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

In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.

Term Rewriting Systems

This paper presents a comprehensive introduction to the ELAN rule-based programming language We describe the main features of the language, the ELAN environment, and introduce bibliographic references to various papers addressing foundations, implementation and applications of ELAN

An Overview of ELAN

https://link.springer.com/content/pdf/10.1007/BFb0030589.pdf

Specification and proof in membership equational logic

Rewriting logic

From an inductive method for proving weak innermost termination of rule-based programs, we automatically infer, for each successful proof, a finite strategy for data evaluation. The proof principle uses an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we analyse proof trees built from the rewrite system, schematizing the innermost derivations of any ground term. These proof trees are issued from patterns representing sets of ground terms. They are generated using two mechanisms, namely abstraction, introducing variables that represent ground terms in normal form, and narrowing, schematizing rewriting on ground terms. From the proof trees, we can extract for any given ground term, a rewriting strategy to compute one of its normal form, even if the ground term admits infinite rewriting derivations.

Proving weak termination also provides the right way to terminate

We propose a formal approach for the detection of high-level program behaviors. These behaviors, defined as combinations of patterns in a signature, are detected by model-checking on abstracted forms of program traces. Our approach works on unbounded sets of traces, which makes our technique useful not only for dynamic analysis, considering one trace at a time, but also for static analysis, considering a set of traces inferred from a control flow graph. Our technique uses a rewriting-based abstraction mechanism, producing a high-level representation of the program behavior, independent of the program implementation. It allows us to handle similar behaviors in a generic way and thus to be robust with respect to variants. Successfully applied to malware detection, our approach allows us in particular to model and detect information leak.

https://hal.inria.fr/inria-00594396/document

Behavior Analysis of Malware by Rewriting-based Abstraction - Extended Version

We propose a transformation based method for proving termination of ELAN strategies. We first give a sufficient criterion for ELAN strategies to terminate, only lying on rewrite rules involved in the strategy. We then give a simplification process of strategies, itself described by rewriting, to empower the previous criterion. This simplification, beyond easing termination proof of strategies, can both facilitate elaboration of specifications and ease proofs of other program properties.

https://hal.inria.fr/inria-00107743/document

Termination of ELAN strategies by simplification - Extended version -

We present here an algorithm for proving termination of term rewriting systems by \gpo ordering constraint solving. The algorithm gives, as automatically as possible, an appropriate instance of the gpo generic ordering proving termination of a given system. Constraint solving is done efficiently thanks to a DAG shared term data structure.

https://hal.inria.fr/inria-00073604/document

Termination Proofs Using gpo Ordering Constraints : Extended Version

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modelized by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. The generic method is then instantiated for the innermost strategy, as an example.

https://hal.inria.fr/inria-00172897/document

Isabelle Gnaedig

Papers

Proving weak termination also provides the right way to terminate

Behavior Analysis of Malware by Rewriting-based Abstraction - Extended Version

Termination of ELAN strategies by simplification - Extended version -

Termination Proofs Using gpo Ordering Constraints : Extended Version

Termination of rewriting strategies: a generic approach