Petri Games: Synthesis of Distributed Systems with Causal Memory

We present a new multiplayer game model for the interaction and the flow of information in a distributed system. The players are tokens on a Petri net. As long as the players move in independent parts of the net, they do not know of each other; when they synchronize at a joint transition, each player gets informed of the causal history of the other player. We show that for Petri games with a single environment player and an arbitrary bounded number of system players, deciding the existence of a safety strategy for the system players is EXPTIME-complete.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Practical Distributed Control Synthesis

Classic distributed control problems have an interesting dichotomy: they are either trivial or undecidable. If we allow the controllers to fully synchronize, then synthesis is trivial. In this case, controllers can effectively act as a single controller with complete information, resulting in a trivial control problem. But when we eliminate communication and restrict the supervisors to locally available information, the problem becomes undecidable. In this paper we argue in favor of a middle way. Communication is, in most applications, expensive, and should hence be minimized. We therefore study a solution that tries to communicate only scarcely and, while allowing communication in order to make joint decision, favors local decisions over joint decisions that require communication.Comment: In Proceedings INFINITY 2011, arXiv:1111.267

Cops and Robbers is EXPTIME-complete

We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point

The Complexity of Escaping Labyrinths and Enchanted Forests

The board games The aMAZEing Labyrinth (or simply Labyrinth for short) and Enchanted Forest published by Ravensburger are seemingly simple family games. In Labyrinth, the players move though a labyrinth in order to collect specific items. To do so, they shift the tiles making up the labyrinth in order to open up new paths (and, at the same time, close paths for their opponents). We show that, even without any opponents, determining a shortest path (i.e., a path using the minimum possible number of turns) to the next desired item in the labyrinth is strongly NP-hard. Moreover, we show that, when competing with another player, deciding whether there exists a strategy that guarantees to reach one\u27s next item faster than one\u27s opponent is PSPACE-hard. In Enchanted Forest, items are hidden under specific trees and the objective of the players is to report their locations to the king in his castle. Movements are performed by rolling two dice, resulting in two numbers of fields one has to move, where each of the two movements must be executed consecutively in one direction (but the player can choose the order in which the two movements are performed). Here, we provide an efficient polynomial-time algorithm for computing a shortest path between two fields on the board for a given sequence of die rolls, which also has implications for the complexity of problems the players face in the game when future die rolls are unknown

Fine-grained Lower Bounds on Cops and Robbers

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel\u27s conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time. In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game

Synthesis in Distributed Environments

Most approaches to the synthesis of reactive systems study the problem in terms of a two-player game with complete observation. In many applications, however, the system\u27s environment consists of several distinct entities, and the system must actively communicate with these entities in order to obtain information available in the environment. In this paper, we model such environments as a team of players and keep track of the information known to each individual player. This allows us to synthesize programs that interact with a distributed environment and leverage multiple interacting sources of information. The synthesis problem in distributed environments corresponds to solving a special class of Petri games, i.e., multi-player games played over Petri nets, where the net has a distinguished token representing the system and an arbitrary number of tokens representing the environment. While, in general, even the decidability of Petri games is an open question, we show that the synthesis problem in distributed environments can be solved in polynomial time for nets with up to two environment tokens. For an arbitrary but fixed number of three or more environment tokens, the problem is NP-complete. If the number of environment tokens grows with the size of the net, the problem is EXPTIME-complete

The complexity of searching implicit graphs

AbstractThe standard complexity classes of complexity theory do not allow for direct classification of most of the problems solved by heuristic search algorithms. The reason is that, almost always, these are defined in terms of implicit graphs of state or problem reduction spaces, while the standard definitions of all complexity classes are specifically tailored to explicit inputs.To allow for more precise comparisons with standard complexity classes, we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures. It is based on previously studied concepts of “succinct representation” techniques, and allows us to prove PSPACE-completeness or EXPTIME-completeness of specific, natural problems on implicit graphs, such as those solved by A∗, AO∗, and other best-first search strategies

Quantitative Games under Failures

We study a generalisation of sabotage games, a model of dynamic network games introduced by van Benthem. The original definition of the game is inherently finite and therefore does not allow one to model infinite processes. We propose an extension of the sabotage games in which the first player (Runner) traverses an arena with dynamic weights determined by the second player (Saboteur). In our model of quantitative sabotage games, Saboteur is now given a budget that he can distribute amongst the edges of the graph, whilst Runner attempts to minimise the quantity of budget witnessed while completing his task. We show that, on the one hand, for most of the classical cost functions considered in the literature, the problem of determining if Runner has a strategy to ensure a cost below some threshold is EXPTIME-complete. On the other hand, if the budget of Saboteur is fixed a priori, then the problem is in PTIME for most cost functions. Finally, we show that restricting the dynamics of the game also leads to better complexity