partial recursive function
Quick Summary
A partial recursive function is a function that can be computed by a procedure that repeatedly applies a given relation or routine operation to known values. This concept originates in logic and mathematics, developed to avoid paradoxes associated with infinite classes by specifying function ranges without reference to infinite sets.1
The Turing machine is a theoretical model that defines computability. It consists of an infinite tape, a read/write head, and a finite set of internal states. The machine performs basic operations like moving the tape, changing cell states, and altering its internal state based on its current state and the cell it's reading. A Turing machine defines a partial function; if it halts after a finite number of steps for a given input, it computes the function's value. If it doesn't halt, the function is undefined for that input.2
The class of functions computable by Turing machines is equivalent to the class of partial recursive functions. This equivalence is fundamental to the theory of computability and has played a significant role in the development of computer science.32
Ask Britannica AI about this topic
Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.
automata theoryThe generalized automaton and Turing’s machine…to machines, namely, the so-called partial recursive functions, has the same membership as the class of computable functions. For the present purposes, then, no effort need be made to define the partial recursive functions.
Read More…to machines, namely, the so-called partial recursive functions, has the same membership as the class of computable functions. For the present purposes, then, no effort need be made to define the partial recursive functions.
Read More- automata theoryRecursively enumerable grammars and Turing acceptors
As noted above, an elementary result of automata theory is that every recursively enumerable set constitutes an accepted set. Generally speaking, acceptors are two-way unbounded tape automata. On the other hand, a grammar consisting of rules g → g′,…
Read MoreAs noted above, an elementary result of automata theory is that every recursively enumerable set constitutes an accepted set. Generally speaking, acceptors are two-way unbounded tape automata. On the other hand, a grammar consisting of rules g → g′,…
Read More