arXiv subjects

Krishnendu Chatterjee

Publications and source records attributed to Krishnendu Chatterjee.

234 records · Page 13Linked to original sources

arXiv2010

Mean-payoff Automaton Expressions

Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic mean-payoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.

cs.LO↗

arXiv2010

Discounting in Games across Time Scales

We introduce two-level discounted games played by two players on a perfect-information stochastic game graph. The upper level game is a discounted game and the lower level game is an undiscounted reachability game. Two-level games model hierarchical and sequential decision making under uncertainty across different time scales. We show the existence of pure memoryless optimal strategies for both players and an ordered field property for such games. We show that if there is only one player (Markov decision processes), then the values can be computed in polynomial time. It follows that whether the value of a player is equal to a given rational constant in two-level discounted games can be decided in NP intersected coNP. We also give an alternate strategy improvement algorithm to compute the value.

cs.LO↗

arXiv2010

GIST: A Solver for Probabilistic Games

Gist is a tool that (a) solves the qualitative analysis problem of turn-based probabilistic games with ω-regular objectives; and (b) synthesizes reasonable environment assumptions for synthesis of unrealizable specifications. Our tool provides the first and efficient implementations of several reduction-based techniques to solve turn-based probabilistic games, and uses the analysis of turn-based probabilistic games for synthesizing environment assumptions for unrealizable specifications.

cs.LO↗

arXiv2010

Synthesis of AMBA AHB from Formal Specification

The standard procedure for hardware design consists of describing circuit in a hardware description language at logic level followed by extensive verification and logic-synthesis. However, this process consumes significant time and needs a lot of effort. An alternative is to use formal specification language as a high-level hardware description language and synthesize hardware from formal specification. Bloem et.al. gave formal specifications and synthesize the AMBA AHB Arbiter. Our contributions are as follows:(1) We present more complete and compact formal specifications for the AMBA AHB Arbiter, and obtain significant (order of magnitude) improvement in synthesis results (both with respect to time and the number of gates of the synthesize circuit); (2) we present formal specification and synthesize to generate compact circuits for the remaining two components of the AMBA AHB protocol, namely, the AMBA AHB Master and AMBA AHB Slave; and (3) from the lessons learnt we present few principles for writing formal specifications for efficient hardware synthesis. Thus with intelligently written complete formal specifications we are able to automatically synthesize an important and widely used industrial protocol.

cs.LO↗

arXiv2009

Probabilistic Weighted Automata

Nondeterministic weighted automata are finite automata with numerical weights on transitions. They define quantitative languages L that assign to each word w a real number L(w). The value of an infinite word w is computed as the maximal value of all runs over w, and the value of a run as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We introduce probabilistic weighted automata, in which the transitions are chosen in a randomized (rather than nondeterministic) fashion. Under almost-sure semantics (resp. positive semantics), the value of a word w is the largest real v such that the runs over w have value at least v with probability 1 (resp. positive probability). We study the classical questions of automata theory for probabilistic weighted automata: emptiness and universality, expressiveness, and closure under various operations on languages. For quantitative languages, emptiness and universality are defined as whether the value of some (resp. every) word exceeds a given threshold. We prove some of these questions to be decidable, and others undecidable. Regarding expressive power, we show that probabilities allow us to define a wide variety of new classes of quantitative languages, except for discounted-sum automata, where probabilistic choice is no more expressive than nondeterminism. Finally, we give an almost complete picture of the closure of various classes of probabilistic weighted automata for the following pointwise operations on quantitative languages: max, min, sum, and numerical complement.

cs.LO↗

arXiv2009

Expressiveness and Closure Properties for Quantitative Languages

Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages $L$ that assign to each word $w$ a real number $L(w)$. In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-$ω$-regular for deterministic limit-average and discounted-sum automata, while this set is always $ω$-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $ω$-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights 0 or 1 and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages $L_1$ and $L_2$, we consider the operations $\max(L_1,L_2)$, $\min(L_1,L_2)$, and $1-L_1$, which generalize the boolean operations on languages, as well as the sum $L_1 + L_2$. We establish the closure properties of all classes of quantitative languages with respect to these four operations.

cs.LO↗

arXiv2009

Qualitative Logics and Equivalences for Probabilistic Systems

We investigate logics and equivalence relations that capture the qualitative behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized CTL (QRCTL): formulas of this logic can express the fact that certain temporal properties hold over all paths, or with probability 0 or 1, but they do not distinguish among intermediate probability values. We present a symbolic, polynomial time model-checking algorithm for QRCTL on MDPs. The logic QRCTL induces an equivalence relation over states of an MDP that we call qualitative equivalence: informally, two states are qualitatively equivalent if the sets of formulas that hold with probability 0 or 1 at the two states are the same. We show that for finite alternating MDPs, where nondeterministic and probabilistic choices occur in different states, qualitative equivalence coincides with alternating bisimulation, and can thus be computed via efficient partition-refinement algorithms. On the other hand, in non-alternating MDPs the equivalence relations cannot be computed via partition-refinement algorithms, but rather, they require non-local computation. Finally, we consider QRCTL*, that extends QRCTL with nested temporal operators in the same manner in which CTL* extends CTL. We show that QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs, while on non-alternating MDPs, the equivalence arising from QRCTL* can be strictly finer. We also provide a full characterization of the relation between qualitative equivalence, bisimulation, and alternating bisimulation, according to whether the MDPs are finite, and to whether their transition relations are finitely-branching.

cs.LO↗

arXiv2009

Alpaga: A Tool for Solving Parity Games with Imperfect Information

Alpaga is a solver for two-player parity games with imperfect information. Given the description of a game, it determines whether the first player can ensure to win and, if so, it constructs a winning strategy. The tool provides a symbolic implementation of a recent algorithm based on antichains.

cs.GT↗

arXiv2008

Termination Criteria for Solving Concurrent Safety and Reachability Games

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player~1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. Our result is significant because the strategy improvement algorithm provides, for the first time, a way to approximate the value of a concurrent safety game from below. Since a value iteration algorithm, or a strategy improvement algorithm for reachability games, can be used to approximate the same value from above, the combination of both algorithms yields a method for computing a converging sequence of upper and lower bounds for the values of concurrent reachability and safety games. Previous methods could approximate the values of these games only from one direction, and as no rates of convergence are known, they did not provide a practical way to solve these games.

cs.GT↗

arXiv2008

Probabilistic Systems with LimSup and LimInf Objectives

We give polynomial-time algorithms for computing the values of Markov decision processes (MDPs) with limsup and liminf objectives. A real-valued reward is assigned to each state, and the value of an infinite path in the MDP is the limsup (resp. liminf) of all rewards along the path. The value of an MDP is the maximal expected value of an infinite path that can be achieved by resolving the decisions of the MDP. Using our result on MDPs, we show that turn-based stochastic games with limsup and liminf objectives can be solved in NP \cap coNP.

cs.GT↗

arXiv2008

Timed Parity Games: Complexity and Robustness

We consider two-player games played in real time on game structures with clocks and parity objectives. The games are concurrent in that at each turn, both players independently propose a time delay and an action, and the action with the shorter delay is chosen. To prevent a player from winning by blocking time, we restrict each player to strategies that ensure that the player cannot be responsible for causing a zeno run. First, we present an efficient reduction of these games to turn-based (i.e., nonconcurrent) finite-state (i.e., untimed) parity games. The states of the resulting game are pairs of clock regions of the original game. Our reduction improves the best known complexity for solving timed parity games. Moreover, the rich class of algorithms for classical parity games can now be applied to timed parity games. Second, we consider two restricted classes of strategies for the player that represents the controller in a real-time synthesis problem, namely, limit-robust and bounded-robust strategies. Using a limit-robust strategy, the controller cannot choose an exact real-valued time delay but must allow for some nonzero jitter in each of its actions. If there is a given lower bound on the jitter, then the strategy is bounded-robust. We show that exact strategies are more powerful than limit-robust strategies, which are more powerful than bounded-robust strategies for any bound. For both kinds of robust strategies, we present efficient reductions to standard timed automaton games. These reductions provide algorithms for the synthesis of robust real-time controllers.

cs.LO↗

arXiv2008

Environment Assumptions for Synthesis

The synthesis problem asks to construct a reactive finite-state system from an $ω$-regular specification. Initial specifications are often unrealizable, which means that there is no system that implements the specification. A common reason for unrealizability is that assumptions on the environment of the system are incomplete. We study the problem of correcting an unrealizable specification $ϕ$ by computing an environment assumption $ψ$ such that the new specification $ψ\toϕ$ is realizable. Our aim is to construct an assumption $ψ$ that constrains only the environment and is as weak as possible. We present a two-step algorithm for computing assumptions. The algorithm operates on the game graph that is used to answer the realizability question. First, we compute a safety assumption that removes a minimal set of environment edges from the graph. Second, we compute a liveness assumption that puts fairness conditions on some of the remaining environment edges. We show that the problem of finding a minimal set of fair edges is computationally hard, and we use probabilistic games to compute a locally minimal fairness assumption.

cs.GT↗

arXiv2008

Algorithms for Büchi Games

The classical algorithm for solving Büchi games requires time $O(n\cdot m)$ for game graphs with $n$ states and $m$ edges. For game graphs with constant outdegree, the best known algorithm has running time $O(n^2/\log n)$. We present two new algorithms for Büchi games. First, we give an algorithm that performs at most $O(m)$ more work than the classical algorithm, but runs in time O(n) on infinitely many graphs of constant outdegree on which the classical algorithm requires time $O(n^2)$. Second, we give an algorithm with running time $O(n\cdot m\cdot\logδ(n)/\log n)$, where $1\leδ(n)\le n$ is the outdegree of the game graph. Note that this algorithm performs asymptotically better than the classical algorithm if $δ(n)=O(\log n)$.

cs.GT↗

arXiv2008

Stochastic Limit-Average Games are in EXPTIME

The value of a finite-state two-player zero-sum stochastic game with limit-average payoff can be approximated to within $ε$ in time exponential in a polynomial in the size of the game times polynomial in logarithmic in $\frac{1}ε$, for all $ε>0$.

cs.GT↗

arXiv2008

Linear Time Algorithm for Weak Parity Games

We consider games played on graphs with the winning conditions for the players specified as weak-parity conditions. In weak-parity conditions the winner of a play is decided by looking into the set of states appearing in the play, rather than the set of states appearing infinitely often in the play. A naive analysis of the classical algorithm for weak-parity games yields a quadratic time algorithm. We present a linear time algorithm for solving weak-parity games.

cs.LO↗

arXiv2008

The Complexity of Coverage

We study the problem of generating a test sequence that achieves maximal coverage for a reactive system under test. We formulate the problem as a repeated game between the tester and the system, where the system state space is partitioned according to some coverage criterion and the objective of the tester is to maximize the set of partitions (or coverage goals) visited during the game. We show the complexity of the maximal coverage problem for non-deterministic systems is PSPACE-complete, but is NP-complete for deterministic systems. For the special case of non-deterministic systems with a re-initializing ``reset'' action, which represent running a new test input on a re-initialized system, we show that the complexity is again co-NP-complete. Our proof technique for reset games uses randomized testing strategies that circumvent the exponentially large memory requirement in the deterministic case.

cs.PL↗

arXiv2008

Strategy Improvement for Concurrent Safety Games

We consider concurrent games played on graphs. At every round of the game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective: ``stay forever in a set F of states'', and its dual, the reachability objective, ``reach a set R of states''. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player 1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. The significance of the result is twofold. First, while strategy improvement algorithms were known for Markov decision processes and turn-based games, as well as for concurrent reachability games, this is the first strategy improvement algorithm for concurrent safety games. Second, and most importantly, the improvement algorithm provides a way to approximate the value of a concurrent safety game from below (the known value-iteration algorithms approximate the value from above). Thus, when used together with value-iteration algorithms, or with strategy improvement algorithms for reachability games, our algorithm leads to the first practical algorithm for computing converging upper and lower bounds for the value of reachability and safety games.

cs.GT↗

arXiv2007

Algorithms for Omega-Regular Games with Imperfect Information

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.

cs.LO↗