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Unformatted text preview: µRecursiveFunctionsHere we investigate the relationship between Turing machines and computable functions.For convenience we will restrict ourselves to only look at numeric computations, this does not reflectany loss of generality since all computational problems can be encoded as numbers (think ASCIIcode). Kurt Gödel used this fact in his famousincompleteness proof.We will show that,The functions computable by a Turing machine are exactly theµrecursive functions.µrecursive functions were developed by Gödel and Stephen Kleene.So, between Turing, Church, Gödel, and Kleene we obtain the following equivalence relation:Algorithms⇔Turing Machines⇔µRecursive Functions⇔λCalculusIn order to work towards a proof of this equivalence we start withprimitive recursive functions.– p. 1/2FunctionCompositionA more general view of function composition in order to define primitive recursive functions,Letg1, g2, . . . , gnbekvariable functions and lethbe annvariable function,then thekvariable functionfdefined byf(x1, . . . , xk) =h(g1(x1, . . . , xk), . . . , gn(x1, . . . , xk))is called thecompositionofhwithg1, g2, . . . , gnand is written asf=h◦(g1, . . . , gk).NOTE:The functionf(x1, . . . , xk)is undefined orf(x1, . . . , xk)↑if either1.gi(x1, . . . , xk)↑for some1≤i≤n, or2.gi(x1, . . . , xk) =yifor1≤i≤nandh(y1, . . . , yn)↑.NOTE:Hereg(·)↑means thatgis undefined.NOTE:Composition is strict in the sense that if any of the arguments of a function are undefined thenso is the whole function.– p. 2/2FunctionCompositionA functionfis called atotal functionif it is completely defined over its domain, that is,∀x, f(x)↓.aA functionfis called apartial functionif it is undefined for at least one element in itsdomain, that is,∃x, f(x)↑.aYou guessed it, the↓indicates that the function is defined.– p. 3/2Primitive RecursiveFunctionsDefinition:The basicprimitive recursive functionsare defined as follows:zero function:z(x) = 0is primitive recursivesuccessor function:s(x) =x+ 1is primitive recursiveprojection function:p(n)i(x1, . . . , xn) =xi,1≤i≤nis primitive recursiveMore complex primitive recursive function can be constructed by a finite number ofapplications of,composition:letg1, g2, . . . , gnbekvariable primitive recursive functions and lethbe annvariable primitive recursive function, then thekvariable functionfdefined byf(x1, . . . , xk) =h(g1(x1, . . . , xk), . . . , gn(x1, . . . , xk))is also primitive recursive.primitive recursion:letgandhbe primitive recursive functions withnandn+ 2variables, respectively, then then+ 1variable functionfdefined by1.f(x1, . . . , xn,) =g(x1, . . . , xn)2.f(x1, . . . , xn,s(y)) =h(x1, . . . , xn,y, f(x1, . . . , xn,y)is also primitive recursive. Here, the variableyis called therecursive variable....
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 Spring '11
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 Turing Machines, Kurt Gödel, Functions and mappings, primitive recursive functions

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