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Classical Descriptive Set Theory

In 1995, HyTech broke new ground as a potentially powerful tool for verifying hybrid systems – yet it has remained severely limited in its applicability to more complex systems. We address the main problems of HyTech with PHAVer, a new tool for the exact verification of safety properties of hybrid systems with piecewise constant bounds on the derivatives. Affine dynamics are handled by on-the-fly overapproximation and by partitioning the state space based on user-definable constraints and the dynamics of the system. PHAVer's exact arithmetic is robust due to the use of the Parma Polyhedra Library, which supports arbitrarily large numbers. To manage the complexity of the polyhedral computations, we propose methods to conservatively limit the number of bits and constraints of polyhedra. Experimental results for a navigation benchmark and a tunnel diode circuit show the effectiveness of the approach.

/pdf/phaver-algorithmic-verification-of-hybrid-systems-past-54obela629.pdf

PHAVer: algorithmic verification of hybrid systems past hytech

https://www.georgejpappas.org/papers/TCS-FP09.pdf

Robustness of temporal logic specifications for continuous-time signals

This paper presents algorithms for the automatic synthesis of real-time controllers by finding a winning strategy for certain games defined by the timed-automata of Alur and Dill. In such games, the outcome depends on the players' actions as well as on their timing. We believe that these results will pave the way for the application of program synthesis techniques to the construction of real-time embedded systems from their specifications.

On the synthesis of discrete controllers for timed systems

https://www.cs.unipr.it/~bagnara/Papers/PDF/SCP08.pdf

The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems

We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Buechi objective. This set is of interest because in the absence of perfect information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds.

Algorithms for Omega-Regular Games with Imperfect Information

Quantitative generalizations of classical languages, which assign to each word a real number instead of a Boolean value, have applications in modeling resource-constrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of the language-inclusion problem remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted automata cannot be determinized.

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Quantitative languages

In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Generalized mean-payoff and energy games replace individual weights by tuples, and the limit average (resp. running sum) of each coordinate must be (resp. remain) nonnegative. These games have applications in the synthesis of resource-bounded processes with multiple resources.

We prove the finite-memory determinacy of generalized energy games and show the inter-reducibility of generalized mean-payoff and energy games for finite-memory strategies. We also improve the computational complexity for solving both classes of games with finite-memory strategies: while the previously best known upper bound was EXPSPACE, and no lower bound was known, we give an optimal coNP-complete bound. For memoryless strategies, we show that the problem of deciding 
the existence of a winning strategy for the protagonist is NP-complete.

/pdf/generalized-mean-payoff-and-energy-games-2r8pj9uhm1.pdf

Generalized Mean-payoff and Energy Games

We propose and evaluate a new algorithm for checking the universality of nondeterministic finite automata. In contrast to the standard algorithm, which uses the subset construction to explicitly determinize the automaton, we keep the determinization step implicit. Our algorithm computes the least fixed point of a monotone function on the lattice of antichains of state sets. We evaluate the performance of our algorithm experimentally using the random automaton model recently proposed by Tabakov and Vardi. We show that on the difficult instances of this probabilistic model, the antichain algorithm outperforms the standard one by several orders of magnitude. We also show how variations of the antichain method can be used for solving the language-inclusion problem for nondeterministic finite automata, and the emptiness problem for alternating finite automata.

/pdf/antichains-a-new-algorithm-for-checking-universality-of-1jlryvbe8t.pdf

Antichains: a new algorithm for checking universality of finite automata

In this paper, we study algorithmic problems for quantitative models that are motivated by the applications in modeling embedded systems. We consider two-player games played on a weighted graph with mean-payoff objective and with energy constraints. We present a new pseudopolynomial algorithm for solving such games, improving the best known worst-case complexity for pseudopolynomial mean-payoff algorithms. Our algorithm can also be combined with the procedure by Andersson and Vorobyov to obtain a randomized algorithm with currently the best expected time complexity. The proposed solution relies on a simple fixpoint iteration to solve the log-space equivalent problem of deciding the winner of energy games. Our results imply also that energy games and mean-payoff games can be reduced to safety games in pseudopolynomial time.

/pdf/faster-algorithms-for-mean-payoff-games-37vlehc8ok.pdf

Laurent Doyen

Papers

Algorithms for Omega-Regular Games with Imperfect Information

Quantitative languages

Generalized Mean-payoff and Energy Games

Antichains: a new algorithm for checking universality of finite automata

Faster algorithms for mean-payoff games