REM (Reduce Elan Machine): Core of the New ELAN Compiler

ELAN is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. It offers a natural and simple logical framework for the combination of the computation and deduction paradigms. It supports the design of theorem provers, logic programming languages, constraint solvers and decision procedures. ELAN takes from functional programming the concept of abstract data types and the function evaluation principle based on rewriting. In ELAN, a program is a set of labelled conditional rewrite rules l : l ⇒ r if c. Informally, rewriting a ground term t consists of selecting a rule whose left-hand side (also called pattern) matches the current term (t), or a subterm (tω), computing a substitution σ that gives the instantiation of rule variables (lσ = tω), and applying it to the right-hand side to build the reduced term (when instantiated conditions are satisfied). In general the normalisation of a term may not terminate, or terminate with different results corresponding to different selected rules, selected sub-terms and non-unicity of the substitution σ (in Associative and Commutative (AC) theories for example). So, evaluation by rewriting is essentially non-deterministic and backtracking may be needed to generate all results. One of the main originalities of the ELAN language is to provide AC operators allowing a simpler and more concise specification, and a strategy language allowing the programmer to specify the control on rule applications. This is in contrast to many existing rewriting-based languages where the term reduction strategy is hard-wired and not accessible to the designer of an application. The strategy language offers primitives for sequential composition, iteration, deterministic and non-deterministic choices of elementary strategies that are labelled rules. From these primitives, more complex strategies can be expressed. In addition, the user can introduce new strategy operators and define them by rewrite rules. Evaluation of strategy application is itself based on rewriting. Moreover, it should be emphasised that ELAN has logical foundations based on rewriting logic [4] and detailed in [1]. So the simple and well-known paradigm of rewriting provides both the evaluation mechanism of the language and the logical framework in which deduction systems can be expressed and combined. The full ELAN system (interpreter, compiler, standard library, examples, applications and documentation) is available through the ELAN web page.