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

We present an approach for proactive malware detection working by abstraction of program behaviors. Our technique consists in abstracting program traces, by rewriting given subtraces into abstract symbols representing their functionality. Traces are captured dynamically by code instrumentation, which allows us to handle packed or self-modifying malware. Suspicious behaviors are detected by comparing trace abstractions to reference malicious behaviors. The expressive power of abstraction allows us to handle general suspicious behaviors rather than specific malware code and then, to detect malware mutations. We present and discuss an implementation validating our approach.

/pdf/behavior-abstraction-in-malware-analysis-zkpuw0je6b.pdf

Behavior abstraction in malware analysis

A termination proof method for rewriting under strategies, based on an explicit induction on the termination property, is presented and instantiated for the innermost, outermost, and local strategies. Rewriting trees are simulated by proof trees generated with an abstraction mechanism, narrowing and constraints representing sets of ground terms. Abstraction introduces variables to represent normal forms without computing them and to control the narrowing mechanism, well known to easily diverge. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. It is established that termination under strategy is equivalent to the construction of finite proof trees schematizing terminating rewriting trees. Sufficient effective conditions to ensure finiteness are studied and the method is illustrated on several examples for each specific strategy.

/pdf/termination-of-rewriting-under-strategies-189pjubjz0.pdf

Termination of rewriting under strategies

We describe Cariboo, the implementation of an inductive process recently proposed to prove termination of rewriting under strategies on ground term algebras. The method is based on an abstraction mechanism, introducing variables that represent ground terms in normal form, and on narrowing, schematizing reductions on ground terms. It applies in particular to non-terminating systems which are terminating with innermost or local strategies. The narrowing process, well known to easily diverge, is controlled by using appropriate abstraction constraints. The proof mechanism lies on abstraction and ordering constraints satisfiability problems. Thanks to the power of induction, ordering constraints are often simple and automatically solved by our system. Otherwise, they can be treated by the user or any external automatic solver. On many examples, Cariboo even enables to succeed without considering any constraint at all; the process is then completely automatic. Cariboo offers a graphical view of the proof process. It is implemented in ELAN, a rule based programming environment, and so can be used for proving termination of ELAN programs.

https://hal.inria.fr/inria-00099426/document

Isabelle Gnaedig

Papers

Behavior abstraction in malware analysis

Termination of rewriting under strategies

CARIBOO: An Induction Based Proof Tool for Termination with Strategies -- Extended version--

Induction for termination with local strategies

Equational completion in order-sorted algebras