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On Ordering Constraints for Deduction with BuiltIn Abelian Semigroups, Monoids and Groups
 In: LICS, 16th IEEE Symposium on Logic in Computer Science, IEEE Computer
, 2001
"... . It is crucial for the performance of ordered resolution or paramodulationbased deduction systems that they incorporate specialized techniques to work efficiently with standard algebraic theories E. Essential ingredients for this purpose are term orderings that are Ecompatible, for the given E ..."
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. It is crucial for the performance of ordered resolution or paramodulationbased deduction systems that they incorporate specialized techniques to work efficiently with standard algebraic theories E. Essential ingredients for this purpose are term orderings that are Ecompatible, for the given E, and algorithms deciding constraint satisfiability for such orderings. Here we introduce a uniform technique providing the first such algorithms for some orderings for abelian semigroups, abelian monoids and abelian groups, which we believe will lead to reasonably efficient techniques for practice. The algorithms are optimal since we show that, for any wellfounded Ecompatible ordering for these E, the constraint satisfiability problem is NPhard even for conjunctions of inequations, and our algorithms are in NP. Keywords: symbolic constraints, term orderings, automated deduction. ? Both authors are partially supported by the ESPRIT Basic Research Action CCLII, ref. WG # 22457...
Superposition modulo a Shostak theory
 AUTOMATED DEDUCTION (CADE19), VOLUME 2741 OF LNAI
, 2003
"... We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatic ..."
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We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatically, much more of the program semantics can be captured. Combining the Shostakstyle components for deciding the clausal validity problem with the ordering and saturation techniques developed for equational reasoning, we derive a refutationally complete calculus on mixed ground clauses which result for example from CNF transforming arbitrary universally quantied formulae. The calculus works modulo a Shostak theory in the sense that it operates on canonizer normalforms. For the Shostak solvers that we study, coherence comes for free: no coherence pairs need to be considered.
Deriving Theory Superposition Calculi from Convergent Term Rewriting Systems
, 1999
"... We show how to derive refutationally complete ground superposition calculi systematically from convergent term rewriting systems for equational theories, in order to make automated theorem proving in these theories more eective. In particular we consider abelian groups and commutative rings. Thes ..."
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We show how to derive refutationally complete ground superposition calculi systematically from convergent term rewriting systems for equational theories, in order to make automated theorem proving in these theories more eective. In particular we consider abelian groups and commutative rings. These are dicult for automated theorem provers, since their axioms of associativity, commutativity, distributivity and the inverse law can generate many variations of the same equation. For these theories ordering restrictions can be strengthened so that inferences apply only to maximal summands, and superpositions into the inverse law that move summands from one side of an equation to the other can be replaced by an isolation rule that isolates the maximal terms on one side. Additional inferences arise from superpositions of extended clauses, but we can show that most of these are redundant. In particular, none are needed in the case of abelian groups, and at most one for any pair of ...