In this paper, we present an Isabelle/HOL formalization of co-rewrite pairs for non-reachability analysis in term rewriting. In particular, we formalize polynomial interpretations over negative integers as well as the weighted path order (WPO) and its variant co-WPO. With this formalization, the verified certifier CeTA is now able to check such non-reachability proofs, including those for non-reachability problems of a database where existing tools fail to provide certified proofs.

This paper presents a new framework for constructing congruence closure of a finite set of ground equations over uninterpreted symbols and interpreted symbols for the group axioms. In this framework, ground equations are flattened into certain forms by introducing new constants, and a completion procedure is performed on ground flat equations. The proposed completion procedure uses equational inference rules and constructs a ground convergent rewrite system for congruence closure with such interpreted symbols. If the completion procedure terminates, then it yields a decision procedure for the word problem for a finite set of ground equations with respect to the group axioms. This paper also provides a sufficient terminating condition of the completion procedure for constructing a ground convergent rewrite system from ground flat equations containing interpreted symbols for the group axioms. In addition, this paper presents a new method for constructing congruence closure of a finite set of ground equations containing interpreted symbols for the semigroup, monoid, and the multiple disjoint sets of group axioms, respectively, using the proposed framework.

Parallel critical pairs (PCPs) have been used to design sufficient criteria for confluence of term rewrite systems. In this work we formalize PCPs and the criteria of Gramlich, Toyama, and Shintani and Hirokawa in the proof assistant Isabelle. In order to reduce the amount of bureaucracy we deviate from the paper-definition of PCPs, i.e., we switch from a position-based definition to a context-based definition. This switch not only simplifies the formalization task, but also gives rise to a simple recursive algorithm to compute PCPs. We further generalize all mentioned criteria from confluence to commutation and integrate them in the certifier CeTA, so that it can now validate confluence- and commutation-proofs based on PCPs. Because of our results, CeTA is now able to certify proofs by the automatic confluence tool Hakusan, which makes heavy use of PCPs. These proofs include term rewrite systems for which no previous certified confluence proof was known.

Abstract Although reasoning about equations over strings has been extensively studied for several decades, little research has been done for equational reasoning on general clauses over strings. This paper introduces a new superposition calculus with strings and present an equational theorem proving framework for clauses over strings. It provides a saturation procedure for clauses over strings and show that the proposed superposition calculus with contraction rules is refutationally complete. In particular, this paper presents a new decision procedure for solving word problems over strings and provides a new method of solving unification problems over strings w.r.t. a set of conditional equations R over strings if R can be finitely saturated under the proposed inference system with contraction rules.

Abstract Unlike other methods for theorem proving modulo with constrained clauses [12, 13], equational theorem proving modulo with constrained clauses along with its simplification techniques has not been well studied. We introduce a basic paramodulation calculus modulo equational theories E satisfying certain properties of E and present a new framework for equational theorem proving modulo E with constrained clauses. We propose an inference rule called Generalized E-Parallel for constrained clauses, which makes our inference system completely basic, meaning that we do not need to allow any paramodulation in the constraint part of a constrained clause for refutational completeness. We present a saturation procedure for constrained clauses based on relative reducibility and show that our inference system including our contraction rules is refutationally complete.

Rewriting modulo equations has been researched for several decades but due to the lack of suitable orderings, there are some limitations to rewriting modulo permutation equations. Given a finite set of permutation equations E, we present a new RPO-based ordering modulo E using (permutation) group actions and their associated orbits. It is an E-compatible reduction ordering on terms with the subterm property and is E-total on ground terms. We also present a completion and ground completion method for rewriting modulo a finite set of permutation equations E using our ordering modulo E. We show that our ground completion modulo E always admits a finite ground convergent (modulo E) rewrite system, which allows us to obtain the decidability of the word problem of ground theories modulo E.