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.[1]

Definition

Example call of T1. The 1st argument gives the source code (in C rather than as a Gödel number e) of a computable function, viz. the Collatz function f. The 2nd argument gives the natural number i to which f is to be applied. The 3rd argument gives a sequence x of computation steps simulating the evaluation of f on i (as an equation chain rather than a Gödel sequence number). The predicate call evaluates to true since x is actually the correct computation sequence for the call f(5), and ends with an expression not involving f anymore. Function U, applied to the sequence x, will return its final expression, viz. 1.

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 ${\displaystyle T}$ predicate is obtained by formalizing this simulation.

The ternary relation ${\displaystyle T_{1}(e,i,x)}$ takes three natural numbers as arguments. ${\displaystyle T_{1}(e,i,x)}$ is true if ${\displaystyle x}$ encodes a computation history of the computable function with index ${\displaystyle e}$ when run with input ${\displaystyle i}$, and the program halts as the last step of this computation history. That is,

• ${\displaystyle T_{1}}$ first asks whether ${\displaystyle x}$ is the Gödel number of a finite sequence ${\displaystyle \langle x_{j}\rangle }$ of complete configurations of the Turing machine with index ${\displaystyle e}$, running a computation on input ${\displaystyle i}$.
• If so, ${\displaystyle T_{1}}$ 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, ${\displaystyle T_{1}}$ finally asks whether the sequence ${\displaystyle \langle x_{j}\rangle }$ ends with the machine in a halting state.

If all three of these questions have a positive answer, then ${\displaystyle T_{1}(e,i,x)}$ is true, otherwise, it is false.

The ${\displaystyle T_{1}}$ 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 ${\displaystyle U}$ such that if ${\displaystyle T_{1}(e,i,x)}$ is true then ${\displaystyle U(x)}$ returns the output of the function with index ${\displaystyle e}$ on input ${\displaystyle i}$.

Because Kleene's formalism attaches a number of inputs to each function, the predicate ${\displaystyle T_{1}}$ can only be used for functions that take one input. There are additional predicates for functions with multiple inputs; the relation

${\displaystyle T_{k}(e,i_{1},\ldots ,i_{k},x)}$

is true if ${\displaystyle x}$ encodes a halting computation of the function with index ${\displaystyle e}$ on the inputs ${\displaystyle i_{1},\ldots ,i_{k}}$.

Like ${\displaystyle T_{1}}$, all functions ${\displaystyle T_{k}}$ are primitive recursive. Because of this, any theory of arithmetic that is able to represent every primitive recursive function is able to represent ${\displaystyle T}$ and ${\displaystyle U}$. Examples of such arithmetical theories include Robinson arithmetic and stronger theories such as Peano arithmetic.

Normal form theorem

The ${\displaystyle T_{k}}$ predicates can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15; Kleene 1943, p. 52—53). This states there exists a fixed primitive recursive function ${\displaystyle U}$ such that a function ${\displaystyle f:\mathbb {N} ^{k}\rightarrow \mathbb {N} }$ is computable if and only if there is a number ${\displaystyle e}$ such that for all ${\displaystyle n_{1},\ldots ,n_{k}}$ one has

${\displaystyle f(n_{1},\ldots ,n_{k})\simeq U(\mu x\,T_{k}(e,n_{1},\ldots ,n_{k},x))}$,

where μ is the μ operator (${\displaystyle \mu x\,\phi (x)}$ is the smallest natural number for which ${\displaystyle \phi (x)}$ is true) and ${\displaystyle \simeq }$ 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 μ operator is used only once, viz. immediately below the topmost ${\displaystyle U}$, which is independent of the computable function ${\displaystyle f}$.

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

${\displaystyle K=\{e{\mbox{ }}:{\mbox{ }}\exists xT_{1}(e,0,x)\}}$

which is of the same Turing degree as the halting problem, is a ${\displaystyle \Sigma _{1}^{0}}$ complete unary relation (Soare 1987, pp. 28, 41). More generally, the set

${\displaystyle K_{n+1}=\{\langle e,a_{1},\ldots ,a_{n}\rangle :\exists xT_{n}(e,a_{1},\ldots ,a_{n},x)\}}$

is a ${\displaystyle \Sigma _{1}^{0}}$ complete (n+1)-ary predicate. Thus, once a representation of the Tn predicate is obtained in a theory of arithmetic, a representation of a ${\displaystyle \Sigma _{1}^{0}}$-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 ${\displaystyle A\subseteq \mathbb {N} ^{k+1}}$ is ${\displaystyle \Sigma _{n}^{0}}$ complete then the set

${\displaystyle \{\langle a_{1},\ldots ,a_{k}\rangle :\forall x(\langle a_{1},\ldots ,a_{k},x)\in A)\}}$

is ${\displaystyle \Pi _{n+1}^{0}}$ 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 (Jan 1943). "Recursive predicates and quantifiers" (PDF). Transactions of the American Mathematical Society. 53 (1): 41–73. doi:10.1090/S0002-9947-1943-0007371-8. 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