In
computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...
, the mortality problem is a
decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...
which can be stated as follows:
:''Given a
Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
, decide whether it halts when run on any configuration (not necessarily a starting one)''
In the statement above, the configuration is a pair , where q is one of the machine's states (not necessarily its initial state) and w is an infinite sequence of symbols representing the initial content of the tape. Note that while we usually assume that in the starting configuration all but finitely many cells on the tape are blanks, in the mortality problem the tape can have arbitrary content, including infinitely many non-blank symbols written on it.
Philip K. Hooper proved in 1966 that the mortality problem is
undecidable. However, it can be shown that the set of Turing machines which are mortal (i.e. halt on every starting configuration) is
recursively enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
.
Theory of computation
Undecidable problems
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