3.2.8: Summary

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Turing machines are determined by their instruction sets, which are finite sets of quintuples (for every state and symbol read, specify new state, symbol written, and movement of the head). The finite sets of quintuples are enumerable, so there is a way of associating a number with each Turing machine instruction set. The index of a Turing machine is the number associated with its instruction set under a fixed such schema. In this way we can “talk about” Turing machines indirectly—by talking about their indices.

One important problem about the behavior of Turing machines is whether they eventually halt. Let \(h(e, n)\) be the function which \(= 1\) if the Turing machine with index \(e\) halts when started on input \(n\), and \(=0\) otherwise. It is called the halting function. The question of whether the halting function is itself Turing computable is called the halting problem. The answer is no: the halting problem is unsolvable. This is established using a diagonal argument.

The halting problem is only one example of a larger class of problems of the form “can \(X\) be accomplished using Turing machines.” Another central problem of logic is the decision problem for first-order logic: is there a Turing machine that can decide if a given sentence is valid or not. This famous problem was also solved negatively: the decision problem is unsolvable. This is established by a reduction argument: we can associate with each Turing machine \(M\) and input \(w\) a first-order sentence \(T(M, w) \lif E(M, w)\) which is valid iff \(M\) halts when started on input \(w\). If the decision problem were solvable, we could thus use it to solve the halting problem.