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Additional resources for Computability and incompleteness. Lecture notes

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As a model of computation, the lambda calculus is a rather simple calculus; the only operations are lambda abstraction and application! From these meager resources, however, it is possible to implement any computational procedure. The “if” part of the theorem is easier to see: suppose a function, f , is represented by a lambda term X. Let us describe an informal procedure to compute f . On input m0 , . . , mn−1 , write down the term Xm0 . . mn−1 . e. the twostep reductions of the original term); and keep going.

G k−1 , respectively, we need to find a term f representing f . But we can simply define f by f (x0 , . . , xl−1 ) = h(g 0 (x0 , . . , xl−1 ), . . , g k−1 (x0 , . . , xl−1 )). In other words, the language of the lambda calculus is well suited to represent composition as well. When it comes to primitive recursion, we finally need to do some work. We will have to proceed in stages. As before, on the assumption that we already have terms g and h representing functions g and h, respectively, we want a term f representing the function f defined by f (0, z) = g(z) f (x + 1, z) = h(z, f (x, z), z).

Each sm n is then just a primitive recursive function that finds a code for the appropriate Turing machine. 4 Every partial computable function has infinitely many indices. Again, this is intuitively clear. Given any Turing machine, M , one can design another Turing machine M that twiddles its thumbs for a while, and then acts like M . Throughout this chapter, we will reason about what types of things are computable. 2. COMPUTABLY ENUMERABLE SETS 59 1. Rigorously: describe a Turing machine or partial recursive function explicitly, and show that it computes the function you have in mind; 2.