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E. languages L such that, for all T , L(T ) = L? Since the class of all ∗ languages over Σ, that is, 2Σ , is not countable while the set of Turing machines is countable (remember how we can encode a Turing machine as a word in a given alphabet), the answer is that non-acceptable languages must exist. Indeed, the diagonalization technique allows us to define one of them. For any i, let Ti be the ith Turing machine and xi be the ith word over Σ. The diagonal language thus contains all words xi such that xi ∈ L(Ti ) and it is obviously not acceptable.

Notes Turing machines were introduced in Turing (1936) in order to formally define the concept of computable numbers (equivalently, that of computable functions or predicates): a number is computable if its digits can be written down by such a machine. In particular, it is shown that certain large classes of numbers are computable although definable numbers exist which are not computable. Furthermore, 32 Elements of computability theory it is shown that there is no general method for ascertaining whether a given formula is provable in Hilbert’s functional calculus.

3 Oracle Turing machines Let us now consider a further variation of the Turing machine model which plays an important part in complexity theory. An oracle deterministic (nondeterministic) Turing machine is a k-tape machine with an additional special-purpose tape called an oracle tape and three additional special-purpose states called query state, yes state, and no state. Whenever an oracle Turing machine enters the query state after having written a given word 22 Elements of computability theory on the oracle tape, the next state will be the yes or the no state, depending on whether that word belongs to the oracle language or not.

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