11 research outputs found
Perfect half space games
We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multidimensional weights diverge in a direction which is consistent with a chosen sequence of perfect half spaces (chosen dynamically by Player 2). We establish that the bounding games of Jurdzinski et al. (ICALP 2015) can be reduced to perfect half space games, which in turn can be translated to the lexicographic energy games of Colcombet and Niwinski, and are positionally determined in a strong sense (Player 2 can play without knowing the current perfect half space). We finally show how perfect half space games and bounding games can be employed to solve multidimensional energy parity games in pseudo-polynomial time when both the numbers of energy dimensions and of priorities are fixed, regardless of whether the initial credit is given as part of the input or existentially quantified. This also yields an optimal 2-EXPTIME complexity with given initial credit, where the best known upper bound was non-elementary
Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS
Vector Addition Systems with States (VASS) provide a well-known and
fundamental model for the analysis of concurrent processes, parameterized
systems, and are also used as abstract models of programs in resource bound
analysis. In this paper we study the problem of obtaining asymptotic bounds on
the termination time of a given VASS. In particular, we focus on the
practically important case of obtaining polynomial bounds on termination time.
Our main contributions are as follows: First, we present a polynomial-time
algorithm for deciding whether a given VASS has a linear asymptotic complexity.
We also show that if the complexity of a VASS is not linear, it is at least
quadratic. Second, we classify VASS according to quantitative properties of
their cycles. We show that certain singularities in these properties are the
key reason for non-polynomial asymptotic complexity of VASS. In absence of
singularities, we show that the asymptotic complexity is always polynomial and
of the form , for some integer , where is the
dimension of the VASS. We present a polynomial-time algorithm computing the
optimal . For general VASS, the same algorithm, which is based on a complete
technique for the construction of ranking functions in VASS, produces a valid
lower bound, i.e., a such that the termination complexity is .
Our results are based on new insights into the geometry of VASS dynamics, which
hold the potential for further applicability to VASS analysis.Comment: arXiv admin note: text overlap with arXiv:1708.0925
LNCS
We study turn-based stochastic zero-sum games with lexicographic preferences over reachability and safety objectives. Stochastic games are standard models in control, verification, and synthesis of stochastic reactive systems that exhibit both randomness as well as angelic and demonic non-determinism. Lexicographic order allows to consider multiple objectives with a strict preference order over the satisfaction of the objectives. To the best of our knowledge, stochastic games with lexicographic objectives have not been studied before. We establish determinacy of such games and present strategy and computational complexity results. For strategy complexity, we show that lexicographically optimal strategies exist that are deterministic and memory is only required to remember the already satisfied and violated objectives. For a constant number of objectives, we show that the relevant decision problem is in NP∩coNP , matching the current known bound for single objectives; and in general the decision problem is PSPACE -hard and can be solved in NEXPTIME∩coNEXPTIME . We present an algorithm that computes the lexicographically optimal strategies via a reduction to computation of optimal strategies in a sequence of single-objectives games. We have implemented our algorithm and report experimental results on various case studies
Playing with Repetitions in Data Words Using Energy Games
We introduce two-player games which build words over infinite alphabets, and
we study the problem of checking the existence of winning strategies. These
games are played by two players, who take turns in choosing valuations for
variables ranging over an infinite data domain, thus generating
multi-attributed data words. The winner of the game is specified by formulas in
the Logic of Repeating Values, which can reason about repetitions of data
values in infinite data words. We prove that it is undecidable to check if one
of the players has a winning strategy, even in very restrictive settings.
However, we prove that if one of the players is restricted to choose valuations
ranging over the Boolean domain, the games are effectively equivalent to
single-sided games on vector addition systems with states (in which one of the
players can change control states but cannot change counter values), known to
be decidable and effectively equivalent to energy games.
Previous works have shown that the satisfiability problem for various
variants of the logic of repeating values is equivalent to the reachability and
coverability problems in vector addition systems. Our results raise this
connection to the level of games, augmenting further the associations between
logics on data words and counter systems
Stochastic Games with Lexicographic Reachability-Safety Objectives
We study turn-based stochastic zero-sum games with lexicographic preferences
over reachability and safety objectives. Stochastic games are standard models
in control, verification, and synthesis of stochastic reactive systems that
exhibit both randomness as well as angelic and demonic non-determinism.
Lexicographic order allows to consider multiple objectives with a strict
preference order over the satisfaction of the objectives. To the best of our
knowledge, stochastic games with lexicographic objectives have not been studied
before. We establish determinacy of such games and present strategy and
computational complexity results. For strategy complexity, we show that
lexicographically optimal strategies exist that are deterministic and memory is
only required to remember the already satisfied and violated objectives. For a
constant number of objectives, we show that the relevant decision problem is in
NP coNP, matching the current known bound for single objectives; and in
general the decision problem is PSPACE-hard and can be solved in NEXPTIME
coNEXPTIME. We present an algorithm that computes the lexicographically
optimal strategies via a reduction to computation of optimal strategies in a
sequence of single-objectives games. We have implemented our algorithm and
report experimental results on various case studies.Comment: Full version (33 pages) of CAV20 conference paper; including an
appendix with technical proof
On the complexity of resource-bounded logics
We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB+-ATL is 2EXPTIME-complete by using recent results on alternating VASS (and in EXPTIME when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is EXPSPACE-complete. The problem is decidable and of the same complexity for RBTL*, proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in PSPACE. We also establish that the model checking problem for RB+-ATL*, the extension of RB+-ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations
On decidability and complexity of low-dimensional robot games
A robot game, also known as a Z-VAS game, is a two-player vector addition game played on the integer lattice Zn, where one of the players, Adam, aims to avoid the origin while the other player, Eve, aims to reach the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the two-dimensional robot game closing the gap between undecidable and decidable cases. We also prove that deciding the winner in a robot game with states in dimension one is EXPSPACE-complete and study a subclass of robot games where deciding the winner is in EXPTIME