From LTL to unambiguous Büchi automata via disambiguation of alternating automata

Due to the high complexity of translating linear temporal logic (LTL) to deterministic automata, several forms of “restricted” nondeterminism have been considered with the aim of maintaining some of the benefits of deterministic automata, while at the same time allowing more efficient translations from LTL. One of them is the notion of unambiguity. This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike other approaches it is based on a known translation from very weak alternating automata (VWAA) to NBA. A notion of unambiguity for alternating automata is introduced and it is shown that the VWAA-to-NBA translation preserves unambiguity. Checking unambiguity of VWAA is determined to be PSPACE-complete, both for the explicit and symbolic encodings of alternating automata. The core of the LTL-to-UBA translation is an iterative disambiguation procedure for VWAA. Several heuristics are introduced for different stages of the procedure. We report on an implementation of our approach in the tool and compare it to an existing LTL-to-UBA implementation in the tool set. Our experiments cover model checking of Markov chains, which is an important application of UBA.

Vacuity in synthesis

In reactive synthesis, one begins with a temporal specification $$\varphi $$ φ , and automatically synthesizes a system $$M$$ M such that $$M\models \varphi $$ M ⊧ φ . As many systems can satisfy a given specification, it is natural to seek ways to force the synthesis tool to synthesize systems that are of a higher quality, in some well-defined sense. In this article we focus on a well-known measure of the way in which a system satisfies its specification, namely vacuity. Our conjecture is that if the synthesized system M satisfies $$\varphi $$ φ non-vacuously, then M is likely to be closer to the user’s intent, because it satisfies $$\varphi $$ φ in a more “meaningful” way. Narrowing the gap between the formal specification and the designer’s intent in this way, automatically, is the topic of this article. Specifically, we propose a bounded synthesis method for achieving this goal. The notion of vacuity as defined in the context of model checking, however, is not necessarily refined enough for the purpose of synthesis. Hence, even when the synthesized system is technically non-vacuous, there are yet more interesting (equivalently, less vacuous) systems, and we would like to be able to synthesize them. To that end, we cope with the problem of synthesizing a system that is as non-vacuous as possible, given that the set of interesting behaviours with respect to a given specification induce a partial order on transition systems. On the theoretical side we show examples of specifications for which there is a single maximal element in the partial order (i.e., the most interesting system), a set of equivalent maximal elements, or a number of incomparable maximal elements. We also show examples of specifications that induce infinite chains of increasingly interesting systems. These results have implications on how non-vacuous the synthesized system can be. We implemented the new procedure in our synthesis tool PARTY. For this purpose we added to it the capability to synthesize a system based on a property which is a conjunction of universal and existential LTL formulas.

Two SAT solvers for solving quantified Boolean formulas with an arbitrary number of quantifier alternations

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the $$\forall $$ ∀ Exp+Res calculus that is the formal foundation of expansion-based solving, it is conceptually simpler than present recursive approaches. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

Temporal prophecy for proving temporal properties of infinite-state systems

Various verification techniques for temporal properties transform temporal verification to safety verification. For infinite-state systems, these transformations are inherently imprecise. That is, for some instances, the temporal property holds, but the resulting safety property does not. This paper introduces a mechanism for tackling this imprecision. This mechanism, which we call temporal prophecy, is inspired by prophecy variables. Temporal prophecy refines an infinite-state system using first-order linear temporal logic formulas, via a suitable tableau construction. For a specific liveness-to-safety transformation based on first-order logic, we show that using temporal prophecy strictly increases the precision. Furthermore, temporal prophecy leads to robustness of the proof method, which is manifested by a cut elimination theorem. We integrate our approach into the Ivy deductive verification system, and show that it can handle challenging temporal verification examples.