Abstract
In general, first-order predicate logic extended with linear integer arithmetic is undecidable. We show that the Bernays-Schönfinkel-Ramsey fragment (\(\exists ^* \forall ^*\)-sentences) extended with a restricted form of linear integer arithmetic is decidable via finite ground instantiation. The identified ground instances can be employed to restrict the search space of existing automated reasoning procedures considerably, e.g., when reasoning about quantified properties of array data structures formalized in Bradley, Manna, and Sipma’s array property fragment. Typically, decision procedures for the array property fragment are based on an exhaustive instantiation of universally quantified array indices with all the ground index terms that occur in the formula at hand. Our results reveal that one can get along with significantly fewer instances.
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Notes
- 1.
For any free-sort variable v that occurs in a clause \((\varLambda \,\Vert \, \varGamma \rightarrow \varDelta ) \in N\) exclusively in equations, we pretend that \(\varDelta \) contains an atom \(\text {False}_v(v)\), for a fresh predicate symbol \(\text {False}_v : \mathcal {S}\). This is merely a technical assumption. Without it, we would have to treat such variables v as a separate case in all definitions. The atom \(\text {False}_v(v)\) is not added “physically” to any clause.
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Horbach, M., Voigt, M., Weidenbach, C. (2017). On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic. In: de Moura, L. (eds) Automated Deduction – CADE 26. CADE 2017. Lecture Notes in Computer Science(), vol 10395. Springer, Cham. https://doi.org/10.1007/978-3-319-63046-5_6
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