System Description: RDL Rewrite and Decision Procedure Laboratory

Abstract

RDL¹ simplifies clauses in a quantifier-free first-order logic with equality using a tight integration between rewriting and decision procedures. On the one hand, this kind of integration is considered the key ingredient for the success of state-of-the-art verification systems, such as ACL2 [10], STeP [8], Tecton [9], and Simplify [7]. On the other hand, obtaining a principled and effective integration poses some difficult problems. Firstly, there are no formal accounts of the incorporation of decision procedures in rewriting. This makes it difficult to reason about basic properties such as soundness and termination of the implementation of the proposed schema. Secondly, most integration schemas are targeted to a given decision procedure and they do not allow to easily plug new decision procedures in the rewriting activity. Thirdly, only a tiny portion of the proof obligations arising in many practical verifition efforts falls exactly into the theory decided by the available decision procedure. RDL solves the problems above as follows:

• 1 RDL is based on CCR (Constraint Contextual Rewriting) [1 2], a formally specified integration schema between (ordered) conditional rewriting and a satisfability decision procedure [11]. RDL inherits the properties of soundness [1] and termination [2] of CCR. It is also fully automatic.

• 2 RDL is an open system which can be modularly extended with new decision procedures provided these offer certain interface functionalities (see [2] for details). In its current version, RDL offers ‘plug-and-play’ decision procedures for the theories of Universal Presburger Arithmetic over Integers (UPAI), Universal Theory of Equality (UTE), and UPAI extended with uninterpreted function symbols [13].

• 3 RDL implements instances of a generic extension schema for decision procedures [3]. The key ingredient of such a schema is a lemma speculation mechanism which ‘reduces’ the validity problem of a given theory to the validity problem of one of its sub-theories for which a decision procedure is available. The proposed mechanism is capable of generating lemmas which are entailed by the union of the theory decided by the available decision procedure and the facts stored in the current context. Three instances of the extension schema lifting a decision procedure for UPAI are available. First, augmentation copes with user-defined functions whose properties can be ex- pressed by conditional lemmas. Second, affinization is a mechanism for the ‘on-the-fly’ generation of lemmas to handle a significant class of formulae in the theory of Universal Arithmetic over Integers (UAI). Third, a combination of augmentation and affinization puts together the flexibility of the former with the automation of the latter. Finally, RDL can be extended with new lemma speculation mechanisms provided these meet certain requirements (see [3] for details).