Model-checking Games Research Articles

Verifiable strategy synthesis for multiple autonomous agents: a scalable approach

Path planning and task scheduling are two challenging problems in the design of multiple autonomous agents. Both problems can be solved by the use of exhaustive search techniques such as model checking and algorithmic game theory. However, model checking suffers from the infamous state-space explosion problem that makes it inefficient at solving the problems when the number of agents is large, which is often the case in realistic scenarios. In this paper, we propose a new version of our novel approach called MCRL that integrates model checking and reinforcement learning to alleviate this scalability limitation. We apply this new technique to synthesize path planning and task scheduling strategies for multiple autonomous agents. Our method is capable of handling a larger number of agents if compared to what is feasibly handled by the model-checking technique alone. Additionally, MCRL also guarantees the correctness of the synthesis results via post-verification. The method is implemented in UPPAAL STRATEGO and leverages our tool MALTA for model generation, such that one can use the method with less effort of model construction and higher efficiency of learning than those of the original MCRL. We demonstrate the feasibility of our approach on an industrial case study: an autonomous quarry, and discuss the strengths and weaknesses of the methods.

On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

Model checking games and a genome sequence search

The paper is considered a concept of model checking games to solve algorithmic puzzles. We describe current approaches in this field and move to a game between a user and a software model checker with the goal to provide a solution to a problem, encoded in a transition system and an LTL formula with a requirement. We show how to encode and solve some problems using this approach. Then we move to the problem of searching a pattern in a genome sequence. We implement the Z-function search method in Promela, construct the model, provide the input of real viral data, and then play a model checking game with SPIN verifier. We created a fuzzy substring search method using the non-deterministic choice operator. Based on experiments we made, we discuss that the problem to find a pattern with some deviations is only solvable using the swarm verification and hash compact methods.

Provenance analysis for logic and games

A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. In database theory, provenance analysis by interpretations in commutative semirings has been rather succesful for positive query languages (such a unions of conjunctive queries, positive relational algebra, or datalog). However, it did not really offer an adequate treatment of negation or missing information. Here we propose a new approach for the provenance analysis of logics with negation, such as first-order logic and fixed-point logics. It is closely related to a provenance analysis of the associated model-checking games, and based on new semirings of dual-indeterminate polynomials or dual-indeterminate formal power series. These are obtained by taking quotients of traditional provenance semirings by congruences that are generated by products of positive and negative provenance tokens. Beyond the use for model-checking problems in logics, provenance analysis of games is of independent interest. Provenance values in games provide detailed information about the number and properties of the strategies of the players, far beyond the question whether or not a player has a winning strategy from a given position.

Simple Fixpoint Iteration To Solve Parity Games

A naive way to solve the model-checking problem of the mu-calculus uses fixpoint iteration. Traditionally however mu-calculus model-checking is solved by a reduction in linear time to a parity game, which is then solved using one of the many algorithms for parity games. We now consider a method of solving parity games by means of a naive fixpoint iteration. Several fixpoint algorithms for parity games have been proposed in the literature. In this work, we introduce an algorithm that relies on the notion of a distraction. The idea is that this offers a novel perspective for understanding parity games. We then show that this algorithm is in fact identical to two earlier published fixpoint algorithms for parity games and thus that these earlier algorithms are the same. Furthermore, we modify our algorithm to only partially recompute deeper fixpoints after updating a higher set and show that this modification enables a simple method to obtain winning strategies. We show that the resulting algorithm is simple to implement and offers good performance on practical parity games. We empirically demonstrate this using games derived from model-checking, equivalence checking and reactive synthesis and show that our fixpoint algorithm is the fastest solution for model-checking games.

The Descriptive Complexity of Modal μ Model-checking Games

This paper revisits the well-established relationship between the modal mu calculus and parity games to show that it is even more robust than previously known. It addresses the question of whether the descriptive complexity of modal mu model-checking games, previously known to depend on the syntactic complexity of a formula, depends in fact on its semantic complexity. It shows that up to formulas of semantic co-B\"uchi complexity, the descriptive complexity of their model-checking games coincides exactly with their semantic complexity. Beyond co-B\"uchi, the descriptive complexity of the model-checking parity games of a formula is shown to be an upper bound on its semantic complexity; whether it is also a lower bound remains an open question.

Model-checking games for logics of imperfect information

Logics of dependence and independence have semantics that, unlike Tarski semantics, are not based on single assignments (mapping variables to elements of a structure) but on sets of assignments. Sets of assignments are called teams and the semantics is called team semantics. We design model-checking games for logics with team semantics in a general and systematic way. The construction works for any extension of first-order logic by atomic formulae on teams, as long as certain natural conditions are observed which are satisfied by all team properties considered so far in the literature, including dependence, independence, constancy, inclusion, and exclusion.The second-order features of team semantics are reflected by the notion of a consistent winning strategy which is also a second-order notion in the sense that it depends not on single plays but on the space of all plays that are compatible with the strategy. Beyond the application to logics with team semantics, we isolate an abstract, purely combinatorial definition of such games, which may be viewed as second-order reachability games, and study their algorithmic properties.A number of examples are provided that show how logics with team semantics express familiar combinatorial problems in a somewhat unexpected way. Based on our games, we provide a complexity analysis of logics with team semantics.

Model Checking the Quantitative mu-Calculus on Linear Hybrid Systems

We study the model-checking problem for a quantitative extension of the modal mu-calculus on a class of hybrid systems. Qualitative model checking has been proved decidable and implemented for several classes of systems, but this is not the case for quantitative questions that arise naturally in this context. Recently, quantitative formalisms that subsume classical temporal logics and allow the measurement of interesting quantitative phenomena were introduced. We show how a powerful quantitative logic, the quantitative mu-calculus, can be model checked with arbitrary precision on initialised linear hybrid systems. To this end, we develop new techniques for the discretisation of continuous state spaces based on a special class of strategies in model-checking games and present a reduction to a class of counter parity games.

Model-checking games for fixpoint logics with partial order models

In this paper, we introduce model-checking games that allow local second-order power on sets of independent transitions in the underlying partial order models where the games are played. Since the interleaving semantics of such models is not considered, some problems that may arise when using interleaving representations are avoided and new decidability results for partial order models of concurrency are achieved. The games are shown to be sound and complete, and therefore determined. While in the interleaving case they coincide with the local model-checking games for the μ -calculus, in a partial order setting they verify properties of a number of fixpoint modal logics that can specify, in concurrent systems with partial order semantics, several properties not expressible with the μ -calculus. The games underpin a novel decision procedure for model-checking all temporal properties of a class of infinite and regular event structures, thus improving, in terms of temporal expressive power, previous results in the literature.

When not losing is better than winning: Abstraction and refinement for the full μ-calculus

This work presents a novel game-based approach to abstraction-refinement for the full μ-calculus, interpreted over 3-valued semantics. A novel notion of non-losing strategy is introduced and exploited for refinement. Previous works on refinement in the context of 3-valued semantics require a direct algorithm for solving a 3-valued model checking game. This was necessary in order to have the information needed for refinement available on one game board. In contrast, while still considering a 3-valued model checking game, here we reduce the problem of solving the game to solving two 2-valued model checking (parity) games. In case the result is indefinite ( don’t know ), the corresponding non-losing strategies, when combined, hold all the information needed for refinement. This approach is beneficial since it can use any solver for 2-valued parity games. Thus, it can take advantage of newly developed such algorithms with improved complexity.

Backtracking games and inflationary fixed points

We define a new class of games, called backtracking games. Backtracking games are essentially parity games with an additional rule allowing players, under certain conditions, to return to an earlier position in the play and revise a choice or to force a countback of the number of moves. This new feature makes backtracking games more powerful than parity games. As a consequence, winning strategies become more complex objects and computationally harder. The corresponding increase in expressiveness allows us to use backtracking games as model-checking games for inflationary fixed-point logics such as IFP or MIC. We identify a natural subclass of backtracking games, the simple games, and show that these are the “right” model-checking games for IFP by (a) giving a translation of formulae φ and structures A into simple games such that A ⊨ φ if, and only if, Player 0 wins the corresponding game and (b) showing that the winner of simple backtracking games can again be defined in IFP.

Fixed-Point Logics and Solitaire Games

The model-checking games associated with fixed-point logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study Solitaire-LFP, a fragment of least fixed- point logic, whose evaluation games are nested soltaire games. This means that on each strongly connected component of the game, only one player can make nontrivial moves. Winning sets of nested solitaire games can be computed efficiently. The model-checking problem for Solitaire-LFP is PSPACE-complete in general and PTIME-complete for formulae of bounded width. On finite structures (but not on infinite ones), Solitaire-LFP is equivalent to transitive closure logic. We also consider the solitaire fragment of guarded fixed-point logics. Due to the restricted quantification pattern of these logics, the associated games are small and therefore admit more efficient model-checking algorithms.