Logically constrained term rewriting is a relatively new rewriting formalism that naturally supports built-in data structures, such as integers and bit vectors. In the analysis of logically constrained term rewrite systems (LCTRSs), rewriting constrained terms plays a crucial role. However, this combines rewrite rule applications and equivalence transformations in a closely intertwined way. This intertwining makes it difficult to establish useful theoretical properties for this kind of rewriting and causes problems in implementations—namely, that impractically large search spaces are often required. To address this issue, we propose in this paper a novel notion of most general constrained rewriting, which operates on existentially constrained terms, a concept recently introduced by the authors. We define a class of left-linear, left-value-free LCTRSs that are general enough to simulate all left-linear LCTRSs and exhibit the desired key property: most general constrained rewriting commutes with equivalence. This property ensures that equivalence transformations can be deferred until after the application of rewrite rules, which helps mitigate the issue of large search spaces in implementations. In addition to that, we show that the original rewriting formalism on constrained terms can be embedded into our new rewriting formalism on existentially constrained terms. Thus, our results are expected to have significant implications for achieving correct and efficient implementations in tools operating on LCTRSs.

Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity analysis and confluence analysis for logically constrained term rewriting, these works solely focus on the syntactic side of the formalism lacking detailed investigations on semantics. In this paper, we investigate a semantic side of logically constrained term rewriting. To this end, we first define constrained equations, constrained equational theories and validity of the former based on the latter. After presenting the relationship of validity and conversion of rewriting, we then construct a sound inference system to prove validity of constrained equations in constrained equational theories. Finally, we give an algebraic semantics, which enables one to establish invalidity of constrained equations in constrained equational theories. This algebraic semantics derive a new notion of consistency for constrained equational theories.

A term rewriting system (TRS) is said to be sufficiently complete when each function yields some value for any input. Proof methods for sufficient completeness of terminating TRSs have been well studied. In this paper, we introduce a simple derivation system for proving sufficient completeness of possibly non-terminating TRSs. The derivation system consists of rules to manipulate a set of guarded terms, and sufficient completeness of a TRS holds if there exists a successful derivation for each function symbol. We also show that variations of the derivation system are useful for proving special cases of local sufficient completeness of TRSs, which is a generalised notion of sufficient completeness.

Uniqueness of normal forms w.r.t. conversion (UNC) of term rewriting systems (TRSs) guarantees that there are no distinct convertible normal forms. It was recently shown that the UNC property of TRSs is decidable for shallow TRSs (Radcliffe et al., 2010). The existing procedure mainly consists of testing whether there exists a counterexample in a finite set of candidates; however, the procedure suffers a bottleneck of having a sheer number of such candidates. In this paper, we propose a new procedure which consists of checking a smaller number of such candidates and enumerating such candidates more efficiently. Correctness of the proposed procedure is proved and its complexity is analyzed. Furthermore, these two procedures have been implemented and it is experimentally confirmed that the proposed procedure runs much faster than the existing procedure.

We present a framework for proving inductive theorems of first-order equational theories, using techniques of implicit induction developed in the field of term rewriting. In this framework, we make use of automated confluence provers, which have recently been developed intensively, as well as a novel condition of sufficient completeness, called local sufficient completeness. The condition is a key to automated proof of inductive theorems of term rewriting systems that include non-terminating functions. We also apply the technique to showing the correctness of program transformation that is realised as an equivalence transformation of term rewriting systems.

Nominal rewriting systems (Fernandez, Gabbay, Mackie, 2004; Fernandez, Gabbay, 2007) have been introduced as a new framework of higher-order rewriting systems based on the nominal approach (Gabbay, Pitts, 2002; Pitts, 2003), which deals with variable binding via permutations and freshness conditions on atoms. Confluence of orthogonal nominal rewriting systems has been shown in (Fernandez, Gabbay, 2007). However, their definition of (non-trivial) critical pairs has a serious weakness so that the orthogonality does not actually hold for most of standard nominal rewriting systems in the presence of binders. To overcome this weakness, we divide the notion of overlaps into the self-rooted and proper ones, and introduce a notion of alpha-stability which guarantees alpha-equivalence of peaks from the self-rooted overlaps. Moreover, we give a sufficient criterion for uniformity and alpha-stability. The new definition of orthogonality and the criterion offer a novel confluence condition effectively applicable to many standard nominal rewriting systems. We also report on an implementation of a confluence prover for orthogonal nominal rewriting systems based on our framework.

Automated inductive reasoning for term rewriting has been extensively studied in the literature. Classes of equations and term rewriting systems (TRSs) with decidable inductive validity have been identified and used to automatize the inductive reasoning. We give procedures for deciding the inductive validity of equations in some standard TRSs on natural numbers and lists. Contrary to previous decidability results, our procedures can automatically decide without involving induction reasoning the inductive validity of arbitrary equations for these TRSs, that is, without imposing any syntactical restrictions on the form of equations. We also report on the complexity of our decision procedures. These decision procedures are implemented in our automated provers for inductive theorems of TRSs and experiments are reported.

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and usual termination. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into terminating part and possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system.

The notion of inductive theorems is well-established in first-order term rewriting. In higher-order term rewriting, in contrast, it is not straightforward to extend this notion because of extensionality (Meinke, 1992). When extending the term rewriting based program transformation of Chiba et al. (2005) to higher-order term rewriting, we need extensibility, a property stating that inductive theorems are preserved by adding new functions via macros. In this paper, we propose and study a new notion of inductive theorems for higher-order rewriting, natural inductive theorems. This allows to incorporate properties such as extensionality and extensibility, based on simply typed S-expression rewriting (Yamada, 2001).

Decreasing diagrams technique (van Oostrom, 1994) is a technique that can be widely applied to prove confluence of rewrite systems. To directly apply the decreasing diagrams technique to prove confluence of rewrite systems, rule-labelling heuristic has been proposed by van Oostrom (2008). We show how constraints for ensuring confluence of term rewriting systems constructed based on the rule-labelling heuristic are encoded as linear arithmetic constraints suitable for solving the satisfiability of them by external SMT solvers. We point out an additional constraint omitted in (van Oostrom, 2008) that is needed to guarantee the soundness of confluence proofs based on the rule-labelling heuristic extended to deal with non-right-linear rules. We also present several extensions of the rule-labelling heuristic by which the applicability of the technique is enlarged.

In many automated methods for proving inductive theorems, finding a suitable generalization of a conjecture is a key for the success of proof attempts. On the other hand, an obtained generalized conjecture may not be a theorem, and in this case hopeless proof attempts for the incorrect conjecture are made, which is against the success and efficiency of theorem proving. Urso and Kounalis (2004) proposed a generalization method for proving inductive validity of equations, called sound generalization, that avoids such an over-generalization. Their method guarantees that if the original conjecture is an inductive theorem then so is the obtained generalization. In this paper, we revise and extend their method. We restore a condition on one of the characteristic argument positions imposed in their previous paper and show that otherwise there exists a counterexample to their main theorem. We also relax a condition imposed in their framework and add some flexibilities to some of other characteristic argument positions so as to enlarge the scope of the technique.

Huet and Lang (1978) presented a framework of automated program transformation based on lambda calculus in which programs are transformed according to a given program transformation template. They introduced a second-order matching algorithm of simply-typed lambda calculus to verify whether the input program matches the template. They also showed how to validate the correctness of the program transformation using the denotational semantics.We propose in this paper a framework of program transformation by templates based on term rewriting. In our new framework, programs are given by term rewriting systems. To automate our program transformation, we introduce a term pattern matching problem and present a sound and complete algorithm that solves this problem.We also discuss how to validate the correctness of program transformation in our framework. We introduce a notion of developed templates and a simple method to construct such templates without explicit use of induction. We then show that in any program transformation by developed templates the correctness of the transformation can be verified automatically. In our framework the correctness of the program transformation is discussed based on the operational semantics. This is a sharp contrast to Huet and Lang's framework.