The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer Scienc

Accelerated Approximation for Stochastic Reachability Games : Extended version of paper New algorithms for solving simple stochastic games

In this paper new algorithms for finding optimal values and strategies inturn-based stochastic games with reachability objectives are presented,whose special case are the simple stochastic games considered elsewhere [4,11]. The general idea of these algorithms is to accelerate the successive approximation scheme for solving stochastic games [13] in which node values are updated in each iteration so that they converge to the optimal values of the game. This scheme is extended with a pair of positional strategies which are updated to remain greedy with respect to the current approximation. This way optimal strategies can be discovered before the current values get close to the optimal ones. The approximation process is accelerated, by predicting an approximate result of several updates of the current valuation and jumping directly to the predicted values. New algorithms are based on three different acceleration techniques: iterative squaring, linear extrapolation, and linear programming; with different difficulty of performing single iteration and different acceleration level achieved by each of them. For each of these algorithms the complexity of a single iteration is polynomial. It is shown that accelerated algorithms will never perform worse than the basic, non-accelerated one and exponential upper bounds on the number of iterations required to solve a game in the worst case is given. It is also proven that new algorithms increase the frequency with which the greedy strategies are updated. The more often strategies are updated, the higher chances that the algorithm will terminate early. It is proven that the algorithm based on linear programming updates the greedy strategies in every iteration, which makes it similar to the strategy improvement method, where also a new strategy is found in each iteration and this also at the cost of solving linear constraint problems Paper is concluded with presentation of results of experiments in which new algorithms were run on a sample of randomly generated games. It could be observed that the proposed acceleration techniques reduce the number of iterations of the basic algorithm by an order of magnitude and that they substantially increase frequency with which the greedy strategies are updated. The algorithms based on linear programming and linear extrapolation displayed similar efficiency as the ones based on strategy improvement. This makes the algorithm based on linear extrapolation especially attractive because it uses much simpler computations than linear constraint solving.The original paper was published in Proceedings of the Workshop on Games in Design and Verification (GDV 2004), volume 119 of Electronic Notes in Theoretical Computer Science, pages 51–65. Elsevier. http://dx.doi.org/10.1016/j.entcs.2004.07.008</p

Logics and Algorithms for Verification of Concurrent Systems

In this thesis we investigate how the known framework of automatic formal verification by model checking can be extended in different directions. One extension is to go beyond the common limitation of the existing specification formalisms, that they can describe only regular properties of components. This can be achieved using logics capable of expressing non-regular properties, such as the Propositional Dynamic Logic of Context-free Programs (PDLCF), Fixpoint Logic with Chop (FLC) or the Higher-order Fixpoint Logic (HFL). Our main result in this area is proving that the problem of model checking HFL formulas of order bounded by k is k-EXPTIME complete. In the proofs we demonstrate two model checking algorithms for that logic. We also show that PDLCF is equivalent to a proper fragment of FLC. The standard model checking algorithms, which are run on a single computer, are severely limited by the amount of available computing resources. A way to overcome this limitation is to develop distributed algorithms, which can be run on a cluster of computers and use their joint resources. In this thesis we show how a distributed model checking algorithm for the alternation-free fragment of the modal μ-calculus can be extended to handle formulas with one level of alternation. This is an important extension, since Lμ formulas with one level of alternation can express the same properties as logics LTL and CTL commonly used in formal verification. Finally, we investigate stochastic games which can be used to model additional aspects of components, such as their interaction with environment and their quantitative properties. We describe new algorithms for finding optimal values and strategies in turn-based stochastic games with reachability winning conditions. We prove their correctness and report on experiments where we compare them against each other and against other known algorithms, such as value iteration and strategy improvement.UPMAR

Equalizing Morphisms of Petri Nets (Extended Abstract)

The problem of finite completeness of categories of Petri nets is studied. Since general Petri nets have finite products, the problem reduces to the issue of the existence of equalizers. We show that the general category of Petri nets has no equalizers, and hence it is not fintely complete. On the other hand, its full subcategory of reachable safe nets is finitely complete