Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Decomposing GR(1) Games with Singleton Liveness Guarantees for Efficient Synthesis

Temporal logic based synthesis approaches are often used to find trajectories that are correct-by-construction for tasks in systems with complex behavior. Some examples of such tasks include synchronization for multi-agent hybrid systems, reactive motion planning for robots. However, the scalability of such approaches is of concern and at times a bottleneck when transitioning from theory to practice. In this paper, we identify a class of problems in the GR(1) fragment of linear-time temporal logic (LTL) where the synthesis problem allows for a decomposition that enables easy parallelization. This decomposition also reduces the alternation depth, resulting in more efficient synthesis. A multi-agent robot gridworld example with coordination tasks is presented to demonstrate the application of the developed ideas and also to perform empirical analysis for benchmarking the decomposition-based synthesis approach

A Random Walk Perspective on Hide-and-Seek Games

We investigate hide-and-seek games on complex networks using a random walk framework. Specifically, we investigate the efficiency of various degree-biased random walk search strategies to locate items that are randomly hidden on a subset of vertices of a random graph. Vertices at which items are hidden in the network are chosen at random as well, though with probabilities that may depend on degree. We pitch various hide and seek strategies against each other, and determine the efficiency of search strategies by computing the average number of hidden items that a searcher will uncover in a random walk of $n$ steps. Our analysis is based on the cavity method for finite single instances of the problem, and generalises previous work of De Bacco et al. [1] so as to cover degree-biased random walks. We also extend the analysis to deal with the thermodynamic limit of infinite system size. We study a broad spectrum of functional forms for the degree bias of both the hiding and the search strategy and investigate the efficiency of families of search strategies for cases where their functional form is either matched or unmatched to that of the hiding strategy. Our results are in excellent agreement with those of numerical simulations. We propose two simple approximations for predicting efficient search strategies. One is based on an equilibrium analysis of the random walk search strategy. While not exact, it produces correct orders of magnitude for parameters characterising optimal search strategies. The second exploits the existence of an effective drift in random walks on networks, and is expected to be efficient in systems with low concentration of small degree nodes.Comment: 31 pages, 10 (multi-part) figure

Physics-based Motion Planning with Temporal Logic Specifications

One of the main foci of robotics is nowadays centered in providing a great degree of autonomy to robots. A fundamental step in this direction is to give them the ability to plan in discrete and continuous spaces to find the required motions to complete a complex task. In this line, some recent approaches describe tasks with Linear Temporal Logic (LTL) and reason on discrete actions to guide sampling-based motion planning, with the aim of finding dynamically-feasible motions that satisfy the temporal-logic task specifications. The present paper proposes an LTL planning approach enhanced with the use of ontologies to describe and reason about the task, on the one hand, and that includes physics-based motion planning to allow the purposeful manipulation of objects, on the other hand. The proposal has been implemented and is illustrated with didactic examples with a mobile robot in simple scenarios where some of the goals are occupied with objects that must be removed in order to fulfill the task.Comment: The 20th World Congress of the International Federation of Automatic Control, 9-14 July 201

Solving Imperfect Information Games Using Decomposition

Decomposition, i.e. independently analyzing possible subgames, has proven to be an essential principle for effective decision-making in perfect information games. However, in imperfect information games, decomposition has proven to be problematic. To date, all proposed techniques for decomposition in imperfect information games have abandoned theoretical guarantees. This work presents the first technique for decomposing an imperfect information game into subgames that can be solved independently, while retaining optimality guarantees on the full-game solution. We can use this technique to construct theoretically justified algorithms that make better use of information available at run-time, overcome memory or disk limitations at run-time, or make a time/space trade-off to overcome memory or disk limitations while solving a game. In particular, we present an algorithm for subgame solving which guarantees performance in the whole game, in contrast to existing methods which may have unbounded error. In addition, we present an offline game solving algorithm, CFR-D, which can produce a Nash equilibrium for a game that is larger than available storage.Comment: 7 pages by 2 columns, 5 figures; April 21 2014 - expand explanations and theor