+            result = self.mem[key]
+            self.hits += 1
+        except KeyError:
+            result = self.mem[key] = self.f(key)
+        return result
+
    
+
    
+
    
+
    
+
    With “Compaction”
    
+
    This version uses the rules above to perform compaction. It keeps the
+expressions from growing too large.
    
+
    def D_compaction(symbol):
+
+    @Memo
+    def derv(R):
+
+        # ∂a(a) → λ
+        if R == {symbol}:
+            return y
+
+        # ∂a(λ) → ϕ
+        # ∂a(ϕ) → ϕ
+        # ∂a(¬a) → ϕ
+        if R == y or R == phi or R in syms:
+            return phi
+
+        tag = R[0]
+
+        # ∂a(R*) → ∂a(R)∘R*
+        if tag == KSTAR:
+            return _cons(derv(R[1]), R)
+
+        # ∂a(¬R) → ¬∂a(R)
+        if tag == NOT:
+            return (NOT, derv(R[1]))
+
+        r, s = R[1:]
+
+        # ∂a(R∘S) → ∂a(R)∘S ∨ δ(R)∘∂a(S)
+        if tag == CONS:
+            A = _cons(derv(r), s)  # A = ∂a(r)∘s
+            # A ∨ δ(R) ∘ ∂a(S)
+            # A ∨  λ   ∘ ∂a(S) → A ∨ ∂a(S)
+            # A ∨  ϕ   ∘ ∂a(S) → A ∨ ϕ → A
+            return _or(A, derv(s)) if nully(r) else A
+
+        # ∂a(R ∧ S) → ∂a(R) ∧ ∂a(S)
+        # ∂a(R ∨ S) → ∂a(R) ∨ ∂a(S)
+        dr, ds = derv(r), derv(s)
+        return _and(dr, ds) if tag == AND else _or(dr, ds)
+
+    return derv
+
    
+
    
+
    
+
    
+
    
+
    Let’s try it out…
    
+
    (FIXME: redo.)
    
+
    o, z = D_compaction('0'), D_compaction('1')
+REs = set()
+N = 5
+names = list(product(*(N * [(0, 1)])))
+dervs = list(product(*(N * [(o, z)])))
+for name, ds in zip(names, dervs):
+    R = it
+    ds = list(ds)
+    while ds:
+        R = ds.pop()(R)
+        if R == phi or R == I:
+            break
+        REs.add(R)
+
+print stringy(it) ; print
+print o.hits, '/', o.calls
+print z.hits, '/', z.calls
+print
+for s in sorted(map(stringy, REs), key=lambda n: (len(n), n)):
+    print s
+
    
+
    
+
    (.111.) & ((.01 | 11*)')
+
+92 / 122
+92 / 122
+
+(.01)'
+(.01 | 1)'
+(.01 | ^)'
+(.01 | 1*)'
+(.111.) & ((.01 | 1)')
+(.111. | 11.) & ((.01 | ^)')
+(.111. | 11. | 1.) & ((.01)')
+(.111. | 11.) & ((.01 | 1*)')
+(.111. | 11. | 1.) & ((.01 | 1*)')
+
    
+
    
+
    Should match:
    
+
    (.111.) & ((.01 | 11*)')
+
+92 / 122
+92 / 122
+
+(.01     )'
+(.01 | 1 )'
+(.01 | ^ )'
+(.01 | 1*)'
+(.111.)            & ((.01 | 1 )')
+(.111. | 11.)      & ((.01 | ^ )')
+(.111. | 11.)      & ((.01 | 1*)')
+(.111. | 11. | 1.) & ((.01     )')
+(.111. | 11. | 1.) & ((.01 | 1*)')
+
    
+
    
+
    
+
    
+
    Larger Alphabets
    
+
    We could parse larger alphabets by defining patterns for e.g. each byte
+of the ASCII code. Or we can generalize this code. If you study the code
+above you’ll see that we never use the “set-ness” of the symbols O
+and l. The only time Python set operators (& and |) appear
+is in the nully() function, and there they operate on (recursively
+computed) outputs of that function, never O and l.
    
+
    What if we try:
    
+
    (OR, O, l)
+
+∂1((OR, O, l))
+                            ∂a(R ∨ S) → ∂a(R) ∨ ∂a(S)
+∂1(O) ∨ ∂1(l)
+                            ∂a(¬a) → ϕ
+ϕ ∨ ∂1(l)
+                            ∂a(a) → λ
+ϕ ∨ λ
+                            ϕ ∨ R = R
+λ
+
    
+
    
+
    And compare it to:
    
+
    {'0', '1')
+
+∂1({'0', '1'))
+                            ∂a(R ∨ S) → ∂a(R) ∨ ∂a(S)
+∂1({'0')) ∨ ∂1({'1'))
+                            ∂a(¬a) → ϕ
+ϕ ∨ ∂1({'1'))
+                            ∂a(a) → λ
+ϕ ∨ λ
+                            ϕ ∨ R = R
+λ
+
    
+
    
+
    This suggests that we should be able to alter the functions above to
+detect sets and deal with them appropriately. Exercise for the Reader
+for now.
    
+
    
+
    
+
    State Machine
    
+
    We can drive the regular expressions to flesh out the underlying state
+machine transition table.
    
+
    .111. & (.01 + 11*)'
+
    
+
    
+
    Says, “Three or more 1’s and not ending in 01 nor composed of all 1’s.”
    
+
    
+State Machine Diagram
    
+
    Start at a and follow the transition arrows according to their
+labels. Accepting states have a double outline. (Graphic generated with
+Dot from Graphviz.) You’ll see that only
+paths that lead to one of the accepting states will match the regular
+expression. All other paths will terminate at one of the non-accepting
+states.
    
+
    There’s a happy path to g along 111:
    
+
    a→c→e→g
+
    
+
    
+
    After you reach g you’re stuck there eating 1’s until you see a 0,
+which takes you to the i→j→i|i→j→h→i “trap”. You can’t reach any
+other states from those two loops.
    
+
    If you see a 0 before you see 111 you will reach b, which forms
+another “trap” with d and f. The only way out is another happy
+path along 111 to h:
    
+
    b→d→f→h
+
    
+
    
+
    Once you have reached h you can see as many 1’s or as many 0’ in a
+row and still be either still at h (for 1’s) or move to i (for
+0’s). If you find yourself at i you can see as many 0’s, or
+repetitions of 10, as there are, but if you see just a 1 you move to
+j.
    
+
    
+
    RE to FSM
    
+
    So how do we get the state machine from the regular expression?
    
+
    It turns out that each RE is effectively a state, and each arrow points
+to the derivative RE in respect to the arrow’s symbol.
    
+
    If we label the initial RE a, we can say:
    
+
    a --0--> ∂0(a)
+a --1--> ∂1(a)
+
    
+
    
+
    And so on, each new unique RE is a new state in the FSM table.
    
+
    Here are the derived REs at each state:
    
+
    a = (.111.) & ((.01 | 11*)')
+b = (.111.) & ((.01 | 1)')
+c = (.111. | 11.) & ((.01 | 1*)')
+d = (.111. | 11.) & ((.01 | ^)')
+e = (.111. | 11. | 1.) & ((.01 | 1*)')
+f = (.111. | 11. | 1.) & ((.01)')
+g = (.01 | 1*)'
+h = (.01)'
+i = (.01 | 1)'
+j = (.01 | ^)'
+
    
+
    
+
    You can see the one-way nature of the g state and the hij “trap”
+in the way that the .111. on the left-hand side of the &
+disappears once it has been matched.
    
+
    from collections import defaultdict
+from pprint import pprint
+from string import ascii_lowercase
+
    
+
    
+
    d0, d1 = D_compaction('0'), D_compaction('1')
+
    
+
    
+
    
+
    
+
    explore()
    
+
    def explore(re):
+
+    # Don't have more than 26 states...
+    names = defaultdict(iter(ascii_lowercase).next)
+
+    table, accepting = dict(), set()
+
+    to_check = {re}
+    while to_check:
+
+        re = to_check.pop()
+        state_name = names[re]
+
+        if (state_name, 0) in table:
+            continue
+
+        if nully(re):
+            accepting.add(state_name)
+
+        o, i = d0(re), d1(re)
+        table[state_name, 0] = names[o] ; to_check.add(o)
+        table[state_name, 1] = names[i] ; to_check.add(i)
+
+    return table, accepting
+
    
+
    
+
    table, accepting = explore(it)
+table
+
    
+
    
+
    {('a', 0): 'b',
+ ('a', 1): 'c',
+ ('b', 0): 'b',
+ ('b', 1): 'd',
+ ('c', 0): 'b',
+ ('c', 1): 'e',
+ ('d', 0): 'b',
+ ('d', 1): 'f',
+ ('e', 0): 'b',
+ ('e', 1): 'g',
+ ('f', 0): 'b',
+ ('f', 1): 'h',
+ ('g', 0): 'i',
+ ('g', 1): 'g',
+ ('h', 0): 'i',
+ ('h', 1): 'h',
+ ('i', 0): 'i',
+ ('i', 1): 'j',
+ ('j', 0): 'i',
+ ('j', 1): 'h'}
+
    
+
    
+
    accepting
+
    
+
    
+
    {'h', 'i'}
+
    
+
    
+
    
+
    
+
    Generate Diagram
    
+
    Once we have the FSM table and the set of accepting states we can
+generate the diagram above.
    
+
    _template = '''\
+digraph finite_state_machine {
+  rankdir=LR;
+  size="8,5"
+  node [shape = doublecircle]; %s;
+  node [shape = circle];
+%s
+}
+'''
+
+def link(fr, nm, label):
+    return '  %s -> %s [ label = "%s" ];' % (fr, nm, label)
+
+
+def make_graph(table, accepting):
+    return _template % (
+        ' '.join(accepting),
+        '\n'.join(
+          link(from_, to, char)
+          for (from_, char), (to) in sorted(table.iteritems())
+          )
+        )
+
    
+
    
+
    print make_graph(table, accepting)
+
    
+
    
+
    digraph finite_state_machine {
+  rankdir=LR;
+  size="8,5"
+  node [shape = doublecircle]; i h;
+  node [shape = circle];
+  a -> b [ label = "0" ];
+  a -> c [ label = "1" ];
+  b -> b [ label = "0" ];
+  b -> d [ label = "1" ];
+  c -> b [ label = "0" ];
+  c -> e [ label = "1" ];
+  d -> b [ label = "0" ];
+  d -> f [ label = "1" ];
+  e -> b [ label = "0" ];
+  e -> g [ label = "1" ];
+  f -> b [ label = "0" ];
+  f -> h [ label = "1" ];
+  g -> i [ label = "0" ];
+  g -> g [ label = "1" ];
+  h -> i [ label = "0" ];
+  h -> h [ label = "1" ];
+  i -> i [ label = "0" ];
+  i -> j [ label = "1" ];
+  j -> i [ label = "0" ];
+  j -> h [ label = "1" ];
+}
+
    
+
    
+
    
+
    
+
    Drive a FSM
    
+
    There are lots of FSM libraries already. Once you have the state
+transition table they should all be straightforward to use. State
+Machine code is very simple. Just for fun, here is an implementation in
+Python that imitates what “compiled” FSM code might look like in an
+“unrolled” form. Most FSM code uses a little driver loop and a table
+datastructure, the code below instead acts like JMP instructions
+(“jump”, or GOTO in higher-level-but-still-low-level languages) to
+hard-code the information in the table into a little patch of branches.
    
+
    
+
    Trampoline Function
    
+
    Python has no GOTO statement but we can fake it with a “trampoline”
+function.
    
+
    def trampoline(input_, jump_from, accepting):
+    I = iter(input_)
+    while True:
+        try:
+            bounce_to = jump_from(I)
+        except StopIteration:
+            return jump_from in accepting
+        jump_from = bounce_to
+
    
+
    
+
    
+
    
+
    Stream Functions
    
+
    Little helpers to process the iterator of our data (a “stream” of “1”
+and “0” characters, not bits.)
    
+
    getch = lambda I: int(next(I))
+
+
+def _1(I):
+    '''Loop on ones.'''
+    while getch(I): pass
+
+
+def _0(I):
+    '''Loop on zeros.'''
+    while not getch(I): pass
+
    
+
    
+
    
+
    
+
    A Finite State Machine
    
+
    With those preliminaries out of the way, from the state table of
+.111. & (.01 + 11*)' we can immediately write down state machine
+code. (You have to imagine that these are GOTO statements in C or
+branches in assembly and that the state names are branch destination
+labels.)
    
+
    a = lambda I: c if getch(I) else b
+b = lambda I: _0(I) or d
+c = lambda I: e if getch(I) else b
+d = lambda I: f if getch(I) else b
+e = lambda I: g if getch(I) else b
+f = lambda I: h if getch(I) else b
+g = lambda I: _1(I) or i
+h = lambda I: _1(I) or i
+i = lambda I: _0(I) or j
+j = lambda I: h if getch(I) else i
+
    
+
    
+
    Note that the implementations of h and g are identical ergo
+h = g and we could eliminate one in the code but h is an
+accepting state and g isn’t.
    
+
    def acceptable(input_):
+    return trampoline(input_, a, {h, i})
+
    
+
    
+
    for n in range(2**5):
+    s = bin(n)[2:]
+    print '%05s' % s, acceptable(s)
+
    
+
    
+
        0 False
+    1 False
+   10 False
+   11 False
+  100 False
+  101 False
+  110 False
+  111 False
+ 1000 False
+ 1001 False
+ 1010 False
+ 1011 False
+ 1100 False
+ 1101 False
+ 1110 True
+ 1111 False
+10000 False
+10001 False
+10010 False
+10011 False
+10100 False
+10101 False
+10110 False
+10111 True
+11000 False
+11001 False
+11010 False
+11011 False
+11100 True
+11101 False
+11110 True
+11111 False
+
    
+
    
+
    
+
    
+
    
+
    
+
    Reversing the Derivatives to Generate Matching Strings
    
+
    (UNFINISHED) Brzozowski also shewed how to go from the state machine to
+strings and expressions…
    
+
    Each of these states is just a name for a Brzozowskian RE, and so, other
+than the initial state a, they can can be described in terms of the
+derivative-with-respect-to-N of some other state/RE:
    
+
    c = d1(a)
+b = d0(a)
+b = d0(c)
+...
+i = d0(j)
+j = d1(i)
+
    
+
    
+
    Consider:
    
+
    c = d1(a)
+b = d0(c)
+
    
+
    
+
    Substituting:
    
+
    b = d0(d1(a))
+
    
+
    
+
    Unwrapping:
    
+
    b = d10(a)
+
    
+
    
+
    ‘’‘
    
+
    j = d1(d0(j))
+
    
+
    
+
    Unwrapping:
    
+
    j = d1(d0(j)) = d01(j)
+
    
+
    
+
    We have a loop or “fixed point”.
    
+
    j = d01(j) = d0101(j) = d010101(j) = ...
+
    
+
    
+
    hmm…
    
+
    j = (01)*
+
    
+
    
+
    
+
    
+
+
+          
    
+        
    
+      
    
+      
+      
    
+    
    
+    
+
+  
+
\ No newline at end of file
diff --git a/docs/sphinx_docs/_build/html/notebooks/The_Four_Operations.html b/docs/sphinx_docs/_build/html/notebooks/The_Four_Operations.html
new file mode 100644
index 0000000..5e6b79e
--- /dev/null
+++ b/docs/sphinx_docs/_build/html/notebooks/The_Four_Operations.html
@@ -0,0 +1,391 @@
+
+
+
+
+  
+    
+    
+    The Four Fundamental Operations of Definite Action — Thun 0.2.0 documentation
+    
+    
+    
+    
+    
+    
+    
+    
+    
+   
+  
+  
+  
+  
+
+  
+  
+
+    
    
+      
    
+        
    
+          
    
+            
+  
    
+
    The Four Fundamental Operations of Definite Action
    
+
    All definite actions (computer program) can be defined by four
+fundamental patterns of combination:
    
+
      
+
    1. Sequence
      2. 
+
    2. Branch
      3. 
+
    3. Loop
      4. 
+
    4. Parallel
      5. 
+
    
+
    
+
    Sequence
    
+
    Do one thing after another. In joy this is represented by putting two
+symbols together, juxtaposition:
    
+
    foo bar
+
    
+
    
+
    Operations have inputs and outputs. The outputs of foo must be
+compatible in “arity”, type, and shape with the inputs of bar.
    
+
    
+
    
+
    Branch
    
+
    Do one thing or another.
    
+
    boolean [F] [T] branch
+
+
+   t [F] [T] branch
+----------------------
+          T
+
+
+   f [F] [T] branch
+----------------------
+      F
+
+
+branch == unit cons swap pick i
+
+boolean [F] [T] branch
+boolean [F] [T] unit cons swap pick i
+boolean [F] [[T]] cons swap pick i
+boolean [[F] [T]] swap pick i
+[[F] [T]] boolean pick i
+[F-or-T] i
+
    
+
    
+
    Given some branch function G:
    
+
    G == [F] [T] branch
+
    
+
    
+
    Used in a sequence like so:
    
+
    foo G bar
+
    
+
    
+
    The inputs and outputs of F and T must be compatible with the
+outputs for foo and the inputs of bar, respectively.
    
+
    foo F bar
+
+foo T bar
+
    
+
    
+
    
+
    ifte
    
+
    Often it will be easier on the programmer to write branching code with
+the predicate specified in a quote. The ifte combinator provides
+this (T for “then” and E for “else”):
    
+
    [P] [T] [E] ifte
+
    
+
    
+
    Defined in terms of branch:
    
+
    ifte == [nullary not] dip branch
+
    
+
    
+
    In this case, P must be compatible with the stack and return a
+Boolean value, and T and E both must be compatible with the
+preceeding and following functions, as described above for F and
+T. (Note that in the current implementation we are depending on
+Python for the underlying semantics, so the Boolean value doesn’t have
+to be Boolean because Python’s rules for “truthiness” will be used to
+evaluate it. I reflect this in the structure of the stack effect comment
+of branch, it will only accept Boolean values, and in the definition
+of ifte above by including not in the quote, which also has the
+effect that the subject quotes are in the proper order for branch.)
    
+
    
+
    
+
    
+
    Loop
    
+
    Do one thing zero or more times.
    
+
    boolean [Q] loop
+
+
+   t [Q] loop
+----------------
+   Q [Q] loop
+
+
+   ... f [Q] loop
+--------------------
+   ...
+
    
+
    
+
    The loop combinator generates a copy of itself in the true branch.
+This is the hallmark of recursive defintions. In Thun there is no
+equivalent to conventional loops. (There is, however, the x
+combinator, defined as x == dup i, which permits recursive
+constructs that do not need to be directly self-referential, unlike
+loop and genrec.)
    
+
    loop == [] swap [dup dip loop] cons branch
+
+boolean [Q] loop
+boolean [Q] [] swap [dup dip loop] cons branch
+boolean [] [Q] [dup dip loop] cons branch
+boolean [] [[Q] dup dip loop] branch
+
    
+
    
+
    In action the false branch does nothing while the true branch does:
    
+
    t [] [[Q] dup dip loop] branch
+      [Q] dup dip loop
+      [Q] [Q] dip loop
+      Q [Q] loop
+
    
+
    
+
    Because loop expects and consumes a Boolean value, the Q
+function must be compatible with the previous stack and itself with a
+boolean flag for the next iteration:
    
+
    Q == G b
+
+Q [Q] loop
+G b [Q] loop
+G Q [Q] loop
+G G b [Q] loop
+G G Q [Q] loop
+G G G b [Q] loop
+G G G
+
    
+
    
+
    
+
    while
    
+
    Keep doing B while some predicate P is true. This is
+convenient as the predicate function is made nullary automatically and
+the body function can be designed without regard to leaving a Boolean
+flag for the next iteration:
    
+
                [P] [B] while
+--------------------------------------
+   [P] nullary [B [P] nullary] loop
+
+
+while == swap [nullary] cons dup dipd concat loop
+
+
+[P] [B] while
+[P] [B] swap [nullary] cons dup dipd concat loop
+[B] [P] [nullary] cons dup dipd concat loop
+[B] [[P] nullary] dup dipd concat loop
+[B] [[P] nullary] [[P] nullary] dipd concat loop
+[P] nullary [B] [[P] nullary] concat loop
+[P] nullary [B [P] nullary] loop
+
    
+
    
+
    
+
    
+
    
+
    Parallel
    
+
    The parallel operation indicates that two (or more) functions do not
+interfere with each other and so can run in parallel. The main
+difficulty in this sort of thing is orchestrating the recombining
+(“join” or “wait”) of the results of the functions after they finish.
    
+
    The current implementaions and the following definitions are not
+actually parallel (yet), but there is no reason they couldn’t be
+reimplemented in terms of e.g. Python threads. I am not concerned with
+performance of the system just yet, only the elegance of the code it
+allows us to write.
    
+
    
+
    cleave
    
+
    Joy has a few parallel combinators, the main one being cleave:
    
+
                   ... x [A] [B] cleave
+---------------------------------------------------------
+   ... [x ...] [A] infra first [x ...] [B] infra first
+---------------------------------------------------------
+                   ... a b
+
    
+
    
+
    The cleave combinator expects a value and two quotes and it executes
+each quote in “separate universes” such that neither can affect the
+other, then it takes the first item from the stack in each universe and
+replaces the value and quotes with their respective results.
    
+
    (I think this corresponds to the “fork” operator, the little
+upward-pointed triangle, that takes two functions A :: x -> a and
+B :: x -> b and returns a function F :: x -> (a, b), in Conal
+Elliott’s “Compiling to Categories” paper, et. al.)
    
+
    Just a thought, if you cleave two jobs and one requires more time to
+finish than the other you’d like to be able to assign resources
+accordingly so that they both finish at the same time.
    
+
    
+
    
+
    “Apply” Functions
    
+
    There are also app2 and app3 which run a single quote on more
+than one value:
    
+
                    ... y x [Q] app2
+---------------------------------------------------------
+   ... [y ...] [Q] infra first [x ...] [Q] infra first
+
+
+       ... z y x [Q] app3
+---------------------------------
+   ... [z ...] [Q] infra first
+       [y ...] [Q] infra first
+       [x ...] [Q] infra first
+
    
+
    
+
    Because the quoted program can be i we can define cleave in
+terms of app2:
    
+
    cleave == [i] app2 [popd] dip
+
    
+
    
+
    (I’m not sure why cleave was specified to take that value, I may
+make a combinator that does the same thing but without expecting a
+value.)
    
+
    clv == [i] app2
+
+   [A] [B] clv
+------------------
+     a b
+
    
+
    
+
    
+
    
+
    map
    
+
    The common map function in Joy should also be though of as a
+parallel operator:
    
+
    [a b c ...] [Q] map
+
    
+
    
+
    There is no reason why the implementation of map couldn’t distribute
+the Q function over e.g. a pool of worker CPUs.
    
+
    
+
    
+
    pam
    
+
    One of my favorite combinators, the pam combinator is just:
    
+
    pam == [i] map
+
    
+
    
+
    This can be used to run any number of programs separately on the current
+stack and combine their (first) outputs in a result list.
    
+
       [[A] [B] [C] ...] [i] map
+-------------------------------
+   [ a   b   c  ...]
+
    
+
    
+
    
+
    
+
    Handling Other Kinds of Join
    
+
    The cleave operators and others all have pretty brutal join
+semantics: everything works and we always wait for every
+sub-computation. We can imagine a few different potentially useful
+patterns of “joining” results from parallel combinators.
    
+
    
+
    first-to-finish
    
+
    Thinking about variations of pam there could be one that only
+returns the first result of the first-to-finish sub-program, or the
+stack could be replaced by its output stack.
    
+
    The other sub-programs would be cancelled.
    
+
    
+
    
+
    “Fulminators”
    
+
    Also known as “Futures” or “Promises” (by everybody else. “Fulinators”
+is what I was going to call them when I was thinking about implementing
+them in Thun.)
    
+
    The runtime could be amended to permit “thunks” representing the results
+of in-progress computations to be left on the stack and picked up by
+subsequent functions. These would themselves be able to leave behind
+more “thunks”, the values of which depend on the eventual resolution of
+the values of the previous thunks.
    
+
    In this way you can create “chains” (and more complex shapes) out of
+normal-looking code that consist of a kind of call-graph interspersed
+with “asyncronous” … events?
    
+
    In any case, until I can find a rigorous theory that shows that this
+sort of thing works perfectly in Joy code I’m not going to worry about
+it. (And I think the Categories can deal with it anyhow? Incremental
+evaluation, yeah?)
    
+
    
+
    
+
    
+
    
+
+
+          
    
+        
    
+      
    
+      
+      
    
+    
    
+    
+
+  
+
\ No newline at end of file
diff --git a/docs/sphinx_docs/_build/html/notebooks/TypeChecking.html b/docs/sphinx_docs/_build/html/notebooks/TypeChecking.html
new file mode 100644
index 0000000..cd3ed20
--- /dev/null
+++ b/docs/sphinx_docs/_build/html/notebooks/TypeChecking.html
@@ -0,0 +1,209 @@
+
+
+
+
+  
+    
+    
+    Type Checking — Thun 0.2.0 documentation
+    
+    
+    
+    
+    
+    
+    
+    
+    
+   
+  
+  
+  
+  
+
+  
+  
+
+    
    
+      
    
+        
    
+          
    
+            
+  
    
+
    Type Checking
    
+
    import logging, sys
+
+logging.basicConfig(
+  format='%(message)s',
+  stream=sys.stdout,
+  level=logging.INFO,
+  )
+
    
+
    
+
    from joy.utils.polytypes import (
+    doc_from_stack_effect,
+    infer,
+    reify,
+    unify,
+    FUNCTIONS,
+    JoyTypeError,
+)
+
    
+
    
+
    D = FUNCTIONS.copy()
+del D['product']
+globals().update(D)
+
    
+
    
+
    
+
    An Example
    
+
    fi, fo = infer(pop, swap, rolldown, rrest, ccons)[0]
+
    
+
    
+
    25 (--) ∘ pop swap rolldown rrest ccons
+28 (a1 --) ∘ swap rolldown rrest ccons
+31 (a3 a2 a1 -- a2 a3) ∘ rolldown rrest ccons
+34 (a4 a3 a2 a1 -- a2 a3 a4) ∘ rrest ccons
+37 ([a4 a5 ...1] a3 a2 a1 -- a2 a3 [...1]) ∘ ccons
+40 ([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1]) ∘
+
    
+
    
+
    print doc_from_stack_effect(fi, fo)
+
    
+
    
+
    ([a4 a5 ...1] a3 a2 a1 -- [a2 a3 ...1])
+
    
+
    
+
    from joy.parser import text_to_expression
+from joy.utils.stack import stack_to_string
+
    
+
    
+
    e = text_to_expression('0 1 2 [3 4]')  # reverse order
+print stack_to_string(e)
+
    
+
    
+
    [3 4] 2 1 0
+
    
+
    
+
    u = unify(e, fi)[0]
+u
+
    
+
    
+
    {a1: 0, a2: 1, a3: 2, a4: 3, a5: 4, s2: (), s1: ()}
+
    
+
    
+
    g = reify(u, (fi, fo))
+print doc_from_stack_effect(*g)
+
    
+
    
+
    (... [3 4 ] 2 1 0 -- ... [1 2 ])
+
    
+
    
+
    
+
    
+
    Unification Works “in Reverse”
    
+
    e = text_to_expression('[2 3]')
+
    
+
    
+
    u = unify(e, fo)[0]  # output side, not input side
+u
+
    
+
    
+
    {a2: 2, a3: 3, s2: (), s1: ()}
+
    
+
    
+
    g = reify(u, (fi, fo))
+print doc_from_stack_effect(*g)
+
    
+
    
+
    (... [a4 a5 ] 3 2 a1 -- ... [2 3 ])
+
    
+
    
+
    
+
    
+
    Failing a Check
    
+
    fi, fo = infer(dup, mul)[0]
+
    
+
    
+
    25 (--) ∘ dup mul
+28 (a1 -- a1 a1) ∘ mul
+31 (f1 -- f2) ∘
+31 (i1 -- i2) ∘
+
    
+
    
+
    e = text_to_expression('"two"')
+print stack_to_string(e)
+
    
+
    
+
    'two'
+
    
+
    
+
    try:
+    unify(e, fi)
+except JoyTypeError, err:
+    print err
+
    
+
    
+
    Cannot unify 'two' and f1.
+
    
+
    
+
    
+
    
+
+
+          
    
+        
    
+      
    
+      
+      
    
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\ No newline at end of file
diff --git a/docs/sphinx_docs/_build/html/notebooks/index.html b/docs/sphinx_docs/_build/html/notebooks/index.html
index 8aae107..6a9a901 100644
--- a/docs/sphinx_docs/_build/html/notebooks/index.html
+++ b/docs/sphinx_docs/_build/html/notebooks/index.html
@@ -130,8 +130,30 @@
 
  • Appendix: Joy in the Logical Paradigm
    • 
 
 
+
  • Type Checking
+
    • 
 
  • No Updates
    • 
 
  • Categorical Programming
    • 
+
  • The Four Fundamental Operations of Definite Action
+
    • 
+
  • ∂RE
+
    •
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0.2.0 Documentation","Joy Interpreter","Functions Grouped by, er, Function with Examples","Function Reference","Categorical Programming","\u2202RE","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","Recursion Combinators","Replacing Functions in the Dictionary","The Four Fundamental Operations of Definite Action","Treating Trees II: treestep","Type Checking","The Blissful Elegance of Typing Joy","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","Type Inference of Joy Expressions"],titleterms:{"\u03bb":5,"\u03d5":5,"abstract":[],"case":[9,11],"function":[2,3,5,8,9,11,13,14,15,16,18],"long":14,"new":11,"p\u00f6ial":18,"try":5,"void":2,"while":[2,15],Adding:11,One:[7,11],The:[6,8,11,13,15,16,18],There:8,Using:7,With:[5,16],about:20,action:15,add:[2,11],adding:11,address:19,alphabet:5,altern:16,ana:13,analysi:6,anamorph:[2,13],app1:2,app2:2,app3:2,appendix:[11,13,18],appli:15,approxim:9,argument:18,auto:3,averag:2,base:[9,11],binari:[2,11,16],bliss:18,both:11,branch:[2,11,15],brzozowski:5,can:11,cata:13,catamorph:13,categor:4,chatter:2,check:17,child:11,choic:2,clear:2,cleav:[2,15],cmp:11,code:[8,11],combin:[2,11,13,18],comment:18,compact:5,compar:11,comparison:2,compil:[7,18],compile_:18,compos:18,comput:9,con:[2,18],concat:2,conclus:[13,18],consecut:9,continu:8,current:11,datastructur:[5,8,11,19],deal:18,defin:[11,16],definit:[12,15],delabel:18,delet:11,deriv:[5,12,13,16],design:13,determin:19,develop:6,diagram:5,dialect:0,dictionari:14,dip:[2,19],dipd:2,dipdd:2,direco:7,disenstacken:2,distinguish:18,div:2,doc_from_stack_effect:18,document:0,doe:11,down_to_zero:2,drive:5,drop:2,dup:[2,18],dupd:2,dupdip:2,effect:18,eleg:18,els:11,empti:11,enstacken:2,equal:11,essai:20,euler:[6,7],eval:8,even:7,exampl:[2,8,11,13,16,17],execut:22,explor:5,express:[5,8,21,24],extract:16,factori:13,fail:17,fibonacci:7,filter:6,find:[9,11,13],finish:15,finit:5,first:[2,6,15,18],five:7,flatten:2,flexibl:16,floordiv:2,formula:12,found:11,four:[13,15],fsm:5,fulmin:15,fun:13,fundament:15,further:6,gcd:2,gener:[3,5,6,7,9],genrec:2,get:[11,16],getitem:2,given:16,greater:11,group:2,handl:15,have:[11,16],help:2,highest:11,host:0,how:[6,7],hybrid:18,hylo:13,hylomorph:13,identifi:18,ift:[2,15],iii:18,implement:[5,18],indic:0,infer:[18,24],inferenc:18,inform:0,infra:[2,19],integ:6,interest:7,interlud:11,intern:21,interpret:[1,8,18],item:19,iter:[6,11],joi:[0,1,3,6,8,13,18,19,20,21,22,23,24],join:15,junk:[],just:6,kei:11,kind:15,languag:0,larger:5,least_fract:2,left:11,less:11,let:[5,6],letter:5,librari:[3,8,18],like:11,list:[2,23],literari:8,littl:6,logic:[2,18],loop:[2,8,15],lower:11,lshift:2,machin:5,make:[7,9],mani:6,map:[2,15],match:5,math:2,memoiz:5,method:9,min:2,miscellan:2,mod:2,modifi:18,modulu:2,more:11,most:11,mul:[2,18],multipl:[6,7,18],must:11,name:12,neg:2,newton:9,next:9,node:11,non:11,now:11,nullari:2,nulli:5,number:[13,18],one:8,onli:8,oper:15,order:[11,16],osdn:0,other:15,our:11,out:5,over:2,pack:6,pam:[2,15],para:13,paradigm:18,parallel:15,parameter:[11,16],pars:[2,21],parser:[8,21],part:18,pass:8,path:19,pattern:13,per:11,poly_typ:[],polytyp:24,pop:[2,18],popd:2,popop:2,pow:2,power:7,pred:2,predic:[6,9,11,16],pretty_print:22,primit:13,primrec:2,print:8,problem:[6,7],process:11,product:2,program:[4,6,12,16,20],progress:18,project:[0,6,7],pure:8,put:[11,12,16],python:[8,14,18],quadrat:12,quick:0,quot:[2,23],rang:[2,6,13],range_to_zero:2,read:8,recur:[9,11],recurs:[11,13,16],redefin:[11,16],refactor:[6,11],refer:3,regular:[5,8],reimplement:16,relabel:18,rem:2,remaind:2,remov:2,render:6,repl:8,replac:[11,14],repres:[5,18],represent:5,reset:7,rest:[2,18],revers:[2,5,17],right:[11,19],rightmost:11,roll:[2,18],rolldown:2,rollup:2,rshift:2,rule:[5,18],run:[2,7],second:[2,18],select:2,sequenc:[7,15,18,23],set:[9,11],shorter:14,should:8,shunt:2,simpl:18,simplest:6,size:[2,14],someth:[],sourc:11,special:[13,18],sqr:[2,18],sqrt:[2,12],stack:[2,8,18,23],start:0,state:5,step:[2,13,16],straightforward:12,stream:5,string:5,structur:11,style:8,sub:[2,11],subtyp:18,succ:2,sum:[2,6],swaack:2,swap:[2,18],swon:2,swoncat:2,symbol:[8,13],tabl:0,tail:13,take:2,term:[6,7,16],ternari:2,text:21,than:11,them:12,thi:11,third:[2,18],three:7,thun:[0,8],time:[2,7],togeth:[11,12,16],token:8,toler:9,trace:[14,22],traceprint:8,trampolin:5,travers:[11,16,19],treat:[11,16],tree:[11,16,19],treegrind:16,treestep:16,triangular:13,truediv:2,truthi:2,tuck:2,two:[5,7],type:[17,18,24],unari:2,unbound:18,uncon:[2,18],unif:17,unifi:18,unit:2,unnecessari:6,unquot:2,unstack:2,updat:[10,18],use:18,util:[22,23,24],valu:[7,11],variabl:12,variat:7,version:[6,11,14,18],view:8,vii:18,within:9,word:2,work:[17,18],write:12,xor:2,yin:18,zero:7,zip:2,zipper:19}}) \ No newline at end of file diff --git a/docs/sphinx_docs/_build/html/types.html b/docs/sphinx_docs/_build/html/types.html index e65bec7..d2335dc 100644 --- a/docs/sphinx_docs/_build/html/types.html +++ b/docs/sphinx_docs/_build/html/types.html @@ -291,7 +291,7 @@ guard against being used on invalid types.

    -joy.utils.polytypes.FUNCTIONS = {'_Tree_add_Ee': _Tree_add_Ee, '_Tree_delete_R0': _Tree_delete_R0, '_Tree_delete_clear_stuff': _Tree_delete_clear_stuff, '_Tree_get_E': _Tree_get_E, 'add': add, 'and': and, 'b': b, 'bool': bool, 'branch': branch, 'ccons': ccons, 'clear': clear, 'concat_': concat_, 'cons': cons, 'dip': dip, 'dipd': dipd, 'dipdd': dipdd, 'div': div, 'divmod': divmod, 'dup': dup, 'dupd': dupd, 'dupdd': dupdd, 'dupdip': dupdip, 'eq': eq, 'first': first, 'first_two': first_two, 'floordiv': floordiv, 'fourth': fourth, 'ge': ge, 'gt': gt, 'i': i, 'infra': infra, 'le': le, 'loop': loop, 'lshift': lshift, 'lt': lt, 'modulus': modulus, 'mul': mul, 'ne': ne, 'neg': neg, 'not': not, 'nullary': nullary, 'over': over, 'pm': pm, 'pop': pop, 'popd': popd, 'popdd': popdd, 'popop': popop, 'popopd': popopd, 'popopdd': popopdd, 'pow': pow, 'pred': pred, 'product': product, 'rest': rest, 'rolldown': rolldown, 'rollup': rollup, 'rrest': rrest, 'rshift': rshift, 'second': second, 'sqrt': sqrt, 'stack': stack, 'stuncons': stuncons, 'stununcons': stununcons, 'sub': sub, 'succ': succ, 'sum': sum, 'swaack': swaack, 'swap': swap, 'swons': swons, 'third': third, 'truediv': truediv, 'tuck': tuck, 'uncons': uncons, 'unit': unit, 'unswons': unswons, 'x': x}
    +joy.utils.polytypes.FUNCTIONS = {'!=': ne, '&': and, '*': mul, '+': add, '++': succ, '-': sub, '--': pred, '/': truediv, '<': lt, '<<': lshift, '<=': le, '<>': ne, '=': eq, '>': gt, '>=': ge, '>>': rshift, '_Tree_add_Ee': _Tree_add_Ee, '_Tree_delete_R0': _Tree_delete_R0, '_Tree_delete_clear_stuff': _Tree_delete_clear_stuff, '_Tree_get_E': _Tree_get_E, 'add': add, 'and': and, 'b': b, 'bool': bool, 'branch': branch, 'ccons': ccons, 'clear': clear, 'concat_': concat_, 'cons': cons, 'dip': dip, 'dipd': dipd, 'dipdd': dipdd, 'div': div, 'divmod': divmod, 'dup': dup, 'dupd': dupd, 'dupdd': dupdd, 'dupdip': dupdip, 'eq': eq, 'first': first, 'first_two': first_two, 'floordiv': floordiv, 'fourth': fourth, 'ge': ge, 'gt': gt, 'i': i, 'infra': infra, 'le': le, 'loop': loop, 'lshift': lshift, 'lt': lt, 'modulus': modulus, 'mul': mul, 'ne': ne, 'neg': neg, 'not': not, 'nullary': nullary, 'over': over, 'pop': pop, 'popd': popd, 'popdd': popdd, 'popop': popop, 'popopd': popopd, 'popopdd': popopdd, 'pow': pow, 'pred': pred, 'product': product, 'rest': rest, 'roll<': rolldown, 'roll>': rollup, 'rolldown': rolldown, 'rollup': rollup, 'rrest': rrest, 'rshift': rshift, 'second': second, 'sqrt': sqrt, 'stack': stack, 'stuncons': stuncons, 'stununcons': stununcons, 'sub': sub, 'succ': succ, 'sum': sum, 'swaack': swaack, 'swap': swap, 'swons': swons, 'third': third, 'truediv': truediv, 'truthy': bool, 'tuck': tuck, 'uncons': uncons, 'unit': unit, 'unswons': unswons, 'x': x}

    Docstring for functions in Sphinx?

    @@ -372,6 +372,14 @@ expression is carried along and updated and yielded.

    effects. An expression is carried along and updated and yielded.

    +
    +
    +joy.utils.polytypes.type_check(name, stack)[source]
    +

    Trinary predicate. True if named function type-checks, False if it +fails, None if it’s indeterminate (because I haven’t entered it into +the FUNCTIONS dict yet.)

    +
    +
    diff --git a/docs/sphinx_docs/notebooks/Derivatives_of_Regular_Expressions.rst b/docs/sphinx_docs/notebooks/Derivatives_of_Regular_Expressions.rst index da66e21..a629254 100644 --- a/docs/sphinx_docs/notebooks/Derivatives_of_Regular_Expressions.rst +++ b/docs/sphinx_docs/notebooks/Derivatives_of_Regular_Expressions.rst @@ -266,10 +266,6 @@ derivation. .. code:: ipython2 - # This is the straightforward version with no "compaction". - # It works fine, but does waaaay too much work because the - # expressions grow each derivation. - def D(symbol): def derv(R): @@ -539,9 +535,8 @@ machine transition table. Says, "Three or more 1's and not ending in 01 nor composed of all 1's." .. figure:: omg.svg - :alt: omg.svg + :alt: State Machine Diagram - omg.svg Start at ``a`` and follow the transition arrows according to their labels. Accepting states have a double outline. (Graphic generated with diff --git a/docs/sphinx_docs/notebooks/index.rst b/docs/sphinx_docs/notebooks/index.rst index dc390b9..c14aafa 100644 --- a/docs/sphinx_docs/notebooks/index.rst +++ b/docs/sphinx_docs/notebooks/index.rst @@ -18,6 +18,9 @@ These essays are adapted from Jupyter notebooks. I hope to have those hosted so Newton-Raphson Zipper Types + TypeChecking NoUpdates Categorical + The_Four_Operations + Derivatives_of_Regular_Expressions