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Partial Recursive Functions Computation Theory , L 8 101/171

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Aim Amoreabstract,machine-independentdescriptionofthe collection of computable partial functions than provided by register/Turing machines: they form the smallest collection of partial functions containing some basic functions and closed under some fundamental operations for forming new functions from old—composition, primitive recursion and minimization The characterization is due to Kleene (1936), building on work of Gdel and Herbrand. Computation Theory , L 8 102/171

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Examples of recursive deﬁnitions )+( )= sum of 0,1,2,..., Computation Theory , L 8 103/169

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Examples of recursive deﬁnitions )+( )= sum of 0,1,2,..., )+ )= th Fibonacci number Computation Theory , L 8 103/169

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Examples of recursive deﬁnitions )+( )= sum of 0,1,2,..., )+ )= th Fibonacci number )+ undeﬁned except when Computation Theory , L 8 103/169

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Examples of recursive deﬁnitions )+( )= sum of 0,1,2,..., )+ )= th Fibonacci number )+ undeﬁned except when if x 100 then x 10 else f 11 )) is McCarthy’s "91 function", which maps to 91 if 100 and to 10 otherwise Computation Theory , L 8 103/169

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Primitive recursion Theorem. Given and there is a unique satisfying ,0 )) for all and We write for and call it the partial function deﬁned by primitive recursion from and Computation Theory , L 8 104/171

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Primitive recursion Theorem. Given and there is a unique satisfying ,0 )) for all and Proof (sketch). Existence :theset ,..., )= )= deﬁnes a partial function satisfying Uniqueness : if and both satisfy , then one can prove by induction on that )= )) Computation Theory , L 8 105/171

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Example: addition Addition add satisﬁes: add ,0 add add )+ So add where Note that proj and succ proj ;so add can be built up from basic functions using composition and primitive recursion: add proj succ proj Computation Theory , L 8 106/171

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Example: predecessor Predecessor pred satisﬁes: pred pred So pred where () Thus pred can be built up from basic functions using primitive recursion: pred zero proj Computation Theory , L 8 107/171

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Example: multiplication Multiplication mult satisﬁes: mult ,0 mult mult )+ and thus mult zero add proj proj )) So mult can be built up from basic functions using composition and primitive recursion (since add can be). Computation Theory , L 8 108/170

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Deﬁnition. A[partial]function is primitive recursive PRIM )ifitcanbebuiltupinﬁnitelymanysteps from the basic functions by use of the operations of composition and primitive recursion. In other words, the set PRIM of primitive recursive functions is the smallest set (with respect to subset inclusion) of partial functions containing the basic functions and closed under the operations of composition and primitive recursion. Computation Theory , L 8 109/171

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Deﬁnition. [partial] function is primitive recursive PRIM )ifitcanbebuiltupinﬁnitelymanysteps from the basic functions by use of the operations of composition and primitive recursion. Every PRIM is a total function, because: all the basic functions are total if ,..., are total, then so is ,..., [why?] if and are total, then so is [why?] Computation Theory , L 8 110/171

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Deﬁnition. A[partial]function is primitive recursive PRIM )ifitcanbebuiltupinﬁnitelymanysteps from the basic functions by use of the operations of composition and primitive recursion. Theorem. Every PRIM is computable. Proof. Already proved: basic functions are computable; composition preserves computability. So just have to show: computable if and are. Suppose and are computed by RM programs and (with our usual I/O conventions). Then the RM speciﬁed on the next slide computes .(Weassume ,..., are some registers not mentioned in and ; and that the latter only use registers ,..., , where .) Computation Theory , L 8 111/171

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START ,..., :: =( ,..., ,0 yes no HALT ,..., :: =( ,..., ,..., :: =( 0,0,...,0 Computation Theory , L 8 112/170

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START ,..., :: =( ,..., ,0 yes no HALT ,..., :: =( ,..., ,..., :: =( 0,0,...,0 Computation Theory , L 8 112/170

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Aim Amoreabstract,machine-independentdescriptionofthe collection of computable partial functions than provided by register/Turing machines: they form the smallest collection of partial functions containing some basic functions and closed under some fundamental operations for forming new functions from old—composition, primitive recursion and minimization The characterization is due to Kleene (1936), building on work of Gdel and Herbrand. Computation Theory , L 9 113/171

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Minimization Given a partial function ,deﬁne by least such that )= and for each 0,..., is deﬁned and (undeﬁned if there is no such In other words ,..., )= Computation Theory , L 9 114/171

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Example of minimization integer part of least such that (undeﬁned if Computation Theory , L 9 115/171

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Example of minimization integer part of least such that (undeﬁned if where is if if Computation Theory , L 9 115/170

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Deﬁnition. Apartialfunction is partial recursive PR )ifitcanbebuiltupinﬁnitelymanysteps from the basic functions by use of the operations of composition, primitive recursion and minimization. In other words, the set PR of partial recursive functions is the smallest set (with respect to subset inclusion) of partial functions containing the basic functions and closed under the operations of composition, primitive recursion and minimization. Computation Theory , L 9 116/171

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Deﬁnition. Apartialfunction is partial recursive PR )ifitcanbebuiltupinﬁnitelymanysteps from the basic functions by use of the operations of composition, primitive recursion and minimization. Theorem. Every PR is computable. Proof. Just have to show: is computable if is. Suppose is computed by RM program (with our usual I/O conventions). Then the RM speciﬁed on the next slide computes .(Weassume ,..., are some registers not mentioned in ; and that the latter only uses registers ,..., , where .) Computation Theory , L 9 117/171

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START ,..., :: =( ,..., ,..., :: =( ,..., ,..., :: =( 0,0,...,0 :: HALT Computation Theory , L 9 118/171

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START ,..., :: =( ,..., ,..., :: =( ,..., ,..., :: =( 0,0,...,0 :: HALT Computation Theory , L 9 118/171

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