An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n \mathbf{G}\mathbf{F} \varphi_i \vee \mathbf{F}\mathbf{G} \psi_i$, where $\varphi_i$ and $\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.Comment: This is the extended version of the referenced conference paper and contains an appendix with additional materia

A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning

Synchronous Closing and Flow Analysis for Model Checking Timed Systems

Abstract. Formal methods, in particular model checking, are increas-ingly accepted as integral part of system development. With large soft-ware systems beyond the range of fully automatic verification, however, a combination of decomposition and abstraction techniques is needed. To model check components of a system, a standard approach is to close the component with an abstraction of its environment, as standard model checkers often do not handle open reactive systems directly. To make it useful in practice, the closing of the component should be automatic, both for data and for control abstraction. Specifically for model checking asynchronous open systems, external input queues should be removed, as they are a potential source of a combinatorial state explosion. In this paper we investigate a class of environmental processes for which the asynchronous communication scheme can safely be replaced by a synchronous one. Such a replacement is possible only if the environment is constructed under rather a severe restriction on the behavior, which can be partially softened via the use of a discrete-time semantics. We employ data-flow analysis to detect instances of variables and timers influenced by the data passing between the system and the environment

Module checking

In computer system design, we distinguish between closed and open systems. A closed system is a system whose behavior is completely determined by the state of the system. An open system is a system that interacts with its environment and whose behavior depends on this interaction. The ability of temporal logics to describe an ongoing interaction of a reactive program with its environment makes them particularly appropriate for the specification of open systems. Nevertheless, model-checking algorithms used for the verification of closed systems are not appropriate for the verification of open systems. Correct model checking of open systems should check the system with respect to arbitrary environments and should take into account uncertainty regarding the environment. This is not the case with current model-checking algorithms and tools. In this paper we introduce and examine the problem of model checking of open systems (module checking, for short). We show that while module checking and model checking coincide for the linear-time paradigm, module checking is much harder than model checking for the branching-time paradigm. We prove that the problem of module checking is EXPTIME-complete for specifications in CTL and 2EXPTIME-complete for specifications in CTL*. This bad news is also carried over when we consider the program-complexity of module checking. As good news, we show that for the commonly-used fragment of CTL (universal, possibly, and always possibly properties), current model-checking tools do work correctly, or can be easily adjusted to work correctly, with respect to both closed and open systems. (C) 2001 Academic Press

Synchronous closing and flow analysis for model checking timed systems

Formal methods, in particular model checking, are increasingly accepted as integral part of system development. With large software systems beyond the range of fully automatic verification, however, a combination of decomposition and abstraction techniques is needed. To model check components of a system, a standard approach is to close the component with an abstraction of its environment, as standard model checkers often do not handle open reactive systems directly. To make it useful in practice, the closing of the component should be automatic, both for data and for control abstraction. Specifically for model checking asynchronous open systems, external input queues should be removed, as they are a potential source of a combinatorial state explosion. In this paper we investigate a class of environmental processes for which the asynchronous communication scheme can safely be replaced by a synchronous one. Such a replacement is possible only if the environment is constructed under rather a severe restriction on the behavior, which can be partially softened via the use of a discrete-time semantics. We employ data-flow analysis to detect instances of variables and timers influenced by the data passing between the system and the environment

An automata-theoretic approach to branching-time model checking

Translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear-time model-checking, and for obtaining many extensions and improvements to this verification method. On the other hand, for branching temporal logic, automate-theoretic techniques have long been thought to introduce an exponential penalty, making them essentially useless for model-checking. Recently, Bernholtz and Grumberg [1993] have shown that this exponential penalty can be avoided, though they did not match the linear complexity of non-automata-theoretic algorithms. In this paper, we show that alternating tree automata are the key to a comprehensive automata-theoretic framework for branching temporal logics. Not only can they be used to obtain optimal decision procedures, as was shown by Muller ct al., but, as we show here, they also make it possible to derive optimal model-checking algorithms. Moreover, the simple combinatorial structure that emerges from the automata-theoretic approach opens up new possibilities for the implementation of branching-time model checking, and has enabled us to derive improved space complexity bounds for this long-standing problem