Kleene's T predicate

In computability theory, the T predicate, first studied by mathematician Stephen Cole Kleene, is a particular set of triples of natural numbers that is used to represent computable functions within formal theories of arithmetic. Informally, the T predicate tells whether a particular computer program will halt when run with a particular input, and the corresponding U function is used to obtain the results of the computation if the program does halt. As with the smn theorem, the original notation used by Kleene has become standard terminology for the concept.

Definition


The definition depends on a suitable Gödel numbering that assigns natural numbers to computable functions (given as Turing machines). This numbering must be sufficiently effective that, given an index of a computable function and an input to the function, it is possible to effectively simulate the computation of the function on that input. The $$T$$ predicate is obtained by formalizing this simulation.

The ternary relation $$T_1(e,i,x)$$ takes three natural numbers as arguments. $$T_1(e,i,x)$$ is true if $$x$$ encodes a computation history of the computable function with index $$e$$ when run with input $$i$$, and the program halts as the last step of this computation history. That is, If all three of these questions have a positive answer, then $$T_1(e,i,x)$$ is true, otherwise, it is false.
 * $$T_1$$ first asks whether $$x$$ is the Gödel number of a finite sequence $$\langle x_{j} \rangle$$ of complete configurations of the Turing machine with index $$e$$, running a computation on input $$i$$.
 * If so, $$T_1$$ then asks if this sequence begins with the starting state of the computation and each successive element of the sequence corresponds to a single step of the Turing machine.
 * If it does, $$T_1$$ finally asks whether the sequence $$\langle x_{j} \rangle$$ ends with the machine in a halting state.

The $$T_1$$ predicate is primitive recursive in the sense that there is a primitive recursive function that, given inputs for the predicate, correctly determines the truth value of the predicate on those inputs.

There is a corresponding primitive recursive function $$U$$ such that if $$T_1(e,i,x)$$ is true then $$U(x)$$ returns the output of the function with index $$e$$ on input $$i$$.

Because Kleene's formalism attaches a number of inputs to each function, the predicate $$T_1$$ can only be used for functions that take one input. There are additional predicates for functions with multiple inputs; the relation


 * $$T_k(e, i_1, \ldots, i_k, x)$$

is true if $$x$$ encodes a halting computation of the function with index $$e$$ on the inputs $$i_1,\ldots,i_k$$.

Like $$T_1$$, all functions $$T_k$$ are primitive recursive. Because of this, any theory of arithmetic that is able to represent every primitive recursive function is able to represent $$T$$ and $$U$$. Examples of such arithmetical theories include Robinson arithmetic and stronger theories such as Peano arithmetic.

Normal form theorem
The $$T_k$$ predicates can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15; Kleene 1943, p. 52&mdash;53). This states there exists a fixed primitive recursive function $$U$$ such that a function $$f:\mathbb{N}^{k}\rightarrow\mathbb{N}$$ is computable if and only if there is a number $$e$$ such that for all $$n_1,\ldots,n_k$$ one has
 * $$f(n_1,\ldots,n_k) \simeq U( \mu x\, T_k(e,n_1,\ldots,n_k,x))$$,

where &mu; is the &mu; operator ($$\mu x\, \phi(x)$$ is the smallest natural number for which $$\phi(x)$$ is true) and $$\simeq$$ is true if both sides are undefined or if both are defined and they are equal. By the theorem, the definition of every general recursive function f can be rewritten into a normal form such that the &mu; operator is used only once, viz. immediately below the topmost $$U$$, which is independent of the computable function $$f$$.

Arithmetical hierarchy
In addition to encoding computability, the T predicate can be used to generate complete sets in the arithmetical hierarchy. In particular, the set
 * $$ K = \{ e \mbox{ } : \mbox{ } \exists x T_1(e,0,x) \}$$

which is of the same Turing degree as the halting problem, is a $$\Sigma^0_1$$ complete unary relation (Soare 1987, pp. 28, 41). More generally, the set
 * $$K_{n+1} = \{ \langle e, a_1, \ldots, a_n\rangle : \exists x T_n(e, a_1, \ldots, a_n, x)\}$$

is a $$\Sigma^0_1$$-complete (n+1)-ary predicate. Thus, once a representation of the Tn predicate is obtained in a theory of arithmetic, a representation of a $$\Sigma^0_1$$-complete predicate can be obtained from it.

This construction can be extended higher in the arithmetical hierarchy, as in Post's theorem (compare Hinman 2005, p. 397). For example, if a set $$A \subseteq \mathbb{N}^{k+1}$$ is $$\Sigma^0_{n}$$ complete then the set
 * $$\{ \langle a_1, \ldots, a_k\rangle : \forall x ( \langle a_1, \ldots, a_k, x) \in A)\}$$

is $$\Pi^0_{n+1}$$ complete.