2015
DOI: 10.1016/j.scico.2015.06.001
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Constrained narrowing for conditional equational theories modulo axioms

Abstract: For an unconditional equational theory (Σ, E) whose oriented equations E are confluent and terminating, narrowing provides an E-unification algorithm. This has been generalized by various authors in two directions: (i) by considering unconditional equational theories (Σ, E∪B) where the E are confluent, terminating and coherent modulo axioms B, and (ii) by considering conditional equational theories. Narrowing for a conditional theory (Σ, E ∪ B) has also been studied, but much less and with various restrictions… Show more

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Cited by 15 publications
(6 citation statements)
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“…-The standard narrowing relation u R v is just the special case of (1) where B " H and φpf q " H for each f P Σ, or perhaps the even more special case where, furthermore, Σ is unsorted. -The special instance of (2) where R " pΣ, B, E, Hq is a strongly deterministic, convergent, conditional theory has been studied in detail in [20]. This work is of special interest for symbolic equational reasoning, since it provides a narrowing-based constrained E Z B-unification algorithm.…”
mentioning
confidence: 99%
“…-The standard narrowing relation u R v is just the special case of (1) where B " H and φpf q " H for each f P Σ, or perhaps the even more special case where, furthermore, Σ is unsorted. -The special instance of (2) where R " pΣ, B, E, Hq is a strongly deterministic, convergent, conditional theory has been studied in detail in [20]. This work is of special interest for symbolic equational reasoning, since it provides a narrowing-based constrained E Z B-unification algorithm.…”
mentioning
confidence: 99%
“…However, our construction requires the rewrite rules to introduce extra variables, some of these extra variables may be present in the unification problem prior to rewriting, and not necessarily present in the term upon which rewriting is applied. This automatically implies that we would need to consider narrowing over a conditional rewrite system [16], however, we nonetheless break the freshness condition required by narrowing as "used" variables may be introduced. As with primal grammars, restricting the variables occurring on the right-hand side of rewrite rules is essential to the decidability of its unification problem, something we avoid.…”
Section: Related Workmentioning
confidence: 99%
“…As we shall see in Section 4, the NS and NI inference rules exploit this lifting property of narrowing for inductive reasoning purposes. More precisely, they do so using so-called constrained narrowing with possibly conditional rules E, in the sense of [12].…”
Section: Narrowing In a Nutshellmentioning
confidence: 99%
“…Although several are well-known and some are widely used in superposition-based [6] automatic first-order theorem provers such as, e.g., [67,72], to the best of my knowledge they have not been previously combined for inductive theorem proving purposes with the extensiveness and generality presented here. They include: (1) equationally defined equality predicates [33]; (2) constrained narrowing [12]; (3) constructor variant unification [54,70]; (4) variant satisfiability [54,70]; (5) recursive path orderings [64,32]; (6) order-sorted congruence closure [53]; (7) contextual rewriting [73]; and (8) ordered rewriting, e.g., [47,4,59]). Furthermore, in this work all these techniques work modulo axioms B, for B any combination of associativity and/or commutativity and/or identity axioms.…”
Section: Introductionmentioning
confidence: 99%