TY - JOUR AB - In two-player finite-state stochastic games of partial observation on graphs, in every state of the graph, the players simultaneously choose an action, and their joint actions determine a probability distribution over the successor states. The game is played for infinitely many rounds and thus the players construct an infinite path in the graph. We consider reachability objectives where the first player tries to ensure a target state to be visited almost-surely (i.e., with probability 1) or positively (i.e., with positive probability), no matter the strategy of the second player. We classify such games according to the information and to the power of randomization available to the players. On the basis of information, the game can be one-sided with either (a) player 1, or (b) player 2 having partial observation (and the other player has perfect observation), or two-sided with (c) both players having partial observation. On the basis of randomization, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. Our main results for pure strategies are as follows: (1) For one-sided games with player 2 having perfect observation we show that (in contrast to full randomized strategies) belief-based (subset-construction based) strategies are not sufficient, and we present an exponential upper bound on memory both for almost-sure and positive winning strategies; we show that the problem of deciding the existence of almost-sure and positive winning strategies for player 1 is EXPTIME-complete and present symbolic algorithms that avoid the explicit exponential construction. (2) For one-sided games with player 1 having perfect observation we show that nonelementarymemory is both necessary and sufficient for both almost-sure and positive winning strategies. (3) We show that for the general (two-sided) case finite-memory strategies are sufficient for both positive and almost-sure winning, and at least nonelementary memory is required. We establish the equivalence of the almost-sure winning problems for pure strategies and for randomized strategies with actions invisible. Our equivalence result exhibit serious flaws in previous results of the literature: we show a nonelementary memory lower bound for almost-sure winning whereas an exponential upper bound was previously claimed. AU - Chatterjee, Krishnendu AU - Doyen, Laurent ID - 2211 IS - 2 JF - ACM Transactions on Computational Logic (TOCL) TI - Partial-observation stochastic games: How to win when belief fails VL - 15 ER - TY - JOUR AB - Recently, there has been an effort to add quantitative objectives to formal verification and synthesis. We introduce and investigate the extension of temporal logics with quantitative atomic assertions. At the heart of quantitative objectives lies the accumulation of values along a computation. It is often the accumulated sum, as with energy objectives, or the accumulated average, as with mean-payoff objectives. We investigate the extension of temporal logics with the prefix-accumulation assertions Sum(v) ≥ c and Avg(v) ≥ c, where v is a numeric (or Boolean) variable of the system, c is a constant rational number, and Sum(v) and Avg(v) denote the accumulated sum and average of the values of v from the beginning of the computation up to the current point in time. We also allow the path-accumulation assertions LimInfAvg(v) ≥ c and LimSupAvg(v) ≥ c, referring to the average value along an entire infinite computation. We study the border of decidability for such quantitative extensions of various temporal logics. In particular, we show that extending the fragment of CTL that has only the EX, EF, AX, and AG temporal modalities with both prefix-accumulation assertions, or extending LTL with both path-accumulation assertions, results in temporal logics whose model-checking problem is decidable. Moreover, the prefix-accumulation assertions may be generalized with "controlled accumulation," allowing, for example, to specify constraints on the average waiting time between a request and a grant. On the negative side, we show that this branching-time logic is, in a sense, the maximal logic with one or both of the prefix-accumulation assertions that permits a decidable model-checking procedure. Extending a temporal logic that has the EG or EU modalities, such as CTL or LTL, makes the problem undecidable. AU - Boker, Udi AU - Chatterjee, Krishnendu AU - Henzinger, Thomas A AU - Kupferman, Orna ID - 2038 IS - 4 JF - ACM Transactions on Computational Logic (TOCL) TI - Temporal specifications with accumulative values VL - 15 ER - TY - CONF AB - We study two-player (zero-sum) concurrent mean-payoff games played on a finite-state graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lies in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP ∩ coNP is the long-standing best known bound). We present a variant of the strategy-iteration algorithm by Hoffman and Karp; show that both our algorithm and the classical value-iteration algorithm can approximate the value in exponential time; and identify a subclass where the value-iteration algorithm is a FPTAS. We also show that the exact value can be expressed in the existential theory of the reals, and establish square-root sum hardness for a related class of games. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus ID - 2162 IS - Part 2 TI - The complexity of ergodic mean payoff games VL - 8573 ER - TY - CONF AB - We consider two-player partial-observation stochastic games on finitestate graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are ε-regular conditions specified as parity objectives. The qualitative-analysis problem given a partial-observation stochastic game and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). These qualitative-analysis problems are known to be undecidable. However in many applications the relevant question is the existence of finite-memory strategies, and the qualitative-analysis problems under finite-memory strategies was recently shown to be decidable in 2EXPTIME.We improve the complexity and show that the qualitative-analysis problems for partial-observation stochastic parity games under finite-memory strategies are EXPTIME-complete; and also establish optimal (exponential) memory bounds for finite-memory strategies required for qualitative analysis. AU - Chatterjee, Krishnendu AU - Doyen, Laurent AU - Nain, Sumit AU - Vardi, Moshe ID - 2213 TI - The complexity of partial-observation stochastic parity games with finite-memory strategies VL - 8412 ER - TY - CONF AB - The theory of graph games is the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic processes, we use 2 1/2-player games where some transitions of the game graph are controlled by two adversarial players, the System and the Environment, and the other transitions are determined probabilistically. We consider 2 1/2-player games where the objective of the System is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a mean-payoff condition). We establish that the problem of deciding whether the System can ensure that the probability to satisfy the mean-payoff parity objective is at least a given threshold is in NP ∩ coNP, matching the best known bound in the special case of 2-player games (where all transitions are deterministic). We present an algorithm running in time O(d·n2d·MeanGame) to compute the set of almost-sure winning states from which the objective can be ensured with probability 1, where n is the number of states of the game, d the number of priorities of the parity objective, and MeanGame is the complexity to compute the set of almost-sure winning states in 2 1/2-player mean-payoff games. Our results are useful in the synthesis of stochastic reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective). AU - Chatterjee, Krishnendu AU - Doyen, Laurent AU - Gimbert, Hugo AU - Oualhadj, Youssouf ID - 2212 TI - Perfect-information stochastic mean-payoff parity games VL - 8412 ER - TY - CONF AB - The edit distance between two (untimed) traces is the minimum cost of a sequence of edit operations (insertion, deletion, or substitution) needed to transform one trace to the other. Edit distances have been extensively studied in the untimed setting, and form the basis for approximate matching of sequences in different domains such as coding theory, parsing, and speech recognition. In this paper, we lift the study of edit distances from untimed languages to the timed setting. We define an edit distance between timed words which incorporates both the edit distance between the untimed words and the absolute difference in time stamps. Our edit distance between two timed words is computable in polynomial time. Further, we show that the edit distance between a timed word and a timed language generated by a timed automaton, defined as the edit distance between the word and the closest word in the language, is PSPACE-complete. While computing the edit distance between two timed automata is undecidable, we show that the approximate version, where we decide if the edit distance between two timed automata is either less than a given parameter or more than δ away from the parameter, for δ > 0, can be solved in exponential space and is EXPSPACE-hard. Our definitions and techniques can be generalized to the setting of hybrid systems, and analogous decidability results hold for rectangular automata. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Majumdar, Ritankar ID - 2216 TI - Edit distance for timed automata ER - TY - GEN AB - We consider Markov decision processes (MDPs) which are a standard model for probabilistic systems. We focus on qualitative properties for MDPs that can express that desired behaviors of the system arise almost-surely (with probability 1) or with positive probability. We introduce a new simulation relation to capture the refinement relation of MDPs with respect to qualitative properties, and present discrete graph theoretic algorithms with quadratic complexity to compute the simulation relation. We present an automated technique for assume-guarantee style reasoning for compositional analysis of MDPs with qualitative properties by giving a counter-example guided abstraction-refinement approach to compute our new simulation relation. We have implemented our algorithms and show that the compositional analysis leads to significant improvements. AU - Chatterjee, Krishnendu AU - Daca, Przemyslaw AU - Chmelik, Martin ID - 5413 SN - 2664-1690 TI - CEGAR for qualitative analysis of probabilistic systems ER - TY - GEN AB - We consider Markov decision processes (MDPs) which are a standard model for probabilistic systems. We focus on qualitative properties for MDPs that can express that desired behaviors of the system arise almost-surely (with probability 1) or with positive probability. We introduce a new simulation relation to capture the refinement relation of MDPs with respect to qualitative properties, and present discrete graph theoretic algorithms with quadratic complexity to compute the simulation relation. We present an automated technique for assume-guarantee style reasoning for compositional analysis of MDPs with qualitative properties by giving a counter-example guided abstraction-refinement approach to compute our new simulation relation. We have implemented our algorithms and show that the compositional analysis leads to significant improvements. AU - Chatterjee, Krishnendu AU - Daca, Przemyslaw AU - Chmelik, Martin ID - 5414 SN - 2664-1690 TI - CEGAR for qualitative analysis of probabilistic systems ER - TY - GEN AB - We consider Markov decision processes (MDPs) which are a standard model for probabilistic systems. We focus on qualitative properties for MDPs that can express that desired behaviors of the system arise almost-surely (with probability 1) or with positive probability. We introduce a new simulation relation to capture the refinement relation of MDPs with respect to qualitative properties, and present discrete graph theoretic algorithms with quadratic complexity to compute the simulation relation. We present an automated technique for assume-guarantee style reasoning for compositional analysis of MDPs with qualitative properties by giving a counter-example guided abstraction-refinement approach to compute our new simulation relation. We have implemented our algorithms and show that the compositional analysis leads to significant improvements. AU - Chatterjee, Krishnendu AU - Daca, Przemyslaw AU - Chmelik, Martin ID - 5412 SN - 2664-1690 TI - CEGAR for qualitative analysis of probabilistic systems ER - TY - CONF AB - We consider multi-player graph games with partial-observation and parity objective. While the decision problem for three-player games with a coalition of the first and second players against the third player is undecidable in general, we present a decidability result for partial-observation games where the first and third player are in a coalition against the second player, thus where the second player is adversarial but weaker due to partial-observation. We establish tight complexity bounds in the case where player 1 is less informed than player 2, namely 2-EXPTIME-completeness for parity objectives. The symmetric case of player 1 more informed than player 2 is much more complicated, and we show that already in the case where player 1 has perfect observation, memory of size non-elementary is necessary in general for reachability objectives, and the problem is decidable for safety and reachability objectives. From our results we derive new complexity results for partial-observation stochastic games. AU - Chatterjee, Krishnendu AU - Doyen, Laurent ID - 2163 IS - Part 2 T2 - Lecture Notes in Computer Science TI - Games with a weak adversary VL - 8573 ER - TY - GEN AB - We consider the reachability and shortest path problems on low tree-width graphs, with n nodes, m edges, and tree-width t, on a standard RAM with wordsize W. We use O to hide polynomial factors of the inverse of the Ackermann function. Our main contributions are three fold: 1. For reachability, we present an algorithm that requires O(n·t2·log(n/t)) preprocessing time, O(n·(t·log(n/t))/W) space, and O(t/W) time for pair queries and O((n·t)/W) time for single-source queries. Note that for constant t our algorithm uses O(n·logn) time for preprocessing; and O(n/W) time for single-source queries, which is faster than depth first search/breath first search (after the preprocessing). 2. We present an algorithm for shortest path that requires O(n·t2) preprocessing time, O(n·t) space, and O(t2) time for pair queries and O(n·t) time single-source queries. 3. We give a space versus query time trade-off algorithm for shortest path that, given any constant >0, requires O(n·t2) preprocessing time, O(n·t2) space, and O(n1−·t2) time for pair queries. Our algorithms improve all existing results, and use very simple data structures. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Pavlogiannis, Andreas ID - 5419 SN - 2664-1690 TI - Improved algorithms for reachability and shortest path on low tree-width graphs ER - TY - GEN AB - We consider multi-player graph games with partial-observation and parity objective. While the decision problem for three-player games with a coalition of the first and second players against the third player is undecidable, we present a decidability result for partial-observation games where the first and third player are in a coalition against the second player, thus where the second player is adversarial but weaker due to partial-observation. We establish tight complexity bounds in the case where player 1 is less informed than player 2, namely 2-EXPTIME-completeness for parity objectives. The symmetric case of player 1 more informed than player 2 is much more complicated, and we show that already in the case where player 1 has perfect observation, memory of size non-elementary is necessary in general for reachability objectives, and the problem is decidable for safety and reachability objectives. Our results have tight connections with partial-observation stochastic games for which we derive new complexity results. AU - Chatterjee, Krishnendu AU - Doyen, Laurent ID - 5418 SN - 2664-1690 TI - Games with a weak adversary ER - TY - GEN AB - We consider concurrent mean-payoff games, a very well-studied class of two-player (player 1 vs player 2) zero-sum games on finite-state graphs where every transition is assigned a reward between 0 and 1, and the payoff function is the long-run average of the rewards. The value is the maximal expected payoff that player 1 can guarantee against all strategies of player 2. We consider the computation of the set of states with value 1 under finite-memory strategies for player 1, and our main results for the problem are as follows: (1) we present a polynomial-time algorithm; (2) we show that whenever there is a finite-memory strategy, there is a stationary strategy that does not need memory at all; and (3) we present an optimal bound (which is double exponential) on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy). AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus ID - 5420 SN - 2664-1690 TI - The value 1 problem for concurrent mean-payoff games ER - TY - GEN AB - We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty. AU - Chatterjee, Krishnendu AU - Chmelik, Martin AU - Gupta, Raghav AU - Kanodia, Ayush ID - 5424 SN - 2664-1690 TI - Qualitative analysis of POMDPs with temporal logic specifications for robotics applications ER - TY - GEN AB - We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty. AU - Chatterjee, Krishnendu AU - Chmelik, Martin AU - Gupta, Raghav AU - Kanodia, Ayush ID - 5426 SN - 2664-1690 TI - Qualitative analysis of POMDPs with temporal logic specifications for robotics applications ER - TY - GEN AB - We present a flexible framework for the automated competitive analysis of on-line scheduling algorithms for firm- deadline real-time tasks based on multi-objective graphs: Given a taskset and an on-line scheduling algorithm specified as a labeled transition system, along with some optional safety, liveness, and/or limit-average constraints for the adversary, we automatically compute the competitive ratio of the algorithm w.r.t. a clairvoyant scheduler. We demonstrate the flexibility and power of our approach by comparing the competitive ratio of several on-line algorithms, including D(over), that have been proposed in the past, for various tasksets. Our experimental results reveal that none of these algorithms is universally optimal, in the sense that there are tasksets where other schedulers provide better performance. Our framework is hence a very useful design tool for selecting optimal algorithms for a given application. AU - Chatterjee, Krishnendu AU - Kössler, Alexander AU - Pavlogiannis, Andreas AU - Schmid, Ulrich ID - 5423 SN - 2664-1690 TI - A framework for automated competitive analysis of on-line scheduling of firm-deadline tasks ER - TY - GEN AB - We consider graphs with n nodes together with their tree-decomposition that has b = O ( n ) bags and width t , on the standard RAM computational model with wordsize W = Θ (log n ) . Our contributions are two-fold: Our first contribution is an algorithm that given a graph and its tree-decomposition as input, computes a binary and balanced tree-decomposition of width at most 4 · t + 3 of the graph in O ( b ) time and space, improving a long-standing (from 1992) bound of O ( n · log n ) time for constant treewidth graphs. Our second contribution is on reachability queries for low treewidth graphs. We build on our tree-balancing algorithm and present a data-structure for graph reachability that requires O ( n · t 2 ) preprocessing time, O ( n · t ) space, and O ( d t/ log n e ) time for pair queries, and O ( n · t · log t/ log n ) time for single-source queries. For constant t our data-structure uses O ( n ) time for preprocessing, O (1) time for pair queries, and O ( n/ log n ) time for single-source queries. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Pavlogiannis, Andreas ID - 5427 SN - 2664-1690 TI - Optimal tree-decomposition balancing and reachability on low treewidth graphs ER - TY - GEN AB - Recently there has been a significant effort to add quantitative properties in formal verification and synthesis. While weighted automata over finite and infinite words provide a natural and flexible framework to express quantitative properties, perhaps surprisingly, several basic system properties such as average response time cannot be expressed with weighted automata. In this work, we introduce nested weighted automata as a new formalism for expressing important quantitative properties such as average response time. We establish an almost complete decidability picture for the basic decision problems for nested weighted automata, and illustrate its applicability in several domains. AU - Chatterjee, Krishnendu AU - Henzinger, Thomas A AU - Otop, Jan ID - 5415 SN - 2664-1690 TI - Nested weighted automata ER - TY - GEN AB - Evolution occurs in populations of reproducing individuals. The structure of the population affects the outcome of the evolutionary process. Evolutionary graph theory is a powerful approach to study this phenomenon. There are two graphs. The interaction graph specifies who interacts with whom in the context of evolution. The replacement graph specifies who competes with whom for reproduction. The vertices of the two graphs are the same, and each vertex corresponds to an individual. A key quantity is the fixation probability of a new mutant. It is defined as the probability that a newly introduced mutant (on a single vertex) generates a lineage of offspring which eventually takes over the entire population of resident individuals. The basic computational questions are as follows: (i) the qualitative question asks whether the fixation probability is positive; and (ii) the quantitative approximation question asks for an approximation of the fixation probability. Our main results are: (1) We show that the qualitative question is NP-complete and the quantitative approximation question is #P-hard in the special case when the interaction and the replacement graphs coincide and even with the restriction that the resident individuals do not reproduce (which corresponds to an invading population taking over an empty structure). (2) We show that in general the qualitative question is PSPACE-complete and the quantitative approximation question is PSPACE-hard and can be solved in exponential time. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Nowak, Martin ID - 5421 SN - 2664-1690 TI - The complexity of evolution on graphs ER - TY - CONF AB - Two-player games on graphs provide the theoretical framework for many important problems such as reactive synthesis. While the traditional study of two-player zero-sum games has been extended to multi-player games with several notions of equilibria, they are decidable only for perfect-information games, whereas several applications require imperfect-information games. In this paper we propose a new notion of equilibria, called doomsday equilibria, which is a strategy profile such that all players satisfy their own objective, and if any coalition of players deviates and violates even one of the players objective, then the objective of every player is violated. We present algorithms and complexity results for deciding the existence of doomsday equilibria for various classes of ω-regular objectives, both for imperfect-information games, and for perfect-information games.We provide optimal complexity bounds for imperfect-information games, and in most cases for perfect-information games. AU - Chatterjee, Krishnendu AU - Doyen, Laurent AU - Filiot, Emmanuel AU - Raskin, Jean-François ID - 10885 SN - 0302-9743 T2 - VMCAI 2014: Verification, Model Checking, and Abstract Interpretation TI - Doomsday equilibria for omega-regular games VL - 8318 ER -