Symbolic Model Checking of Infinite-State Systems Using Narrowing

Abstract

Abstract. Rewriting is a general and expressive way of specifying concurrent systems, where concurrent transitions are axiomatized by rewrite rules. Narrowing is a complete symbolic method for model checking reachability properties. We show that this method can be reinterpreted as a lifting simulation relating the original system and the symbolic system associated to the narrowing transitions. Since the narrowing graph can be infinite, this lifting simulation only gives us a semi-decision procedure for the failure of invariants. However, we propose new methods for folding the narrowing tree that can in practice result in finite systems that symbolically simulate the original system and can be used to algorithmically verify its properties. We also show how both narrowing and folding can be used to symbolically model check systems which, in addition, have state predicates, and therefore correspond to Kripke structures on which ACT L * and LT L formulas can be algorithmically verified using such finite symbolic abstractions.

Folding Variant Narrowing and Optimal Variant Termination

“…We have developed in [7] a way of detecting such repetitions Definition 15 (Transition System). [7] A transition system is written A = (A, →), where A is a set of states, and → is a transition relation between states, i.e., →⊆ A × A. We write A = (A, →, I) when I ⊆ A is a set of initial states.…”

“…Definition 16 (Folding Reachable Transition Subsystem [7]). Given a transition system A = (A, →, I) and a relation G ⊆ A × A, the reachable subsystem from I in A with folding G is written…”

“…Narrowing is a fundamental rewriting technique useful for many purposes, including equational unification and equational theorem proving [15], combinations of functional and logic programming [12,13], partial evaluation [2], symbolic reachability analysis of rewrite theories understood as transition systems [19], and symbolic model checking [7].…”

Proving Safety Properties of Rewrite Theories

“…Example 1. Consider the following version of Lamport's bakery protocol, borrowed and slightly adapted from [6], in which there are several processes, each with its internal state and possibly with a natural number, that achieve mutual exclusion by the usual method common in bakeries and deli shops: there is a number dispenser and customers are served in sequential order according to the ticket that they hold. This system can be specified in Maude as a topmost rewrite theory BAKERY with top sort State, as follows:…”

“…Narrowing-based symbolic model checking techniques for topmost rewrite theories R have been previously studied in [6], where the idea is to "fold" the narrowing tree for R that can in practice result in finite-state system that symbolically simulates R. It is worth pursuing an extension of these narrowing symbolic model checking techniques for conditional rewrite theories, so they can be combined with our approach for symbolic model checking and for symbolic simulation (following the idea of Rusu in [18]). …”