Publisher Summary
This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

CHAPTER 6 – Rewrite Systems

Termination of term rewriting using dependency pairs

0~12 is a functional programming language with an underlying formal semantics that is based upon equational logic, and an oprtationnl semantics that is based upon rewrite rules. Four clsssrs of design principles for 01352 ate discussed briefly in this inttoduct,ion, and then in mote detail brlnw: (1) motlulntizntion and patnmcteriantion; (2) subsorts; (3) implcmcntnt.ion IcBchniquc>s; and (4) inlrtaction and flexibility. WC also lrace C)II.l history, current shtus, and future plans, and give n fairly comp1rl.c OnJ bibliography. blast rxnmplc codr hns n-ct,unlly bcbcn run on out currclnt OI%.JZ intcrprc,t (‘r.

Principles of OBJ2

I. Natural and formal logic.- II. The connection method in propositional logic.- III. The connection method in first-order logic.- IV. Variants and improvements.- V. Applications and extensions.- Mnemonics for use in references.- References.- List of Symbols.

Automated Theorem Proving

Narrowing is the operational principle of languages that integrate functional and logic programming. We propose a notion of a needed narrowing step that, for inductively sequential rewrite systems, extends the Huet and Le´vy notion of a needed reduction step. We define a strategy, based on this notion, that computes only needed narrowing steps. Our strategy is sound and complete for a large class of rewrite systems, is optimal w.r.t. the cost measure that counts the number of distinct steps of a derivation, computes only independent unifiers, and is efficiently implemented by pattern matching.

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A needed narrowing strategy

Proofs by induction in equational theories with constructors

We show how to prove (and disprove) theorems in the initial algebra of an equational variety by a simple extension of the Knuth-Bendix completion algorithm. This allows us to prove by purely equational reasoning theorems whose proof usually requires induction. We show applications of this method to proofs of programs computing over data structures, and to proofs of algebraic summation identities. This work extends and simplifies recent results of Musser15 and Goguen6.

https://hal.inria.fr/inria-00076533/document

Jean-Marie Hullot

Papers

Proofs by induction in equational theories with constructors

Proofs by induction in equational theories with constructors