12 research outputs found
Undecidability of Two-dimensional Robot Games
Robot game is a two-player vector addition game played on the integer lattice
. Both players have sets of vectors and in each turn the vector
chosen by a player is added to the current configuration vector of the game.
One of the players, called Eve, tries to play the game from the initial
configuration to the origin while the other player, Adam, tries to avoid the
origin. The problem is to decide whether or not Eve has a winning strategy. In
this paper we prove undecidability of the robot game in dimension two answering
the question formulated by Doyen and Rabinovich in 2011 and closing the gap
between undecidable and decidable cases
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 prove 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.SCOPUS: cp.kinfo:eu-repo/semantics/publishe
LNCS
In the analysis of reactive systems a quantitative objective assigns a real value to every trace of the system. The value decision problem for a quantitative objective requires a trace whose value is at least a given threshold, and the exact value decision problem requires a trace whose value is exactly the threshold. We compare the computational complexity of the value and exact value decision problems for classical quantitative objectives, such as sum, discounted sum, energy, and mean-payoff for two standard models of reactive systems, namely, graphs and graph games
Revisiting Synthesis for One-Counter Automata
We study the (parameter) synthesis problem for one-counter automata with
parameters. One-counter automata are obtained by extending classical
finite-state automata with a counter whose value can range over non-negative
integers and be tested for zero. The updates and tests applicable to the
counter can further be made parametric by introducing a set of integer-valued
variables called parameters. The synthesis problem for such automata asks
whether there exists a valuation of the parameters such that all infinite runs
of the automaton satisfy some omega-regular property. Lechner showed that (the
complement of) the problem can be encoded in a restricted one-alternation
fragment of Presburger arithmetic with divisibility. In this work (i) we argue
that said fragment, called AERPADPLUS, is unfortunately undecidable.
Nevertheless, by a careful re-encoding of the problem into a decidable
restriction of AERPADPLUS, (ii) we prove that the synthesis problem is
decidable in general and in N2EXP for several fixed omega-regular properties.
Finally, (iii) we give a polynomial-space algorithm for the special case of the
problem where parameters can only be used in tests, and not updates, of the
counter
Countdown games, and simulation on (succinct) one-counter nets
We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS
2016) [HLMT16] for simulation preorder of succinct one-counter nets (i.e.,
one-counter automata with no zero tests where counter increments and decrements
are integers written in binary), by showing that all relations between
bisimulation equivalence and simulation preorder are EXPSPACE-hard for these
nets. We describe a reduction from reachability games whose
EXPSPACE-completeness in the case of succinct one-counter nets was shown by
Hunter [RP 2015], by using other results. We also provide a direct
self-contained EXPSPACE-completeness proof for a special case of such
reachability games, namely for a modification of countdown games that were
shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie [LMCS 2008]; in our
modification the initial counter value is not given but is freely chosen by the
first player. We also present a new simplified proof of the belt theorem that
gives a simple graphic presentation of simulation preorder on one-counter nets
and leads to a polynomial-space algorithm; it is an alternative to the proof
from [HLMT16].Comment: A part of this paper elaborates arxiv-paper 1801.01073 and the
related paper presented at Reachability Problems 201
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
Reachability games with relaxed energy constraints
We study games with reachability objectives under energy constraints. We first prove that under strict energy constraints (either only lower-bound constraint or interval constraint), those games are LOGSPACE-equivalent to energy games with the same energy constraints but without reachability objective (i.e., for infinite runs). We then consider two relaxations of the upper-bound constraints (while keeping the lower-bound constraint strict): in the first one, called weak upper bound, the upper bound is absorbing, i.e., when the upper bound is reached, the extra energy is not stored; in the second one, we allow for temporary violations of the upper bound, imposing limits on the number or on the amount of violations. We prove that when considering weak upper bound, reachability objectives require memory, but can still be solved in polynomial-time for one-player arenas ; we prove that they are in coNP in the two-player setting. Allowing for bounded violations makes the problem PSPACE-complete for one-player arenas and EXPTIME-complete for two players. We then address the problem of existence of bounds for a given arena. We show that with reachability objectives, existence can be a simpler problem than the game itself, and conversely that with infinite games, existence can be harder
Countdown games, and simulation on (succinct) one-counter nets
We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS
2016) for simulation preorder on the class of succinct one-counter nets (i.e.,
one-counter automata with no zero tests where counter increments and decrements
are integers written in binary); the problem was known to be PSPACE-hard and in
EXPSPACE. We show that all relations between bisimulation equivalence and
simulation preorder are EXPSPACE-hard for these nets; simulation preorder is
thus EXPSPACE-complete. The result is proven by a reduction from reachability
games whose EXPSPACE-completeness in the case of succinct one-counter nets was
shown by Hunter (RP 2015), by using other results. We also provide a direct
self-contained EXPSPACE-completeness proof for a special case of such
reachability games, namely for a modification of countdown games that were
shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie (LMCS 2008); in our
modification the initial counter value is not given but is freely chosen by the
first player. We also present an alternative proof for the upper bound by
Hofman et al. In particular, we give a new simplified proof of the belt theorem
that yields a simple graphic presentation of simulation preorder on
(non-succinct) one-counter nets and leads to a polynomial-space algorithm
(which is trivially extended to an exponential-space algorithm for succinct
one-counter nets)