Uninterpreted function: Difference between revisions

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{{Short description|Type of function in mathematical logic}}

{{Context|date=October 2009}}

In [[mathematical logic]], an '''uninterpreted function'''<ref>Bryant,{{Cite Lahiri,book Seshia| (2002) "[chapter-url=https://link.springer.com/content/pdf/10.1007/3-540-45657-0_7.pdf |doi = 10.1007/3-540-45657-0_7|chapter = Modeling and verifyingVerifying systemsSystems usingUsing a logicLogic of counterCounter arithmeticArithmetic with lambdaLambda expressionsExpressions and uninterpretedUninterpreted functions]".Functions|title = ''Computer Aided Verification''|volume = '''2404/|pages = 78–92|series = Lecture Notes in Computer Science|year = 2002''',|last1 106&ndash;122= Bryant|first1 = Randal E.|last2 = Lahiri|first2 = Shuvendu K.|last3 = Seshia|first3 = Sanjit A.|isbn = 978-3-540-43997-4| s2cid=9471360 }}</ref> or '''function symbol'''<ref>{{cite book |last1=Baader |first1=Franz |authorlink1=Franz Baader |last2=Nipkow |first2=Tobias |authorlink2=Tobias Nipkow |year=1999 |title=Term Rewriting and All That |publisher=Cambridge University Press |isbn=978-0-521-77920-3 |page=34}}</ref> is one that has no other property than its name and ''[[Arity|n-ary]]'' form. Function symbols are used, together with constants and variables, to form [[term (logic)|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 (mathematical logic)|theory]] having an empty set of [[sentence (mathematical logic)|sentences]] (in analogy to an [[initial algebra]]). Theories with a non-empty set of equations are known as [[equational theory|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 [[E-Unification]] and [[Narrowing (computer science)|Narrowing]].

==Example==

AnAs an example of uninterpreted functions infor [[SMT-LIB]], anif this input standardis forgiven to an [[Satisfiability modulo theories|SMT Solverssolver]]:

<syntaxhighlight lang="text" line="1">

(declare-fun f (Int) Int)

(assertdeclare-fun (=f (f 10Int) 1)Int)

(assert (= (f 10) 1))

This is satisfiable: <code>f</code> is an uninterpreted function. All that is known about <code>f</code> is its signature, so it is possible that <code>f(10) = 1</code>.

</syntaxhighlight>

the SMT solver would return "This input is satisfiable:". That happens because <code>f</code> is an uninterpreted function (i.e., Allall that is known about <code>f</code> is its [[Signature (logic)|signature]]), so it is possible that <code>f(10) = 1</code>. But by applying the input below:

(declare-fun f (Int) Int)

<syntaxhighlight lang="text" line="1">

(assert (= (f 10) 1))

(assertdeclare-fun (=f (f 10Int) 42)Int)

(assert (= (f 10) 1))

(assert (= (f 10) 42))

This is unsatisfiable: although <code>f</code> has no interpretation, it is impossible that it returns different values for the same input.

</syntaxhighlight>

the SMT solver would return "This input is unsatisfiable:". althoughThat happens because <code>f</code> has no interpretation, itbeing isa impossiblefunction, thatcan itnever returnsreturn different values for the same input.

==Discussion==

The [[decision problem]] for free theories is particularly important, asbecause many theories can be reduced toby it.<ref>{{citationcite book |last1=de Moura |first1=Leonardo |last2=Bjørner |first2=Nikolaj |title=Formal methods : foundations and applications : 12th Brazilian Symposium on Formal Methods, SBMF 2009, Gramado, Brazil, August 19-21, 2009 : revised selected papers needed|date=December2009 2019|publisher=Springer |___location=Berlin |isbn=978-3-642-10452-7 |url=https://leodemoura.github.io/files/sbmf09.pdf}}</ref>

Free theories can be solved by searching for [[common subexpression]]s to form the [[congruence closure]].{{clarify|reason=Indicate about solving which problem in free theories the sentence is supposed to speak. E.g. to solve the satisfiability problem of conjunctions of equations, the Martelli-Montanari syntactic unification algorithm suffices, neither common subexpressions nor congruence closures are needed. Maybe, satisfiability of arbitrary booleanBoolean combinations of equations is meant?|date=May 2014}} Solvers include [[satisfiability modulo theories]] solvers.

== See also ==

Line 26 ⟶ 28:

* [[Initial algebra]]

* [[Term algebra]]

* [[Theory of pure equality]]

==Notes==

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==References==

{{Reflist}}

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[[Category:Specification languages]]

{{Mathematical logic}}

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[[Category:Specification languages]]

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