Reasoning About Substructures and Games

Abstract

Studi di Napoli Federico IIMany 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.

Practical verification of multi-agent systems against Slk specifications

“…Instead, we show a direct reduction from the satisfiability problem of quantified Boolean formulas (QBF, for short), which was proved to be complete for PSpace in [36]. This reduction is loosely inspired by the one used to prove a similar result for Substructure Temporal Logic over finite models [37].…”

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

“…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 possible intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games Strategy Logic [12,[33][34][35][36] Substructure Temporal Logic [8,9], and fixed-point extensions of guarded first-order logics [10].Previous exponential solutions essentially divide into two families. The first one collects procedures that attempt to directly build winning strategies for the two players on the entire game.…”

“…Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games Strategy Logic [12,[33][34][35][36] Substructure Temporal Logic [8,9], and fixed-point extensions of guarded first-order logics [10].…”