Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Taming Strategy Logic: Non-Recurrent Fragments

Strategy Logic (SL for short) is one of the prominent languages for reasoning about the strategic abilities of agents in a multi-agent setting. This logic extends LTL with first-order quantifiers over the agent strategies and encompasses other formalisms, such as ATL* and CTL*. The model-checking problem for SL and several of its fragments have been extensively studied. On the other hand, the picture is much less clear on the satisfiability front, where the problem is undecidable for the full logic. In this work, we study two fragments of One-Goal SL, where the nesting of sentences within temporal operators is constrained. We show that the satisfiability problem for these logics, and for the corresponding fragments of ATL* and CTL*, is ExpSpace and PSpace-complete, respectively

Quantifying Over Trees in Monadic Second-Order Logic

Monadic Second-Order Logic (MSO) extends FirstOrder Logic ( FO) with variables ranging over sets and quantifications over those variables. We introduce and study Monadic Tree Logic (MTL), a fragment of MSO interpreted on infinitetree models, where the sets over which the variables range are arbitrary subtrees of the original model. We analyse the expressiveness of MTL compared with variants of MSO and MPL, namely MSO with quantifications over paths. We also discuss the connections with temporal logics, by providing non-trivial fragments of the Graded mu-CALCULUS that can be embedded into MTL and by showing that MTL is enough to encode temporal logics for reasoning about strategies with FO-definable goals

Taming Strategy Logic: Non-Recurrent Fragments

Strategy Logic ([Formula presented] for short) is one of the prominent languages for reasoning about the strategic abilities of agents in a multi-agent setting. This logic extends [Formula presented] with first-order quantifiers over the agent strategies and encompasses other formalisms, such as [Formula presented] and [Formula presented]. The model-checking problem for [Formula presented] and several of its fragments has been extensively studied. On the other hand, the picture is much less clear on the satisfiability front, where the problem is undecidable for the full logic. In this work, we study two fragments of One-Goal [Formula presented], where the nesting of sentences within temporal operators is constrained. We show that the satisfiability problem for these two logics, and for the corresponding fragments of [Formula presented] and [Formula presented], is in [Formula presented] and [Formula presented], respectively

Substructure Temporal Logic

In formal verification and design, reasoning about substructures is a crucial aspect for several fundamental problems, whose solution often requires to select a portion of the model of interest on which to verify a specific property. In this paper, we present a new branching-time temporal logic, called Substructure Temporal Logic (STL, for short), whose distinctive feature is to allow for quantifying over the possible substructure of a given structure. This logic is obtained by adding two new operators to CTL, whose interpretation is given relative to the partial order induced by a suitable substructure relation. STL* turns out to be very expressive and allows to capture in a very natural way many well known problems, such as module checking, reactive synthesis and reasoning about games. A formal account of the model theoretic properties of the new logic and results about (un)decidability and complexity of related decision problems are also provide

Reasoning about substructures and games

Many decision problems in formal verification and design can be suitably formulated in game-theoretic terms. This is the case for the model checking of open and closed systems and both controller and reactive synthesis. Interpreted in this context, these problems require one to find a strategy (i.e., a plan) to force the system to fulfill some desired goal, no matter what the opponent (e.g., the environment) does. A strategy essentially constrains the possible behaviors of the system to those that are compatible with the decisions dictated by the plan itself. Therefore, finding a strategy to meet some goal basically reduces to identifying a portion of the model of interest (i.e., one of its substructures) that satisfies that goal. In this view, the ability to reason about substructures becomes a crucial aspect for several fundamental problems. In this article, we present and study a new branching-time temporal logic, called Substructure Temporal Logic (STL * for short), whose distinctive feature is to allow for quantifying over the possible substructure of a given structure. The logic is obtained by adding four new temporal-like operators to CTL *, whose interpretation is given relative to the partial order induced by a suitable substructure relation. STL * turns out to be very expressive and allows one to capture in a very natural way many well-known problems, such as module checking, reactive synthesis, and reasoning about games in a wide sense. A formal account of the model-theoretic properties of the new logic and results about (un)decidability and complexity of related decision problems are also provided