Model Checking Games for the Quantitative mu-Calculus

We investigate quantitative extensions of modal logic and the modal mu-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative mu-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative mu-calculus

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.Comment: LMCS submissio

A Counting Logic for Structure Transition Systems

Quantitative questions such as "what is the maximum number of tokens in a place of a Petri net?" or "what is the maximal reachable height of the stack of a pushdown automaton?" play a significant role in understanding models of computation. To study such problems in a systematic way, we introduce structure transition systems on which one can define logics that mix temporal expressions (e.g. reachability) with properties of a state (e.g. the height of the stack). We propose a counting logic Qmu[#MSO] which allows to express questions like the ones above, and also many boundedness problems studied so far. We show that Qmu[#MSO] has good algorithmic properties, in particular we generalize two standard methods in model checking, decomposition on trees and model checking through parity games, to this quantitative logic. These properties are used to prove decidability of Qmu[#MSO] on tree-producing pushdown systems, a generalization of both pushdown systems and regular tree grammars

Degrees of Lookahead in Regular Infinite Games

We study variants of regular infinite games where the strict alternation of moves between the two players is subject to modifications. The second player may postpone a move for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This captures situations in distributed systems, e.g. when buffers are present in communication or when signal transmission between components is deferred. We distinguish strategies with different degrees of lookahead, among them being the continuous and the bounded lookahead strategies. In the first case the lookahead is of finite possibly unbounded size, whereas in the second case it is of bounded size. We show that for regular infinite games the solvability by continuous strategies is decidable, and that a continuous strategy can always be reduced to one of bounded lookahead. Moreover, this lookahead is at most doubly exponential in the size of a given parity automaton recognizing the winning condition. We also show that the result fails for non-regular gamesxwhere the winning condition is given by a context-free omega-language.Comment: LMCS submissio

A Unified Approach to Boundedness Properties in MSO

In the past years, extensions of monadic second-order logic (MSO) that can specify boundedness properties by the use of operators referring to the sizes of sets have been considered. In particular, the logics costMSO introduced by T. Colcombet and MSO+U by M. Bojanczyk were analyzed and connections to automaton models have been established to obtain decision procedures for these logics. In this work, we propose the logic quantitative counting MSO (qcMSO for short), which combines aspects from both costMSO and MSO+U. We show that both logics can be embedded into qcMSO in a natural way. Moreover, we provide a decidability proof for the theory of its weak variant (quantification only over finite sets) for the natural numbers with order and the infinite binary tree. These decidability results are obtained using a regular cost function extension of automatic structures called resource-automatic structures

Cardinality and counting quantifiers on omega-automatic structures

We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers `there exist at most $aleph_0$ many\u27, \u27there exist finitely many\u27 and \u27there exist $k$ modulo $m$ many\u27 are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regular set of representatives. This implies Blumensath\u27s conjecture that a countable structure with an $omega$-automatic presentation can be represented using automata on finite words. This also complements a very recent result of Hj"orth, Khoussainov, Montalban and Nies showing that there is an omega-automatic structure which has no injective presentation