Infinite-State Model Checking of LTLR Formulas Using Narrowing

Abstract

The linear temporal logic of rewriting (LTLR) is a simple extension of LTL that adds spatial action patterns to the logic, expressing that a specific instance of an action described by a rewrite rule has been performed. Although the theory and algorithms of LTLR for finite-state model checking are well-developed [2], no theoretical foundations have yet been developed for infinite-state LTLR model checking. The main goal of this paper is to develop such foundations for narrowing-based logical model checking of LTLR properties. A key theme in this paper is the systematic relationship, in the form of a simulation with remarkably good properties, between the concrete state space and the symbolic state space. A related theme is the use of additional state space reduction methods, such as folding and equational abstractions, that can in some cases yield a finite symbolic state space.1 The temporal logics that can be verified by infinite-state model checking techniques are generally less expressive than those supported by finite-state model checkers.

Programming and symbolic computation in Maude

“…Example 5 (LP-Syntax). To define the semantics of logic programs as a system module 8 we first specify an LP-SYNTAX functional module that imports the TERM functional module with sort Term in Example 1. An atomic predicate is defined as a Qid symbol applied to a non-empty list of terms in parentheses:…”

Constrained narrowing for conditional equational theories modulo axioms

“…Much work remains ahead, particularly: (i) on implementing our approach, for which we plan to rely on, and extend, the existing Maude infrastucture for narrowing, variants, and unification (we have developed a preliminary design of an implementation that relies on such a planned extensions of the Core Maude infrastructure); (ii) on extending it from the equational case to, as mentioned in the Introduction and in Section 6, the model checking analysis of concurrent systems specified as conditional rewrite theories; (iii) on experimentally evaluating the effectiveness and performance of our approach, and comparing it with other approaches using such an implementation; and (iv) on developing a substantial body of case studies demonstrating the usefulness and effectiveness of our approach: at the purely equational level, case studies showing how finite complete sets of constrained variants and constrained unifiers can represent infinite sets of actual variants and actual unifiers would be particularly appealing; at the model checking level mentioned in (ii) above, case studies showing how infinite-state systems specified with conditional equations and rules can be model cheked in a way that generalizes the current unconditional narrowing-based model checking in [9,10] would be very useful. All this will be the focus of our work in the near future.…”

“…Various kinds of symbolic techniques such as: (i) automata and grammars, e.g., [1,15,13,14,38,66,5,4,3,33]; (ii) SMT and other forms of constraint solving, e.g., [6,20,35,36,64,73,75,39,11]; and (iii) narrowing [72,30,31,9,10], have been employed for this purpose. All are useful in their own way and can complement each other; and there is great interest in combining the power of these different symbolic approaches to handle a wider range of applications [59,60] (see, e.g., [70] and the survey [58] for combinations of this kind).…”

“…Instead of concrete system states, we can represent a possibly infinite set of such states by a term t with logical variables. We can then use narrowing modulo E to "symbolically execute" state transitions as narrowing steps t ; R,E t , thus obtaining a useful form of symbolic reachability analysis and model checking with many applications [72,30,31,9,10] (see [58] for a survey of this work).…”

“…Since the requirement that (Σ, B, E 0 ) is convergent is de rigueur for the equational part E = E 0 ∪ B of executable rewrite theories R = (Σ, E, R), this provides indeed a very general method of symbolic system analysis for R. Note the interesting fact that narrowing then happens at two levels: at the concurrent system level as narrowing modulo E with a relation t ; R,E t , and at the equational level to perform E-unification by narrowing with the relation t ; E0,B t . Specifically, for narrowing at the equational level, folding variant narrowing [32] is the most effective method currently available, is supported by the Maude tool, and has been proved effective in the above-mentioned applications [30,31,9,10].…”