Undecidability of Two-dimensional Robot Games

Robot game is a two-player vector addition game played on the integer lattice $\mathbb{Z}^n$. 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

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

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted automata operating on infinite words and prove that the universality problem is undecidable for this new class of weighted automata. We show that the universality problem is undecidable by using a non-standard encoding of the infinite Post correspondence problem

Reachability games and related matrix and word problems

In this thesis, we study different two-player zero-sum games, where one player, called Eve, has a reachability objective (i.e., aims to reach a particular configuration) and the other, called Adam, has a safety objective (i.e., aims to avoid the configuration). We study a general class of games, called Attacker-Defender games, where the computational environment can vary from as simple as the integer line to n-dimensional topological braids. Similarly, the moves themselves can be simple vector addition or linear transformations defined by matrices. The main computational problem is to decide whether Eve has a winning strategy to reach the target configuration from the initial configuration, or whether the dual holds, that is, whether Adam can ensure that the target is never reached. The notion of a winning strategy is widely used in game semantics and its existence means that the player can ensure that his or her winning conditions are met, regardless of the actions of the opponent. It general, games provide a powerful framework to model and analyse interactive processes with uncontrollable adversaries. We formulated several Attacker-Defender games played on different mathematical domains with different transformations (moves), and identified classes of games, where the checking for existence of a winning strategy is undecidable. In other classes, where the problem is decidable, we established their computational complexity. In the thesis, we investigate four classes of games where determining the winner is undecidable: word games, where the players' moves are words over a group alphabet together with integer weights or where the moves are pairs of words over group alphabets; matrix games on vectors, where players transform a three-dimensional vector by linear transformations defined by 3×3 integer matrices; braid games, where players braid and unbraid a given braid; and last, but not least, games played on two-dimensional Z-VAS, closing the gap between decidable and undecidable cases and answering an existing open problem of the field. We also identified decidable fragments, such as word games, where the moves are over a single group alphabet, games on one-dimensional Z-VASS. For word games, we provide an upper-bound of EXPTIME , while for games on Z-VASS, tight bounds of EXPTIME-complete or EXPSPACE-complete, depending on the state structure. We also investigate single-player systems such as polynomial iteration and identity problem in matrix semigroups. We show that the reachability problem for polynomial iteration is PSPACE-complete while the identity problem for the Heisenberg group is in PTIME for dimension three and in EXPTIME for higher dimensions