Kleene's T predicate

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.

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 ${\displaystyle f}$.

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 T_n 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.

• 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