Rewrite-based equational theorem proving with selection and simplification

We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutation completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutation completeness. It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space even further, as we demonstrate for so-called quasi-Horn clauses. The results in this paper contain as special cases or generalize many known results about Knuth-Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses

Translating logic programs into conditional rewriting systems

In this paper a translation from a subclass of logic programs consisting of the simply moded logic programs into rewriting systems is defined. In these rewriting systems conditions and explicit substitutions may be present. We argue that our translation is more natural than previously studied ones and establish a result showing its correctness

Applications and extensions of context-sensitive rewriting

[EN] Context-sensitive rewriting is a restriction of term rewriting which is obtained by imposing replacement restrictions on the arguments of function symbols. It has proven useful to analyze computational properties of programs written in sophisticated rewriting-based programming languages such asCafeOBJ, Haskell, Maude, OBJ*, etc. Also, a number of extensions(e.g., to conditional rewritingor constrained equational systems) and generalizations(e.g., controlled rewritingor forbidden patterns) of context-sensitive rewriting have been proposed. In this paper, we provide an overview of these applications and related issues. (C) 2021 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Applications and extensions of context-sensitive rewriting. Journal of Logical and Algebraic Methods in Programming. 121:1-33. https://doi.org/10.1016/j.jlamp.2021.10068013312

Basic paramodulation

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the {\em basic\/} strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences

On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary