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We can describe this argument in terms of Turing machines. Suppose there were a Turing machine F that took as input a description of a Turing machine K and an input x, and decided whether or not K halts on input x. Then we could build another Turing machine G which takes a single input x, calls F to decide if machine x halts on input x, and does the opposite. In other words, if F reports that x halts on input x, G goes into an infinite loop, and if F reports that x doesn’t halt on input x, then F just halts.

26 CHAPTER 2. MODELS OF COMPUTATION One can also define relations using bounded quantification: • Bounded universal quantification: if R(x, z) is a primitive recursive relation, then so is the relation ∀x < y R(x, z) which holds if and only if R(x, z) holds for every x less than y. • Bounded existential quantification: if R(x, z) is a primitive recursive relation, then so is ∃x < y R(x, z). By convention, we take expressions of the form ∀x < 0 R(x, z) to be true (for the trivial reason that there are no x less than 0) and ∃x < 0 R(x, z) to be false.

Let us consider some examples. 1. We have (λx. xxy)λz z 1 (λz z)(λz z)y 1 (λz z)y 1 y 2. “Simplifying” a term can make it more complex: (λx. xxy)(λx. xxy) 1 (λx. xxy)(λx. xxy)y 1 (λx. xxy)(λx. xxy)yy 1 ... 3. It can also leave a term unchanged: (λx. xx)(λx. xx) 1 (λx. xx)(λx. xx) 4. Also, some terms can be reduced in more than one way; for example, (λx (λy yx)z)v 1 (λy yv)z by contracting the outermost application; and (λx (λy yx)z)v 1 (λx zx)v by contracting the innermost one. Note, in this case, however, that both terms further reduce to the same term, zv.

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