Pierre Wolper

Reasoning about Infinite Computations

We investigate extensions of temporal logic by connectives defined by finite automata on infinite words. We consider three different logics, corresponding to three different types of acceptance conditions (finite, looping, and repeating) for the automata. It turns out, however that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem.

Simple on-the-fly automatic verification of linear temporal logic

We present a tableau-based algorithm for obtaining an automaton from a temporal logic formula. The algorithm is geared towards being used in model checking in an “on-the-fly” fashion, that is the automaton can be constructed simultaneously with, and guided by, the generation of the model. In particular, it is possible to detect that a property does not hold by only constructing part of the model and of the automaton. The algorithm can also be used to check the validity of a temporal logic assertion. Although the general problem is PSPACE-complete, experiments show that our algorithm performs quite well on the temporal formulas typically encountered in verification. While basing linear-time temporal logic model-checking upon a transformation to automata is not new, the details of how to do this efficiently, and in “on-the-fly” fashion have never been given.

Temporal Logic Can Be More Expressive

It is first proved that there are properties of sequences that are not expressible in temporal logic, even though they are easily expressible using, for instance, regular expressions. Then, it is shown how temporal logic can be extended to express any property definable by a right-linear grammar and hence a regular expression. Finally, a complete axiomatization and a decision procedure for the extended temporal logic are given and the complexity of the extended logic is examined.

Memory Efficient Algorithms for the Verification of Temporal Properties

This article addresses the problem of designing memory-efficient algorithms for the verification of temporal properties of finite-state programs. Both the programs and their desired temporal properties are modeled as automata on infinite words (Büchi automata). Verification is then reduced to checking the emptiness of the automaton resulting from the product of the program and the property. This problem is usually solved by computing the strongly connected components of the graph representing the product automaton. Here, we present algorithms that solve the emptiness problem without explicitly constructing the strongly connected components of the product graph. By allowing the algorithms to err with some probability, we can implement them with a randomly accessed memory of size O(n) bits, where n is the number of states of the graph, instead of O(n log n) bits that the presently known algorithms require.

Automata-Theoretic techniques for modal logics of programs

Abstract We present a new technique for obtaining decision procedures for modal logics of programs. The technique centers around a new class of finite automata on infinite trees for which the emptiness problem can be solved in polynomial time. The decision procedures then consist of constructing an automaton Af for a given formula f such that Af accepts some tree if and only if f is satisfiable. We illustrate our technique by giving exponential decision procedures for several variants of deterministic propositional dynamic logic.

The complementation problem for Bu¨chi automata with applications to temporal logic

The problem of complementing Buchi automata arises when developing procedures for temporal logics of programs. Unfortunately, previously known constructions for complementing Buchi automata involve a doubly exponential blow-up in the size of the automaton. We present a construction that involves only an exponential blow-up. We use this construction to prove a polynomial space upper bound for the propositional temporal logic of regular events and to prove a complexity hierarchy result for quantified propositional temporal logic.

Using partial orders for the efficient verification of deadlock freedom and safety properties

This article presents an algorithm for detecting deadlocks in concurrent finite-state systems without incurring most of the state explosion due to the modeling of concurrency by interleaving. For systems that have a high level of concurrency, our algorithm can be much more efficient than the classical exploration of the whole state space. Finally, we show that our algorithm can also be used for verifying arbitrary safety properties.

Reasoning about infinite computation paths

We investigate extensions of temporal logic by finite automata on infinite words. There are three different types of acceptance conditions (finite, looping and repeating) that one can give for these finite automata. This gives rise to three different logics. It turns out, however. that these logics have the same expressive power but differ in the complexity of their decision problem. We also investigate the addition of alternation and show that it does not increase the complexity of the decision problem.

A partial approach to model checking

A model-checking method for linear-time temporal logic that avoids the state explosion due to the modeling of concurrency by interleaving is presented. The method relies on the concept of the Mazurkiewicz trace as a semantic basis and uses automata-theoretic techniques, including automata that operate on words of ordinality higher than omega . In particular, automata operating on words of length omega *n, n in omega are defined. These automata are studied, and an efficient algorithm to check whether such automata are nonempty is given. It is shown that when it is viewed as an omega *n automaton, the trace automaton can be substituted for the production automaton in linear-time model checking. The efficiency of the method of P. Godefroid (Proc. Workshop on Computer Aided Verification, 1990) is thus fully available for model checking. >

Expressing interesting properties of programs in propositional temporal logic

We show that the class of properties of programs expressible in propositional temporal logic can be substantially extended if we assume the programs to be <i>data-independent.</i> Basically, a program is data-independent if its behavior does not depend on the specific data it operates upon. Our results significantly extend the applicability of program verification and synthesis methods based on propositional temporal logic.