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Unformatted text preview: RecursiveFunctionsHere 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 Gdel used this fact in his famousincompleteness proof.We will show that,The functions computable by a Turing machine are exactly therecursive functions.recursive functions were developed by Gdel and Stephen Kleene.So, between Turing, Church, Gdel, and Kleene we obtain the following equivalence relation:AlgorithmsTuring MachinesRecursive FunctionsCalculusIn 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 some1in, or2.gi(x1, . . . , xk) =yifor1inandh(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, theindicates 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,1inis 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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