# Uninterpreted function

In mathematical logic, an uninterpreted function[1] or function symbol[2] is one that has no other property than its name and arity. Function symbols are used, together with constants and variables, to form terms.

The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory having an empty set of sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theories. The decision problem for free theories is a satisfiability problem, and is solved by syntactic unification. It is particularly important, as many other theories can be reduced to it. Interpreters for various computer languages, such as Prolog, require algorithms for solving the free theory.

## Example

An array can be specified by the following equational axiom:

select(store(a,i,v),j) = (if i = j then v else select(a,j))

This axiom can be used to deduce

select(store(store(a,1,−1),2,−2),1)
= select(store(a,1,−1),1)
= −1

Note that this reasoning did not use any 'definition' or interpretation for the functions select and store. All that is known is the axiom.

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## Discussion

The decision problem for free theories is particularly important, as many theories can be reduced to it; the above example is the prototypical example of the theory of arrays, where 'select' and 'store' are the canonical array access functions.[3]

Free theories can be solved by searching for common subexpressions to form the congruence closure. Solvers include satisfiability modulo theories solvers.

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