166 research outputs found

    Modal µ-Calculus, Model Checking and Gauß Elimination

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    In this paper we present a novel approach for solving Boolean equation systems with nested minimal and maximal fixpoints. The method works by successively eliminating variables and reducing a Boolean equation system similar to Gauß elimination for linear equation systems. It does not require backtracking techniques. Within one framework we suggest a global and a local algorithm. In the context of model checking in the modal-calculus the local algorithm is related to the tableau methods, but has a better worst case complexity

    Local Model Checking Algorithm Based on Mu-calculus with Partial Orders

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    The propositionalμ-calculus can be divided into two categories, global model checking algorithm and local model checking algorithm. Both of them aim at reducing time complexity and space complexity effectively. This paper analyzes the computing process of alternating fixpoint nested in detail and designs an efficient local model checking algorithm based on the propositional μ-calculus by a group of partial ordered relation, and its time complexity is O(d2(dn)d/2+2) (d is the depth of fixpoint nesting,  is the maximum of number of nodes), space complexity is O(d(dn)d/2). As far as we know, up till now, the best local model checking algorithm whose index of time complexity is d. In this paper, the index for time complexity of this algorithm is reduced from d to d/2. It is more efficient than algorithms of previous research

    Algebraic optimization of recursive queries

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    Over the past few years, much attention has been paid to deductive databases. They offer a logic-based interface, and allow formulation of complex recursive queries. However, they do not offer appropriate update facilities, and do not support existing applications. To overcome these problems an SQL-like interface is required besides a logic-based interface.\ud \ud In the PRISMA project we have developed a tightly-coupled distributed database, on a multiprocessor machine, with two user interfaces: SQL and PRISMAlog. Query optimization is localized in one component: the relational query optimizer. Therefore, we have defined an eXtended Relational Algebra that allows recursive query formulation and can also be used for expressing executable schedules, and we have developed algebraic optimization strategies for recursive queries. In this paper we describe an optimization strategy that rewrites regular (in the context of formal grammars) mutually recursive queries into standard Relational Algebra and transitive closure operations. We also describe how to push selections into the resulting transitive closure operations.\ud \ud The reason we focus on algebraic optimization is that, in our opinion, the new generation of advanced database systems will be built starting from existing state-of-the-art relational technology, instead of building a completely new class of systems

    Modal mu-calculi

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    On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

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    For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1)

    Succinct progress measures for solving parity games

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    The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right

    Three notes on the complexity of model checking fixpoint logic with chop

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    This paper analyses the complexity of model checking fixpoint logic with Chop – an extension of the modal μ-calculus with a sequential composition operator. It uses two known game-based characterisations to derive the following results: the combined model checking complexity as well as the data complexity of FLC are EXPTIME-complete. This is already the case for its alternation-free fragment. The expression complexity of FLC is trivially P-hard and limited from above by the complexity of solving a parity game, i.e. in UP ∩ co-UP. For any fragment of fixed alternation depth, in particular alternation- free formulas it is P-complete

    Decomposing GR(1) Games with Singleton Liveness Guarantees for Efficient Synthesis

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    Temporal logic based synthesis approaches are often used to find trajectories that are correct-by-construction for tasks in systems with complex behavior. Some examples of such tasks include synchronization for multi-agent hybrid systems, reactive motion planning for robots. However, the scalability of such approaches is of concern and at times a bottleneck when transitioning from theory to practice. In this paper, we identify a class of problems in the GR(1) fragment of linear-time temporal logic (LTL) where the synthesis problem allows for a decomposition that enables easy parallelization. This decomposition also reduces the alternation depth, resulting in more efficient synthesis. A multi-agent robot gridworld example with coordination tasks is presented to demonstrate the application of the developed ideas and also to perform empirical analysis for benchmarking the decomposition-based synthesis approach

    The modal mu-calculus alternation hierarchy is strict

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    AbstractOne of the open questions about the modal mu-calculus is whether the alternation hierarchy collapses; that is, whether all modal fixpoint properties can be expressed with only a few alternations of least and greatest fixpoints. In this paper, we resolve this question by showing that the hierarchy does not collapse
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