@inproceedings{3345, abstract = {We consider Markov Decision Processes (MDPs) with mean-payoff parity and energy parity objectives. In system design, the parity objective is used to encode ω-regular specifications, and the mean-payoff and energy objectives can be used to model quantitative resource constraints. The energy condition re- quires that the resource level never drops below 0, and the mean-payoff condi- tion requires that the limit-average value of the resource consumption is within a threshold. While these two (energy and mean-payoff) classical conditions are equivalent for two-player games, we show that they differ for MDPs. We show that the problem of deciding whether a state is almost-sure winning (i.e., winning with probability 1) in energy parity MDPs is in NP ∩ coNP, while for mean- payoff parity MDPs, the problem is solvable in polynomial time, improving a recent PSPACE bound.}, author = {Chatterjee, Krishnendu and Doyen, Laurent}, location = {Warsaw, Poland}, pages = {206 -- 218}, publisher = {Springer}, title = {{Energy and mean-payoff parity Markov Decision Processes}}, doi = {10.1007/978-3-642-22993-0_21}, volume = {6907}, year = {2011}, } @inproceedings{3346, abstract = {We study Markov decision processes (MDPs) with multiple limit-average (or mean-payoff) functions. We consider two different objectives, namely, expectation and satisfaction objectives. Given an MDP with k reward functions, in the expectation objective the goal is to maximize the expected limit-average value, and in the satisfaction objective the goal is to maximize the probability of runs such that the limit-average value stays above a given vector. We show that under the expectation objective, in contrast to the single-objective case, both randomization and memory are necessary for strategies, and that finite-memory randomized strategies are sufficient. Under the satisfaction objective, in contrast to the single-objective case, infinite memory is necessary for strategies, and that randomized memoryless strategies are sufficient for epsilon-approximation, for all epsilon>;0. We further prove that the decision problems for both expectation and satisfaction objectives can be solved in polynomial time and the trade-off curve (Pareto curve) can be epsilon-approximated in time polynomial in the size of the MDP and 1/epsilon, and exponential in the number of reward functions, for all epsilon>;0. Our results also reveal flaws in previous work for MDPs with multiple mean-payoff functions under the expectation objective, correct the flaws and obtain improved results.}, author = {Brázdil, Tomáš and Brožek, Václav and Chatterjee, Krishnendu and Forejt, Vojtěch and Kučera, Antonín}, location = {Toronto, Canada}, publisher = {IEEE}, title = {{Two views on multiple mean payoff objectives in Markov Decision Processes}}, doi = {10.1109/LICS.2011.10}, year = {2011}, } @inproceedings{3347, abstract = {The class of omega-regular languages provides a robust specification language in verification. Every omega-regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens "eventually". Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness. It requires that there exists an unknown, fixed bound b such that something good happens within b transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Buchi, parity and Streett languages. We study languages expressible by such automata: we give their topological complexity and present a regular-expression characterization. We compare the expressive power of finitary automata and give optimal algorithms for classical decisions questions. We show that the finitary languages are Sigma 2-complete; we present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; we show that the languages defined by finitary parity automata exactly characterize the star-free fragment of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete and universality as well as language inclusion are PSPACE-complete for finitary parity and Streett automata.}, author = {Chatterjee, Krishnendu and Fijalkow, Nathanaël}, location = {Tarragona, Spain}, pages = {216 -- 226}, publisher = {Springer}, title = {{Finitary languages}}, doi = {10.1007/978-3-642-21254-3_16}, volume = {6638}, year = {2011}, } @inproceedings{3348, abstract = {We study synthesis of controllers for real-time systems, where the objective is to stay in a given safe set. The problem is solved by obtaining winning strategies in the setting of concurrent two-player timed automaton games with safety objectives. 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. We construct winning strategies for the controller which require access only to (1) the system clocks (thus, controllers which require their own internal infinitely precise clocks are not necessary), and (2) a linear (in the number of clocks) number of memory bits. Precisely, we show that for safety objectives, a memory of size (3 · |C|+lg(|C|+1)) bits suffices for winning controller strategies, where C is the set of clocks of the timed automaton game, significantly improving the previous known exponential bound. We also settle the open question of whether winning region controller strategies require memory for safety objectives by showing with an example the necessity of memory for region strategies to win for safety objectives.}, author = {Chatterjee, Krishnendu and Prabhu, Vinayak}, location = {Chicago, USA}, pages = {221 -- 230}, publisher = {Springer}, title = {{Synthesis of memory efficient real time controllers for safety objectives}}, doi = {10.1145/1967701.1967734}, year = {2011}, } @inproceedings{3349, abstract = {Games on graphs provide a natural model for reactive non-terminating systems. In such games, the interaction of two players on an arena results in an infinite path that describes a run of the system. Different settings are used to model various open systems in computer science, as for instance turn-based or concurrent moves, and deterministic or stochastic transitions. In this paper, we are interested in turn-based games, and specifically in deterministic parity games and stochastic reachability games (also known as simple stochastic games). We present a simple, direct and efficient reduction from deterministic parity games to simple stochastic games: it yields an arena whose size is linear up to a logarithmic factor in size of the original arena.}, author = {Chatterjee, Krishnendu and Fijalkow, Nathanaël}, location = {Minori, Italy}, pages = {74 -- 86}, publisher = {EPTCS}, title = {{A reduction from parity games to simple stochastic games}}, doi = {10.4204/EPTCS.54.6}, volume = {54}, year = {2011}, } @inproceedings{3350, abstract = {A controller for a discrete game with ω-regular objectives requires attention if, intuitively, it requires measuring the state and switching from the current control action. Minimum attention controllers are preferable in modern shared implementations of cyber-physical systems because they produce the least burden on system resources such as processor time or communication bandwidth. We give algorithms to compute minimum attention controllers for ω-regular objectives in imperfect information discrete two-player games. We show a polynomial-time reduction from minimum attention controller synthesis to synthesis of controllers for mean-payoff parity objectives in games of incomplete information. This gives an optimal EXPTIME-complete synthesis algorithm. We show that the minimum attention controller problem is decidable for infinite state systems with finite bisimulation quotients. In particular, the problem is decidable for timed and rectangular automata.}, author = {Chatterjee, Krishnendu and Majumdar, Ritankar}, editor = {Fahrenberg, Uli and Tripakis, Stavros}, location = {Aalborg, Denmark}, pages = {145 -- 159}, publisher = {Springer}, title = {{Minimum attention controller synthesis for omega regular objectives}}, doi = {10.1007/978-3-642-24310-3_11}, volume = {6919}, year = {2011}, } @inproceedings{3351, abstract = {In two-player games on graph, the players construct an infinite path through the game graph and get a reward computed by a payoff function over infinite paths. Over weighted graphs, the typical and most studied payoff functions compute the limit-average or the discounted sum of the rewards along the path. Besides their simple definition, these two payoff functions enjoy the property that memoryless optimal strategies always exist. In an attempt to construct other simple payoff functions, we define a class of payoff functions which compute an (infinite) weighted average of the rewards. This new class contains both the limit-average and the discounted sum functions, and we show that they are the only members of this class which induce memoryless optimal strategies, showing that there is essentially no other simple payoff functions.}, author = {Chatterjee, Krishnendu and Doyen, Laurent and Singh, Rohit}, editor = {Owe, Olaf and Steffen, Martin and Telle, Jan Arne}, location = {Oslo, Norway}, pages = {148 -- 159}, publisher = {Springer}, title = {{On memoryless quantitative objectives}}, doi = {10.1007/978-3-642-22953-4_13}, volume = {6914}, year = {2011}, } @inproceedings{3357, abstract = {We consider two-player graph games whose objectives are request-response condition, i.e conjunctions of conditions of the form "if a state with property Rq is visited, then later a state with property Rp is visited". The winner of such games can be decided in EXPTIME and the problem is known to be NP-hard. In this paper, we close this gap by showing that this problem is, in fact, EXPTIME-complete. We show that the problem becomes PSPACE-complete if we only consider games played on DAGs, and NP-complete or PTIME-complete if there is only one player (depending on whether he wants to enforce or spoil the request-response condition). We also present near-optimal bounds on the memory needed to design winning strategies for each player, in each case.}, author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Horn, Florian}, editor = {Dediu, Adrian-Horia and Inenaga, Shunsuke and Martín-Vide, Carlos}, location = {Tarragona, Spain}, pages = {227 -- 237}, publisher = {Springer}, title = {{The complexity of request-response games}}, doi = {10.1007/978-3-642-21254-3_17}, volume = {6638}, year = {2011}, } @inproceedings{3361, abstract = {In this paper, we investigate the computational complexity of quantitative information flow (QIF) problems. Information-theoretic quantitative relaxations of noninterference (based on Shannon entropy)have been introduced to enable more fine-grained reasoning about programs in situations where limited information flow is acceptable. The QIF bounding problem asks whether the information flow in a given program is bounded by a constant $d$. Our first result is that the QIF bounding problem is PSPACE-complete. The QIF memoryless synthesis problem asks whether it is possible to resolve nondeterministic choices in a given partial program in such a way that in the resulting deterministic program, the quantitative information flow is bounded by a given constant $d$. Our second result is that the QIF memoryless synthesis problem is also EXPTIME-complete. The QIF memoryless synthesis problem generalizes to QIF general synthesis problem which does not impose the memoryless requirement (that is, by allowing the synthesized program to have more variables then the original partial program). Our third result is that the QIF general synthesis problem is EXPTIME-hard.}, author = {Cerny, Pavol and Chatterjee, Krishnendu and Henzinger, Thomas A}, location = {Cernay-la-Ville, France}, pages = {205 -- 217}, publisher = {IEEE}, title = {{The complexity of quantitative information flow problems}}, doi = {10.1109/CSF.2011.21}, year = {2011}, } @unpublished{3363, abstract = {We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, Büchi, coBüchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words.}, author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Tracol, Mathieu}, pages = {19}, publisher = {ArXiv}, title = {{The decidability frontier for probabilistic automata on infinite words}}, doi = {10.48550/arXiv.1104.0127}, year = {2011}, } @inproceedings{3365, abstract = {We present the tool Quasy, a quantitative synthesis tool. Quasy takes qualitative and quantitative specifications and automatically constructs a system that satisfies the qualitative specification and optimizes the quantitative specification, if such a system exists. The user can choose between a system that satisfies and optimizes the specifications (a) under all possible environment behaviors or (b) under the most-likely environment behaviors given as a probability distribution on the possible input sequences. Quasy solves these two quantitative synthesis problems by reduction to instances of 2-player games and Markov Decision Processes (MDPs) with quantitative winning objectives. Quasy can also be seen as a game solver for quantitative games. Most notable, it can solve lexicographic mean-payoff games with 2 players, MDPs with mean-payoff objectives, and ergodic MDPs with mean-payoff parity objectives.}, author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Jobstmann, Barbara and Singh, Rohit}, location = {Saarbrucken, Germany}, pages = {267 -- 271}, publisher = {Springer}, title = {{QUASY: quantitative synthesis tool}}, doi = {10.1007/978-3-642-19835-9_24}, volume = {6605}, year = {2011}, } @inproceedings{3366, abstract = {We present an algorithmic method for the quantitative, performance-aware synthesis of concurrent programs. The input consists of a nondeterministic partial program and of a parametric performance model. The nondeterminism allows the programmer to omit which (if any) synchronization construct is used at a particular program location. The performance model, specified as a weighted automaton, can capture system architectures by assigning different costs to actions such as locking, context switching, and memory and cache accesses. The quantitative synthesis problem is to automatically resolve the nondeterminism of the partial program so that both correctness is guaranteed and performance is optimal. As is standard for shared memory concurrency, correctness is formalized "specification free", in particular as race freedom or deadlock freedom. For worst-case (average-case) performance, we show that the problem can be reduced to 2-player graph games (with probabilistic transitions) with quantitative objectives. While we show, using game-theoretic methods, that the synthesis problem is Nexp-complete, we present an algorithmic method and an implementation that works efficiently for concurrent programs and performance models of practical interest. We have implemented a prototype tool and used it to synthesize finite-state concurrent programs that exhibit different programming patterns, for several performance models representing different architectures. }, author = {Cerny, Pavol and Chatterjee, Krishnendu and Henzinger, Thomas A and Radhakrishna, Arjun and Singh, Rohit}, editor = {Gopalakrishnan, Ganesh and Qadeer, Shaz}, location = {Snowbird, USA}, pages = {243 -- 259}, publisher = {Springer}, title = {{Quantitative synthesis for concurrent programs}}, doi = {10.1007/978-3-642-22110-1_20}, volume = {6806}, year = {2011}, } @article{3315, abstract = {We consider two-player games played in real time on game structures with clocks where the objectives of players are described using parity conditions. 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 play 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., not concurrent) finite-state (i.e., untimed) parity games. 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. The states of the resulting game are based on clock regions of the original game, and the state space of the finite game is linear in the size of the region graph. 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 winning strategies. Using a limit-robust winning 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 winning. We show that exact strategies are more powerful than limit-robust strategies, which are more powerful than bounded-robust winning 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.}, author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Prabhu, Vinayak}, journal = {Logical Methods in Computer Science}, number = {4}, publisher = {International Federation for Computational Logic}, title = {{Timed parity games: Complexity and robustness}}, doi = {10.2168/LMCS-7(4:8)2011}, volume = {7}, year = {2011}, } @inproceedings{3356, abstract = {There is recently a significant effort to add quantitative objectives to formal verification and synthesis. We introduce and investigate the extension of temporal logics with quantitative atomic assertions, aiming for a general and flexible framework for quantitative-oriented specifications. In the heart of quantitative objectives lies the accumulation of values along a computation. It is either the accumulated summation, as with the energy objectives, or the accumulated average, as with the 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 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 of time. We also allow the path-accumulation assertions LimInfAvg(v) ≥ c and LimSupAvg(v) ≥ c, referring to the average value along an entire computation. We study the border of decidability for 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 by prefix-accumulation assertions and extending LTL with path-accumulation assertions, result in temporal logics whose model-checking problem is decidable. The extended logics allow to significantly extend the currently known energy and mean-payoff objectives. Moreover, the prefix-accumulation assertions may be refined 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 the fragment we point to is, in a sense, the maximal logic whose extension with prefix-accumulation assertions permits a decidable model-checking procedure. Extending a temporal logic that has the EG or EU modalities, and in particular CTL and LTL, makes the problem undecidable.}, author = {Boker, Udi and Chatterjee, Krishnendu and Henzinger, Thomas A and Kupferman, Orna}, location = {Toronto, Canada}, publisher = {IEEE}, title = {{Temporal specifications with accumulative values}}, doi = {10.1109/LICS.2011.33}, year = {2011}, } @misc{5385, abstract = {There is recently a significant effort to add quantitative objectives to formal verification and synthesis. We introduce and investigate the extension of temporal logics with quantitative atomic assertions, aiming for a general and flexible framework for quantitative-oriented specifications. In the heart of quantitative objectives lies the accumulation of values along a computation. It is either the accumulated summation, as with the energy objectives, or the accumulated average, as with the 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 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 of time. We also allow the path-accumulation assertions LimInfAvg(v) ≥ c and LimSupAvg(v) ≥ c, referring to the average value along an entire computation. We study the border of decidability for 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 by prefix-accumulation assertions and extending LTL with path-accumulation assertions, result in temporal logics whose model-checking problem is decidable. The extended logics allow to significantly extend the currently known energy and mean-payoff objectives. Moreover, the prefix-accumulation assertions may be refined 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 the fragment we point to is, in a sense, the maximal logic whose extension with prefix-accumulation assertions permits a decidable model-checking procedure. Extending a temporal logic that has the EG or EU modalities, and in particular CTL and LTL, makes the problem undecidable.}, author = {Boker, Udi and Chatterjee, Krishnendu and Henzinger, Thomas A and Kupferman, Orna}, issn = {2664-1690}, pages = {14}, publisher = {IST Austria}, title = {{Temporal specifications with accumulative values}}, doi = {10.15479/AT:IST-2011-0003}, year = {2011}, } @misc{5381, abstract = {In two-player finite-state stochastic games of partial obser- vation on graphs, in every state of the graph, the players simultaneously choose an action, and their joint actions determine a probability distri- bution 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 pos- itively (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 perfect observation we show that (in contrast to full randomized strategies) belief-based (subset-construction based) strate- gies are not sufficient, and present an exponential upper bound on mem- ory 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 algo- rithms that avoid the explicit exponential construction. (2) For one-sided games with player 1 perfect observation we show that non-elementary memory is both necessary and sufficient for both almost-sure and posi- tive 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 non-elementary 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 re- sult exhibit serious flaws in previous results in the literature: we show a non-elementary memory lower bound for almost-sure winning whereas an exponential upper bound was previously claimed.}, author = {Chatterjee, Krishnendu and Doyen, Laurent}, issn = {2664-1690}, pages = {43}, publisher = {IST Austria}, title = {{Partial-observation stochastic games: How to win when belief fails}}, doi = {10.15479/AT:IST-2011-0007}, year = {2011}, } @article{3354, abstract = {We consider two-player games played on a finite state space for an infinite number of rounds. The games are concurrent: in each round, the two players (player 1 and player 2) choose their moves independently and simultaneously; the current state and the two moves determine the successor state. We consider ω-regular winning conditions specified as parity objectives. Both players are allowed to use randomization when choosing their moves. We study the computation of the limit-winning set of states, consisting of the states where the sup-inf value of the game for player 1 is 1: in other words, a state is limit-winning if player 1 can ensure a probability of winning arbitrarily close to 1. We show that the limit-winning set can be computed in O(n2d+2) time, where n is the size of the game structure and 2d is the number of priorities (or colors). The membership problem of whether a state belongs to the limit-winning set can be decided in NP ∩ coNP. While this complexity is the same as for the simpler class of turn-based parity games, where in each state only one of the two players has a choice of moves, our algorithms are considerably more involved than those for turn-based games. This is because concurrent games do not satisfy two of the most fundamental properties of turn-based parity games. First, in concurrent games limit-winning strategies require randomization; and second, they require infinite memory.}, author = {Chatterjee, Krishnendu and De Alfaro, Luca and Henzinger, Thomas A}, journal = {ACM Transactions on Computational Logic}, number = {4}, publisher = {ACM}, title = {{Qualitative concurrent parity games}}, doi = {10.1145/1970398.1970404}, volume = {12}, year = {2011}, } @inproceedings{4388, abstract = {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.}, author = {Chatterjee, Krishnendu and Henzinger, Thomas A and Jobstmann, Barbara and Radhakrishna, Arjun}, location = {Edinburgh, UK}, pages = {665 -- 669}, publisher = {Springer}, title = {{GIST: A solver for probabilistic games}}, doi = {10.1007/978-3-642-14295-6_57}, volume = {6174}, year = {2010}, } @inproceedings{489, abstract = {Graph games of infinite length are a natural model for open reactive processes: one player represents the controller, trying to ensure a given specification, and the other represents a hostile environment. The evolution of the system depends on the decisions of both players, supplemented by chance. In this work, we focus on the notion of randomised strategy. More specifically, we show that three natural definitions may lead to very different results: in the most general cases, an almost-surely winning situation may become almost-surely losing if the player is only allowed to use a weaker notion of strategy. In more reasonable settings, translations exist, but they require infinite memory, even in simple cases. Finally, some traditional problems becomes undecidable for the strongest type of strategies.}, author = {Cristau, Julien and David, Claire and Horn, Florian}, booktitle = {Proceedings of GandALF 2010}, location = {Minori, Amalfi Coast, Italy}, pages = {30 -- 39}, publisher = {Open Publishing Association}, title = {{How do we remember the past in randomised strategies?}}, doi = {10.4204/EPTCS.25.7}, volume = {25}, year = {2010}, } @misc{5388, abstract = {We present an algorithmic method for the synthesis of concurrent programs that are optimal with respect to quantitative performance measures. The input consists of a sequential sketch, that is, a program that does not contain synchronization constructs, and of a parametric performance model that assigns costs to actions such as locking, context switching, and idling. The quantitative synthesis problem is to automatically introduce synchronization constructs into the sequential sketch so that both correctness is guaranteed and worst-case (or average-case) performance is optimized. Correctness is formalized as race freedom or linearizability. We show that for worst-case performance, the problem can be modeled as a 2-player graph game with quantitative (limit-average) objectives, and for average-case performance, as a 2 1/2 -player graph game (with probabilistic transitions). In both cases, the optimal correct program is derived from an optimal strategy in the corresponding quantitative game. We prove that the respective game problems are computationally expensive (NP-complete), and present several techniques that overcome the theoretical difficulty in cases of concurrent programs of practical interest. We have implemented a prototype tool and used it for the automatic syn- thesis of programs that access a concurrent list. For certain parameter val- ues, our method automatically synthesizes various classical synchronization schemes for implementing a concurrent list, such as fine-grained locking or a lazy algorithm. For other parameter values, a new, hybrid synchronization style is synthesized, which uses both the lazy approach and coarse-grained locks (instead of standard fine-grained locks). The trade-off occurs because while fine-grained locking tends to decrease the cost that is due to waiting for locks, it increases cache size requirements.}, author = {Chatterjee, Krishnendu and Cerny, Pavol and Henzinger, Thomas A and Radhakrishna, Arjun and Singh, Rohit}, issn = {2664-1690}, pages = {17}, publisher = {IST Austria}, title = {{Quantitative synthesis for concurrent programs}}, doi = {10.15479/AT:IST-2010-0004}, year = {2010}, }