72 lines
2.0 KiB
Markdown
72 lines
2.0 KiB
Markdown
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## genrec
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Basis Function Combinator
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General Recursion Combinator. :
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[if] [then] [rec1] [rec2] genrec
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---------------------------------------------------------------------
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[if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte
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From \"Recursion Theory and Joy\" (j05cmp.html) by Manfred von Thun:
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\"The genrec combinator takes four program parameters in addition to
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whatever data parameters it needs. Fourth from the top is an if-part,
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followed by a then-part. If the if-part yields true, then the then-part
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is executed and the combinator terminates. The other two parameters are
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the rec1-part and the rec2-part. If the if-part yields false, the
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rec1-part is executed. Following that the four program parameters and
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the combinator are again pushed onto the stack bundled up in a quoted
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form. Then the rec2-part is executed, where it will find the bundled
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form. Typically it will then execute the bundled form, either with i or
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with app2, or some other combinator.\"
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The way to design one of these is to fix your base case \[then\] and the
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test \[if\], and then treat rec1 and rec2 as an else-part
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\"sandwiching\" a quotation of the whole function.
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For example, given a (general recursive) function \'F\': :
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F == [I] [T] [R1] [R2] genrec
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If the \[I\] if-part fails you must derive R1 and R2 from: :
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... R1 [F] R2
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Just set the stack arguments in front, and figure out what R1 and R2
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have to do to apply the quoted \[F\] in the proper way. In effect, the
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genrec combinator turns into an ifte combinator with a quoted copy of
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the original definition in the else-part: :
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F == [I] [T] [R1] [R2] genrec
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== [I] [T] [R1 [F] R2] ifte
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Primitive recursive functions are those where R2 == i. :
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P == [I] [T] [R] tailrec
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== [I] [T] [R [P] i] ifte
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== [I] [T] [R P] ifte
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Gentzen diagram.
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### Definition
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if not basis.
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### Derivation
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if not basis.
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### Source
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if basis
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### Discussion
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Lorem ipsum.
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### Crosslinks
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Lorem ipsum.
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