Takahito Aoto

Proving Confluence of Term Rewriting Systems Automatically

We have developed an automated confluence prover for term rewriting systems (TRSs). This paper presents theoretical and technical ingredients that have been used in our prover. A distinctive feature of our prover is incorporation of several divide---and---conquer criteria such as those for commutative (Toyama, 1988), layer-preserving (Ohlebusch, 1994) and persistent (Aoto & Toyama, 1997) combinations. For a TRS to which direct confluence criteria do not apply, the prover decomposes it into components and tries to apply direct confluence criteria to each component. Then the prover combines these results to infer the (non-)confluence of the whole system. To the best of our knowledge, an automated confluence prover based on such an approach has been unknown.

Termination of simply typed term rewriting by translation and labelling

Simply typed term rewriting proposed by Yamada (RTA 2001) is a framework of term rewriting allowing higher-order functions. In contrast to the usual higher-order term rewriting frameworks, simply typed term rewriting dispenses with bound variables. This paper presents a method for proving termination of simply typed term rewriting systems (STTRSs, for short). We first give a translation of STTRSs into many-sorted first-order TRSs and show that termination problem of STTRSs is reduced to that of many-sorted first-order TRSs. Next, we introduce a labelling method which is applied to first-order TRSs obtained by the translation to facilitate termination proof of them; our labelling employs an extension of semantic labelling where terms are interpreted on a many-sorted algebra.

Dependency pairs for simply typed term rewriting

Simply typed term rewriting proposed by Yamada (RTA, 2001) is a framework of higher-order term rewriting without bound variables. In this paper, the dependency pair method of first-order term rewriting introduced by Arts and Giesl (TCS, 2000) is extended in order to show termination of simply typed term rewriting systems. Basic concepts such as dependency pairs and estimated dependency graph in the simply typed term rewriting framework are clarified. The subterm criterion introduced by Hirokawa and Middeldorp (RTA, 2004) is successfully extended to the case where terms of function type are allowed. Finally, an experimental result for a collection of simply typed term rewriting systems is presented. Our method is compared with the direct application of the first-order dependency pair method to a first-order encoding of simply typed term rewriting systems.

Program transformation by templates based on term rewriting

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 Langs framework.

Uniqueness of Normal Proofs in Implicational Intuitionistic Logic

A minimal theorem in a logic L is an L-theorem which is not a non-trivial substitution instance of another L-theorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique β-normal proof in NJ whenever A is provable without non-prime contraction. The non-prime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komoris problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique βη-normal proof in NJ.

Automated Confluence Proof by Decreasing Diagrams based on Rule-Labelling.

Decr easing 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.

Solution to the Problem of Zantema on a Persistent Property of Term Rewriting Systems

A property P of term rewriting systems is persistent if for any many-sorted term rewriting system R, R has the property P iff its underlying term rewriting system θ(R), which results from R by omitting its sort information, has the property P. It is shown that termination is a persistent property of many-sorted term rewriting systems that contain only variables of the same sort. This is the positive solution to a problem of Zantema, which has been appeared as Rewriting Open Problem 60 in literature.

Disproving Confluence of Term Rewriting Systems by Interpretation and Ordering

In order to disprove confluence of term rewriting systems, we develop new criteria for ensuring non-joinability of terms based on interpretation and ordering. We present some instances of the criteria which are amenable for automation, and report on an implementation of a confluence disproving procedure based on these instances. The experiments reveal that our method is successfully applied to automatically disprove confluence of some term rewriting systems, on which state-of-the-art automated confluence provers fail. A key idea to make our method effective is the introduction of usable rules—this allows one to decompose the constraint on rewrite rules into smaller components that depend on starting terms.

On Composable Properties of Term Rewriting Systems

A property of term rewriting system (TRS, for short) is said to be composable if it is preserved under unions. We present composable properties of TRSs on the base of modularity results for direct sums of TRSs. We propose a decomposition by a naive sort attachment, and show that modular properties for direct sums of TRSs are τ-composable for a naive sort attachment τ. Here, a decomposition of a TRS R is a pair (R1,R2) of (not necessary disjoint) subsets of R such that R = R1 U R2; and for a naive sort attachment T a property o of TRSs is said to be τ-composable if for any TRS R such that τ is consistent with R, o(R1) Λ φ(R2) implies φ(R) where (R1, R2) is the decomposition of R by τ.

Confluence of Orthogonal Nominal Rewriting Systems Revisited

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.