Uninterpreted function

In mathematical logic, an uninterpreted function^[1] or function symbol^[2] is one that has no other property than its name and n-ary form. 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 satisfiability problem for free theories is solved by syntactic unification; algorithms for the latter are used by interpreters for various computer languages, such as Prolog. Syntactic unification is also used in algorithms for the satisfiability problem for certain other equational theories, see Unification (computer science).

Example

As an example of uninterpreted functions for SMT-LIB, if this input is given to an SMT solver:

(declare-fun f (Int) Int)
(assert (= (f 10) 1))

the SMT solver would return "This input is satisfiable". That happens because f is an uninterpreted function (i.e., all that is known about f is its signature), so it is possible that f(10) = 1. But by applying the input below:

(declare-fun f (Int) Int)
(assert (= (f 10) 1))
(assert (= (f 10) 42))

the SMT solver would return "This input is unsatisfiable". That happens because f, being a function, can never return different values for the same input.

Discussion

The decision problem for free theories is particularly important, because many theories can be reduced by it.^[3]

Free theories can be solved by searching for common subexpressions to form the congruence closure.^{[clarification needed]} Solvers include satisfiability modulo theories solvers.

See also

• Algebraic data type
• Initial algebra
• Term algebra
• Theory of pure equality

Notes

References

1. Bryant, Randal E.; Lahiri, Shuvendu K.; Seshia, Sanjit A. (2002). "Modeling and Verifying Systems Using a Logic of Counter Arithmetic with Lambda Expressions and Uninterpreted Functions" (PDF). Computer Aided Verification. Lecture Notes in Computer Science. Vol. 2404. pp. 78–92. doi:10.1007/3-540-45657-0_7. ISBN 978-3-540-43997-4. S2CID 9471360.
2. Baader, Franz; Nipkow, Tobias (1999). Term Rewriting and All That. Cambridge University Press. p. 34. ISBN 978-0-521-77920-3.
3. de Moura, Leonardo; Bjørner, Nikolaj (2009). Formal methods : foundations and applications : 12th Brazilian Symposium on Formal Methods, SBMF 2009, Gramado, Brazil, August 19-21, 2009 : revised selected papers (PDF). Berlin: Springer. ISBN 978-3-642-10452-7.