Mw Maciej Gazda

Zielonka's recursive algorithm : dull, weak and solitaire games and tighter bounds

Dull, weak and nested solitaire games are important classes of parity games, capturing, among others, alternation-free mu-calculus and ECTL* model checking problems. These classes can be solved in polynomial time using dedicated algorithms. We investigate the complexity of Zielonkas Recursive algorithm for solving these special games, showing that the algorithm runs in O(d (n + m)) on weak games, and, somewhat surprisingly, that it requires exponential time to solve dull games and (nested) solitaire games. For the latter classes, we provide a family of games G, allowing us to establish a lower bound of 2^(n/3). We show that an optimisation of Zielonkas algorithm permits solving games from all three classes in polynomial time. Moreover, we show that there is a family of (non-special) games M that permits us to establish a lower bound of 2^(n/3), improving on the previous lower bound for the algorithm.

Consistent consequence for boolean equation systems

Inspired by the concept of a consistent correlation for Boolean equation systems, we introduce and study a novel relation, called consistent consequence . We show that it can be used as an approximation of the solution to an equation system. For the closed, simple and recursive fragment of equation systems we prove that it coincides with direct simulation for parity games. In addition, we show that deciding both consistent consequence and consistent correlations are coNP-complete problems, and we provide a sound and complete proof system for consistent consequence. As an application, we define a novel abstraction mechanism for parameterised Boolean equation systems and we establish its correctness using our theory.

Turning GSOS Rules into Equations for Linear Time-Branching Time Semantics

An existing axiomatization strategy for process algebras modulo bisimulation semantics can be extended so that it can be applied to other behavioural semantics as well. We study term rewriting properties of the resulting axiomatizations.

Expressiveness and completeness in abstraction

We study two notions of expressiveness, which have appeared in abstraction theory for model checking, and find them incomparable in general. In particular, we show that according to the most widely used notion, the class of Kripke Modal Transition Systems is strictly less expressive than the class of Generalised Kripke Modal Transition Systems (a generalised variant of Kripke Modal Transition Systems equipped with hypertransitions). Furthermore, we investigate the ability of an abstraction framework to prove a formula with a finite abstract model, a property known as completeness. We address the issue of completeness from a general perspective: the way it depends on certain abstraction parameters, as well as its relationship with expressiveness.

Improvement in small progress measures

Small Progress Measures is one of the classical parity game solving algorithms. For games with n vertices, m edges and d different priorities, the original algorithm computes the winning regions and a winning strategy for one of the players in O(dm.(n/floor(d/2))^floor(d/2)) time. Computing a winning strategy for the other player requires a re-run of the algorithm on that players winning region, thus increasing the runtime complexity to O(dm.(n/ceil(d/2))^ceil(d/2)) for computing the winning regions and winning strategies for both players. We modify the algorithm so that it derives the winning strategy for both players in one pass. This reduces the upper bound on strategy derivation for SPM to O(dm.(n/floor(d/2))^floor(d/2)). At the basis of our modification is a novel operational interpretation of the least progress measure that we provide.

On Parity Game Preorders and the Logic of Matching Plays

Parity games can be used to solve satisfiability, verification and controller synthesis problems. As part of an effort to better understand their nature, or the nature of the problems they solve, preorders on parity games have been studied. Defining these relations, and in particular proving their transitivity, has proven quite difficult on occasion. We propose a uniform way of lifting certain preorders on Kripke structures to parity games and study the resulting preorders. We explore their relation with parity game preorders from the literature and we study new relations. Finally, we investigate whether these preorders can also be obtained via modal characterisations of the preorders.

Abstraction in Fixpoint Logic

We present a theory of abstraction for the framework of parameterised Boolean equation systems, a first-order fixpoint logic. Parameterised Boolean equation systems can be used to solve a variety of problems in verification. We study the capabilities of the abstraction theory by comparing it to an abstraction theory for Generalised Kripke modal Transition Systems (GTSs). We show that for model checking the modal μ-calculus, our abstractions can be exponentially more succinct than GTSs and our theory is as complete as the GTS framework for abstraction. Furthermore, we investigate the completeness of our theory irrespective of the encoded decision problem. We illustrate the potential of our theory through case studies using the first-order modal μ-calculus and a real-time extension thereof, conducted using a prototype implementation of a new syntactic transformation for parameterised Boolean equation systems.