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 s_mn theorem, the original notation used by Kleene has become standard terminology for the concept.^[1]

Definition

The definition depends on a suitable Gödel numbering that assigns natural numbers to computable functions. 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 T₁(e,i,x) takes three natural numbers as arguments. The triples of numbers (e,i,x) that belong to the relation (the ones for which T₁(e,i,x) is true) are defined to be exactly the triples in which 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, T₁ first asks whether x is the Gödel number of a finite sequence ⟨x_j⟩ of complete configurations of the Turing machine with index e, running a computation on input i. If so, T₁ 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, T₁ finally asks whether the sequence ⟨x_j⟩ ends with the machine in a halting state. If all three of these questions have a positive answer, then T₁(e,i,x) holds (is true). Otherwise, T₁(e,i,x) does not hold (is false).

There is a corresponding function U such that if T(e,i,x) holds 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 T₁ can only be used for functions that take one input. There are additional predicates for functions with multiple inputs; the relation

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

holds if x encodes a halting computation of the function with index e on the inputs i₁,...,i_k.

Normal form theorem

The T predicate can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15). This states there exists a primitive recursive function U such that a function f of one integer argument is computable if and only if there is a number e such that for all n one has

$f(n) \simeq U( \mu x\, T(e,n,x))$,

where μ is the μ operator ($\mu x\, \phi(x)$ is the smallest natural number for which $\phi(x)$ holds) and $\simeq$ holds if both sides are undefined or if both are defined and they are equal. Here U is a universal operation (it is independent of the computable function f) whose purpose is to extract, from the number x (encoding a complete computation history) returned by the operator μ, just the value f(n) that was found at the end of the computation.

Formalization

The T predicate is primitive recursive in the sense that there is a primitive recursive function that, given inputs for the predicate, correctly determine the truth value of the predicate on those inputs. Similarly, the U function is 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.

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

$K = \{ e \mbox{ } : \mbox{ } \exists x T(e,0,x) \}$

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

$K_{n+1} = \{ \langle e, a_1, \ldots, a_n\rangle : \exists x T(e, a_1, \ldots, a_n, x)\}$

is a $\Sigma^0_1$ complete (n+1)-ary predicate. Thus, once a representation of the T 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.

Notes

1. The predicate described here was presented in (Kleene 1943) and (Kleene 1952), and this is what is usually called "Kleene's T predicate". (Kleene 1967) uses the letter T to describe a different predicate related to computable functions, but which cannot be used to obtain Kleene's normal form theorem.

References

• Peter Hinman, 2005, Fundamentals of Mathematical Logic, A K Peters. ISBN 978-1-56881-262-5
• Stephen Cole Kleene, 1943, "Recursive predicates and quantifiers", Transactions of the AMS v. 53 n. 1, pp. 41–73. Reprinted in The Undecidable, Martin Davis, ed., 1965, pp. 255–287.
• —, 1952, Introduction to Metamathematics, North-Holland. Reprinted by Ishi press, 2009, ISBN 0-923891-57-9.
• —, 1967. Mathematical Logic, John Wiley. Reprinted by Dover, 2001, ISBN 0-486-42533-9.
• Robert I. Soare, 1987, Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer. ISBN 0-387-15299-7