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 predicate is obtained by formalizing this simulation. The ternary relation takes three natural numbers as arguments. is true if encodes a computation history of the computable function with index when run with input , and the program halts as the last step of this computation history. That is, first asks whether is the Gödel number of a finite sequence of complete configurations of the Turing machine with index , running a computation on input . If so, 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, finally asks whether the sequence ends with the machine in a halting state. If all three of these questions have a positive answer, then is true, otherwise, it is false. The 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 such that if is true then returns the output of the function with index on input . Because Kleene's formalism attaches a number of inputs to each function, the predicate can only be used for functions that take one input. There are additional predicates for functions with multiple inputs; the relation is true if encodes a halting computation of the function with index on the inputs . Like , all functions are primitive recursive. Because of this, any