arXiv subjects

Martin Zimmermann

Publications and source records attributed to Martin Zimmermann.

At least 73 records · Page 4Linked to original sources

arXiv2017

Games with Costs and Delays

We demonstrate the usefulness of adding delay to infinite games with quantitative winning conditions. In a delay game, one of the players may delay her moves to obtain a lookahead on her opponent's moves. We show that determining the winner of delay games with winning conditions given by parity automata with costs is EXPTIME-complete and that exponential bounded lookahead is both sufficient and in general necessary. Thus, although the parity condition with costs is a quantitative extension of the parity condition, our results show that adding costs does not increase the complexity of delay games with parity conditions. Furthermore, we study a new phenomenon that appears in quantitative delay games: lookahead can be traded for the quality of winning strategies and vice versa. We determine the extent of this tradeoff. In particular, even the smallest lookahead allows to improve the quality of an optimal strategy from the worst possible value to almost the smallest possible one. Thus, the benefit of introducing lookahead is twofold: not only does it allow the delaying player to win games she would lose without, but lookahead also allows her to improve the quality of her winning strategies in games she wins even without lookahead.

cs.GT↗

arXiv2017

Bounding Average-energy Games

We consider average-energy games, where the goal is to minimize the long-run average of the accumulated energy. While several results have been obtained on these games recently, decidability of average-energy games with a lower-bound constraint on the energy level (but no upper bound) remained open; in particular, so far there was no known upper bound on the memory that is required for winning strategies. By reducing average-energy games with lower-bounded energy to infinite-state mean-payoff games and analyzing the density of low-energy configurations, we show an almost tight doubly-exponential upper bound on the necessary memory, and that the winner of average-energy games with lower-bounded energy can be determined in doubly-exponential time. We also prove EXPSPACE-hardness of this problem. Finally, we consider multi-dimensional extensions of all types of average-energy games: without bounds, with only a lower bound, and with both a lower and an upper bound on the energy. We show that the fully-bounded version is the only case to remain decidable in multiple dimensions.

cs.LO↗

arXiv2017

The First-Order Logic of Hyperproperties

We investigate the logical foundations of hyperproperties. Hyperproperties generalize trace properties, which are sets of traces, to sets of sets of traces. The most prominent application of hyperproperties is information flow security: information flow policies characterize the secrecy and integrity of a system by comparing two or more execution traces, for example by comparing the observations made by an external observer on execution traces that result from different values of a secret variable. In this paper, we establish the first connection between temporal logics for hyperproperties and first-order logic. Kamp's seminal theorem (in the formulation due to Gabbay et al.) states that linear-time temporal logic (LTL) is expressively equivalent to first-order logic over the natural numbers with order. We introduce first-order logic over sets of traces and prove that HyperLTL, the extension of LTL to hyperproperties, is strictly subsumed by this logic. We furthermore exhibit a fragment that is expressively equivalent to HyperLTL, thereby establishing Kamp's theorem for hyperproperties.

cs.LO↗

arXiv2016

Limit Your Consumption! Finding Bounds in Average-energy Games

Energy games are infinite two-player games played in weighted arenas with quantitative objectives that restrict the consumption of a resource modeled by the weights, e.g., a battery that is charged and drained. Typically, upper and/or lower bounds on the battery capacity are part of the problem description. Here, we consider the problem of determining upper bounds on the average accumulated energy or on the capacity while satisfying a given lower bound, i.e., we do not determine whether a given bound is sufficient to meet the specification, but if there exists a sufficient bound to meet it. In the classical setting with positive and negative weights, we show that the problem of determining the existence of a sufficient bound on the long-run average accumulated energy can be solved in doubly-exponential time. Then, we consider recharge games: here, all weights are negative, but there are recharge edges that recharge the energy to some fixed capacity. We show that bounding the long-run average energy in such games is complete for exponential time. Then, we consider the existential version of the problem, which turns out to be solvable in polynomial time: here, we ask whether there is a recharge capacity that allows the system player to win the game. We conclude by studying tradeoffs between the memory needed to implement strategies and the bounds they realize. We give an example showing that memory can be traded for bounds and vice versa. Also, we show that increasing the capacity allows to lower the average accumulated energy.

cs.GT↗

arXiv2016

Prompt Delay

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. Recently, such games with quantitative winning conditions in weak MSO with the unbounding quantifier were studied, but their properties turned out to be unsatisfactory. In particular, unbounded lookahead is in general necessary. Here, we study delay games with winning conditions given by Prompt-LTL, Linear Temporal Logic equipped with a parameterized eventually operator whose scope is bounded. Our main result shows that solving Prompt-LTL delay games is complete for triply-exponential time. Furthermore, we give tight triply-exponential bounds on the necessary lookahead and on the scope of the parameterized eventually operator. Thus, we identify Prompt-LTL as the first known class of well-behaved quantitative winning conditions for delay games. Finally, we show that applying our techniques to delay games with ω-regular winning conditions answers open questions in the cases where the winning conditions are given by non-deterministic, universal, or alternating automata.

cs.GT↗

arXiv2016

Distributed PROMPT-LTL Synthesis

We consider the synthesis of distributed implementations for specifications in Prompt Linear Temporal Logic (PROMPT-LTL), which extends LTL by temporal operators equipped with parameters that bound their scope. For single process synthesis it is well-established that such parametric extensions do not increase worst-case complexities. For synchronous systems, we show that, despite being more powerful, the distributed realizability problem for PROMPT-LTL is not harder than its LTL counterpart. For asynchronous systems we have to consider an assume-guarantee synthesis problem, as we have to express scheduling assumptions. As asynchronous distributed synthesis is already undecidable for LTL, we give a semi-decision procedure for the PROMPT-LTL assume-guarantee synthesis problem based on bounded synthesis.

cs.LO↗

arXiv2016

Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time

We consider the optimization variant of the realizability problem for Prompt Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the prompt eventually operator whose scope is bounded by some parameter. In the realizability optimization problem, one is interested in computing the minimal such bound that allows to realize a given specification. It is known that this problem is solvable in triply-exponential time, but not whether it can be done in doubly-exponential time, i.e., whether it is just as hard as solving LTL realizability. We take a step towards resolving this problem by showing that the optimum can be approximated within a factor of two in doubly-exponential time. Also, we report on a proof-of-concept implementation of the algorithm based on bounded LTL synthesis, which computes the smallest implementation of a given specification. In our experiments, we observe a tradeoff between the size of the implementation and the bound it realizes. We investigate this tradeoff in the general case and prove upper bounds, which reduce the search space for the algorithm, and matching lower bounds.

cs.LO↗

arXiv2016

Unbounded Lookahead in WMSO+U Games

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. We consider delay games with winning conditions expressed in weak monadic second order logic with the unbounding quantifier (WMSO+U), which is able to express (un)boundedness properties. It is decidable whether the delaying player is able to win such a game with bounded lookahead, i.e., if she only skips a finite number of moves. However, bounded lookahead is not always sufficient: we present a game that can be won with unbounded lookahead, but not with bounded lookahead. Then, we consider WMSO+U delay games with unbounded lookahead and show that the exact evolution of the lookahead is irrelevant: the winner is always the same, as long as the initial lookahead is large enough and the lookahead tends to infinity.

cs.GT↗

arXiv2016

Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL (full version)

We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all omega-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.

cs.LO↗

arXiv2015

Parametric Linear Dynamic Logic (full version)

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL itself was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all omega-regular specifications while still maintaining many of LTL's desirable properties like intuitive syntax and semantics and a translation into non-deterministic Büchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to express all omega-regular properties and that subsumes parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our main technical contribution is a translation of PLDL formulas into non-deterministic Büchi automata of exponential size via alternating automata. This yields polynomial space algorithms for model checking and assume-guarantee model checking and a realizability algorithm with doubly-exponential running time. All three problems are also shown to be complete for these complexity classes. Moreover, we give tight upper and lower bounds on optimal parameter values for model checking and realizability. Using these bounds, we present a polynomial space procedure for model checking optimization and an algorithm with triply-exponential running time for realizability optimization. Our results show that PLDL model checking and realizability are no harder than their respective (parametric) LTL counterparts.

cs.LO↗

arXiv2015

Delay Games with WMSO+U Winning Conditions

cs.GT↗

arXiv2015

Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL

cs.LO↗

arXiv2015

What are Strategies in Delay Games? Borel Determinacy for Games with Lookahead

We investigate determinacy of delay games with Borel winning conditions, infinite-duration two-player games in which one player may delay her moves to obtain a lookahead on her opponent's moves. First, we prove determinacy of such games with respect to a fixed evolution of the lookahead. However, strategies in such games may depend on information about the evolution. Thus, we introduce different notions of universal strategies for both players, which are evolution-independent, and determine the exact amount of information a universal strategy needs about the history of a play and the evolution of the lookahead to be winning. In particular, we show that delay games with Borel winning conditions are determined with respect to universal strategies. Finally, we consider decidability problems, e.g., "Does a player have a universal winning strategy for delay games with a given winning condition?", for omega-regular and omega-context-free winning conditions.

cs.LO↗

arXiv2014

The Complexity of Counting Models of Linear-time Temporal Logic

We determine the complexity of counting models of bounded size of specifications expressed in Linear-time Temporal Logic. Counting word models is #P-complete, if the bound is given in unary, and as hard as counting accepting runs of nondeterministic polynomial space Turing machines, if the bound is given in binary. Counting tree models is as hard as counting accepting runs of nondeterministic exponential time Turing machines, if the bound is given in unary. For a binary encoding of the bound, the problem is at least as hard as counting accepting runs of nondeterministic exponential space Turing machines. On the other hand, it is not harder than counting accepting runs of nondeterministic doubly-exponential time Turing machines.

cs.LO↗

arXiv2014

Parametric Linear Dynamic Logic

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all $ω$-regular specifications while still maintaining many of LTL's desirable properties like an intuitive syntax and a translation into non-deterministic Büchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to express all $ω$-regular properties and that subsumes parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our main technical contribution is a translation of PLDL formulas into non-deterministic Büchi word automata of exponential size via alternating automata. This yields a PSPACE model checking algorithm and a realizability algorithm with doubly-exponential running time. Furthermore, we give tight upper and lower bounds on optimal parameter values for both problems. These results show that PLDL model checking and realizability are not harder than LTL model checking and realizability.

cs.LO↗

arXiv2014

Parity and Streett Games with Costs

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical omega-regular conditions and the corresponding finitary conditions. For parity games with costs we show that the first player has positional winning strategies and that determining the winner lies in NP and coNP. For Streett games with costs we show that the first player has finite-state winning strategies and that determining the winner is EXPTIME-complete. The second player might need infinite memory in both games. Both types of games with costs can be solved by solving linearly many instances of their classical variants.

cs.LO↗

arXiv2014

Optimal Strategy Synthesis for Request-Response Games

We show the existence and effective computability of optimal winning strategies for request-response games in case the quality of a play is measured by the limit superior of the mean accumulated waiting times between requests and their responses.

cs.FL↗

arXiv2012

Down the Borel Hierarchy: Solving Muller Games via Safety Games

We transform a Muller game with n vertices into a safety game with (n!)^3 vertices whose solution allows to determine the winning regions of the Muller game and to compute a finite-state winning strategy for one player. This yields a novel antichain-based memory structure and a natural notion of permissive strategies for Muller games. Moreover, we generalize our construction by presenting a new type of game reduction from infinite games to safety games and show its applicability to several other winning conditions.

cs.LO↗