3,349 research outputs found

    Reduction of Nondeterministic Tree Automata

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    We present an efficient algorithm to reduce the size of nondeterministic tree automata, while retaining their language. It is based on new transition pruning techniques, and quotienting of the state space w.r.t. suitable equivalences. It uses criteria based on combinations of downward and upward simulation preorder on trees, and the more general downward and upward language inclusions. Since tree-language inclusion is EXPTIME-complete, we describe methods to compute good approximations in polynomial time. We implemented our algorithm as a module of the well-known libvata tree automata library, and tested its performance on a given collection of tree automata from various applications of libvata in regular model checking and shape analysis, as well as on various classes of randomly generated tree automata. Our algorithm yields substantially smaller and sparser automata than all previously known reduction techniques, and it is still fast enough to handle large instances.Comment: Extended version (including proofs) of material presented at TACAS 201

    Reachability in Higher-Order-Counters

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    Higher-order counter automata (\HOCS) can be either seen as a restriction of higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an extension of counter automata to higher levels. We distinguish two principal kinds of \HOCS: those that can test whether the topmost counter value is zero and those which cannot. We show that control-state reachability for level kk \HOCS with 00-test is complete for \mbox{(k−2)(k-2)}-fold exponential space; leaving out the 00-test leads to completeness for \mbox{(k−2)(k-2)}-fold exponential time. Restricting \HOCS (without 00-test) to level 22, we prove that global (forward or backward) reachability analysis is \PTIME-complete. This enhances the known result for pushdown systems which are subsumed by level 22 \HOCS without 00-test. We transfer our results to the formal language setting. Assuming that \PTIME \subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS form strictly interleaving hierarchies. Interestingly, Engelfriet's constructions also allow to conclude immediately that the hierarchy of collapsible pushdown languages is strict level-by-level due to the existing complexity results for reachability on collapsible pushdown graphs. This answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201

    Two-Way Automata Making Choices Only at the Endmarkers

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    The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

    Visibly Pushdown Modular Games

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    Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. In this work, we study for the first time modular strategies with respect to winning conditions that can be expressed by a pushdown automaton. We show that such games are undecidable in general, and become decidable for visibly pushdown automata specifications. Our solution relies on a reduction to modular games with finite-state automata winning conditions, which are known in the literature. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal Buchi or co Buchi visibly pushdown winning condition are EXPTIME-complete, and when the winning condition is given by a CARET or NWTL temporal logic formula the problem is 2EXPTIME-complete, and it remains 2EXPTIME-hard even for simple fragments of these logics. As a further contribution, we present a different solution for modular games with finite-state automata winning condition that runs faster than known solutions for large specifications and many exits.Comment: In Proceedings GandALF 2014, arXiv:1408.556

    Can Nondeterminism Help Complementation?

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    Complementation and determinization are two fundamental notions in automata theory. The close relationship between the two has been well observed in the literature. In the case of nondeterministic finite automata on finite words (NFA), complementation and determinization have the same state complexity, namely Theta(2^n) where n is the state size. The same similarity between determinization and complementation was found for Buchi automata, where both operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing question is whether there exists a type of omega-automata whose determinization is considerably harder than its complementation. In this paper, we show that for all common types of omega-automata, the determinization problem has the same state complexity as the corresponding complementation problem at the granularity of 2^\Theta(.).Comment: In Proceedings GandALF 2012, arXiv:1210.202

    Deterministic Autopoietic Automata

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    This paper studies two issues related to the paper on Computing by Self-reproduction: Autopoietic Automata by Jiri Wiedermann. It is shown that all results presented there extend to deterministic computations. In particular, nondeterminism is not needed for a lineage to generate all autopoietic automata

    The Complexity of Enriched Mu-Calculi

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    The fully enriched μ-calculus is the extension of the propositional μ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched μ-calculus is known to be decidable and ExpTime-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched μ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and ExpTime-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are ExpTime-complete, and then reducing satisfiability in the relevant logics to these problems. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched μ-calculus extends the standard μ-calculus.Comment: A preliminary version of this paper appears in the Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP), 2006. This paper has been selected for a special issue in LMC
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