On sufficient-completeness and related properties of term rewriting systems

Summary The decidability of the sufficient completeness property of equational specifications satisfying certain conditions is shown. In addition, the decidability of the related concept of quasi-reducibility of a term with respect to a set of rules is proved. Other results about irreducible ground terms of a term rewriting system also follow from a key technical lemma used in these decidability proofs; this technical lemma states that there is a finite bound on the substitutions of ground terms that need to be considered in order to check for a given term, whether the result obtained by any substitution of ground terms into the term is irreducible. These results are first shown for untyped systems and are subsequently extended to typed systems.Partially supported by the National Science Foundation Grant no. DCR-8408461     

A Categorical Critical-pair Completion Algorithm

We introduce a general critical-pair/completion algorithm, formulated in the language of category theory. It encompasses the Knuth–Bendix procedure for term rewriting systems (also modulo equivalence relations), the Gröbner basis algorithm for polynomial ideal theory, and the resolution procedure for automated theorem proving. We show how these three procedures fit in the general algorithm, and how our approach relates to other categorical modeling approaches to these algorithms, especially term rewriting.     

Nominal rewriting

Nominal rewriting is based on the observation that if we add support for α-equivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order reduction schemes such as λ-calculus beta-reduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the object-language (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is not capture-avoiding. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem for orthogonal systems.     

Rewrite rule systems for modal propositional logic

This paper explains new results relating modal propositional logic and rewrite rule systems. More precisely, we give complete term rewriting systems for the modal propositional systems known as K, Q, T, and S5. These systems are presented as extensions of Hsiang's system for classical propositional calculus. We have checked local confluence with the rewrite rule system K.B. (cf. the Knuth-Bendix algorithm) developed by the Formel project at INRIA. We prove that these systems are noetherian, and then infer their confluence from Newman's lemma. Therefore each term rewriting system provides a new automated decision procedure and defines a canonical form for the corresponding logic. We also show how to characterize the canonical forms thus obtained.     

Completion of Rewrite Systems with Membership Constraints Part I: Deduction Rules

We consider a constrained equational logic where the constraints are membership conditions t∈swheresis interpreted as a regular tree language. Our logic includes a fragment of second-order equational logic (without projections) where second-order variables range over regular sets of contexts. The problem with constrained equational logics is the failure of the critical pair lemma. This is the reason why we propose new deduction rules for which the critical pair lemma is restored. Computing critical pairs requires, however, solving some constraints in a second-order logic with membership constraints. In a second paper we give a terminating set of transformation rules for these formulas, which decides the existence of a solution, thus showing a new term scheme unification algorithm.Since an order-sorted signature is nothing but a bottom–up tree automaton, order-sorted equational logic falls into the scope of our study; our results show how to perform order-sorted completion without regularity and without sort-decreasingness. It also shows how to perform unification in the order-sorted case, with some higher-order variables (without any regularity assumption).     

A compact fixpoint semantics for term rewriting systems

This work is motivated by the fact that a “compact” semantics for term rewriting systems, which is essential for the development of effective semantics-based program manipulation tools (e.g. automatic program analyzers and debuggers), does not exist. The big-step rewriting semantics that is most commonly considered in functional programming is the set of values/normal forms that the program is able to compute for any input expression. Such a big-step semantics is unnecessarily oversized, as it contains many “semantically useless” elements that can be retrieved from a smaller set of terms. Therefore, in this article, we present a compressed, goal-independent collecting fixpoint semantics that contains the smallest set of terms that are sufficient to describe, by semantic closure, all possible rewritings. We prove soundness and completeness under ascertained conditions. The compactness of the semantics makes it suitable for applications. Actually, our semantics can be finite whereas the big-step semantics is generally not, and even when both semantics are infinite, the fixpoint computation of our semantics produces fewer elements at each step. To support this claim we report several experiments performed with a prototypical implementation.     

Formal Proofs About Rewriting Using ACL2

We present an application of the ACL2 theorem prover to reason about rewrite systems theory. We describe the formalization and representation aspects of our work using the first-order, quantifier-free logic of ACL2 and we sketch some of the main points of the proof effort. First, we present a formalization of abstract reduction systems and then we show how this abstraction can be instantiated to establish results about term rewriting. The main theorems we mechanically proved are Newman's lemma (for abstract reductions) and Knuth–Bendix critical pair theorem (for term rewriting).     

Knuth-Bendix Completion for Non-Symmetric Transitive Relations

We extend the Knuth-Bendix completion procedure from equational rewriting to rewriting with non-symmetric transitive relations and quasi-orderings. The main differences are the following: Specification of the general non-ground case seems beyond first-order logic. It is within that realm when terms are linear or functions non-monotonic. The procedure requires critical-pair computations and need not terminate even in the ground case. Simplification is not don't care non-deterministic, but search based. Applications include ordered resolution and ordered chaining calculi, development of rule-based declarative procedures and algorithms, program and reachability analysis (in rewriting logic) and propagation of inequality constraints.     

External Rewriting for Skeptical Proof Assistants

This paper presents the design, the implementation, and experiments of the integration of syntactic, conditional possibly associative-commutative term rewriting into proof assistants based on constructive type theory. Our approach is called external because it consists in performing term rewriting in a specific and efficient environment and checking the computations later in a proof assistant. Two typical systems are considered in this work: ELAN, based on the rewriting calculus, as the term rewriting-based environment, and Coq, based on the calculus of inductive constructions as the proof assistant. We first formalize the proof terms for deduction by rewriting and strategies in ELAN using the rewriting calculus with explicit substitutions. We then show how these proof terms can soundly be translated into Coq syntax where they can be directly type checked. For the method to be applicable for rewriting modulo associativity and commutativity, we provide an effective method to prove equalities modulo these axioms in Coq using ELAN. These results have been integrated into an ELAN-based rewriting tactic in Coq.     

A Characterisation of Multiply Recursive Functions with Higman's Lemma

We prove that string rewriting systems which reduce by Higman's lemma exhaust the multiply recursive functions. This result provides a full characterisation of the expressiveness of Higman's lemma when applied to rewriting theory. The underlying argument of our construction is to connect the order type and the derivation length via the Hardy hierarchy.     

A class of confluent term rewriting systems and unification

The narrowing mechanism and term rewriting systems are powerful tools for constructing complete and efficient unification algorithms for useful classes of equational theories. This has been shown for the case where term rewriting systems are confluent and noetherian (i.e., terminating). In this paper we show that the narrowing mechanism, combined with ordinary unification, yields a complete unification algorithm for equational theories that can be described by a closed linear term rewriting system with the non-repetition property; this class allows non-terminating rewrite systems. For some special forms of input terms, narrowing generates complete sets of E-unifiers without resorting to the non-repetition property. The key observation underlying the proof is that a reduction sequence in this class of term rewriting system can be transformed into one which possesses properties that enable a completeness proof.     

Formalization of the Resolution Calculus for First-Order Logic

I present a formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs. To prove the calculus sound, I use the substitution lemma, and to prove it complete, I use Herbrand interpretations and semantic trees. The correspondence between unsatisfiable sets of clauses and finite semantic trees is formalized in Herbrand’s theorem. I discuss the difficulties that I had formalizing proofs of the lifting lemma found in the literature, and I formalize a correct proof. The completeness proof is by induction on the size of a finite semantic tree. Throughout the paper I emphasize details that are often glossed over in paper proofs. I give a thorough overview of formalizations of first-order logic found in the literature. The formalization of resolution is part of the IsaFoL project, which is an effort to formalize logics in Isabelle/HOL.     

Modular Church-Rosser Modulo:The Complete Picture

In Ref.[19], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution,was later substantially simplified in Refs.[8,12]. In this paper, we present a further simplification of the proof of Toyama's result for confluence, which shows that the crux of the problem lies on two di?erent properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions and a modularity property of ordered completion that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for arbitrary kinds of rules, and scales up to rewriting modulo arbitrary sets of equations.     

Descendants and Origins in Term Rewriting

In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first-order term rewriting and λ-calculus, their finitary as well as their infinitary variants. A recurrent theme is the parallel moves lemma. Next to the classical notion of descendant, we introduce an extended version, known as origin tracking. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the genericity lemma in λ-calculus, the theorem of Huet and Lévy on needed reductions in first-order term rewriting, and Berry's sequentiality theorem in (infinitary) λ-calculus.     

Proof by consistency

Advances of the past decade in methods and computer programs for showing consistency of proof systems based on first-order equations have made it feasible, in some settings, to use proof by consistency as an alternative to conventional rules of inference. Musser described the method applied to proof of properties of inductively defined objects. Refinements of this inductionless induction method were discussed by Kapur, Goguen, Huet and Hullot, Huet and Oppen, Lankford, Dershowitz, Paul, and more recently by Jouannaud and Kounalis as well as by Kapur, Narendran and Zhang. This paper gives a very general account of proof by consistency and inductionless induction, and shows how previous results can be derived simply from the general theory. New results include a theorem giving characterizations of an unambiguity property that is key to applicability of proof by consistency, and a theorem similar to the Birkhoff's Completeness Theorem for equational proof systems, but concerning inductive proof.     

A Formalization of the Knuth–Bendix(–Huet) Critical Pair Theorem

A mechanical proof of the Knuth–Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet’s well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth–Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.     

Certifying Term Rewriting Proofs in ELAN

Term rewriting has been shown to be a good environment for both programming and proving. For analysing and debugging rule-based programs, we propose in this work a formalism based on the rewriting calculus with explicit substitutions (ρσ-calculus). This formalism also allows us to build the proof terms of rewriting derivations. Therefore, term rewriting proofs can be exported to other systems by translating them into the corresponding syntaxes. That is, using a proof checker, one can certify these proofs and vice versa, this method allows us to get term rewriting in proof assistants using an external system. Our method not only works with syntactic rewriting but also with rewriting modulo a set of axioms (e.g. associativity-commutativity).     

A New Method for the Boolean Ring Based Theorem Proving

A new method for first-order theorem proving based on the Boolean ring approach is proposed. The method is an extension of Hsiang's N-Strategy in two aspects: (1) When the input polynomials are derived from clauses, our method is reduced to a more restricted (but still complete) version of N-Strategy: Only maximal atoms in an N-rule are considered for generating new inferences. (2) When the input polynomials are derived from non-clausal formulas, no new inference rules are needed in our method for ensuring the completeness. Unlike Kapur and Narendran's method which considers every pair of polynomials for superposition, our method restricts the pairs to those one of which consists of an odd number of monomials. The completeness proof of our method with the integration of reduction is also provided and is done by using the technique of semantic trees. The same technique is used to prove the completeness of N-strategy with reduction (using only N-rules and P-rules) for clausal theorem proving, thus it settles a longtime open problem.     

Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation

In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm.