56 research outputs found
Minimal Proof Search for Modal Logic K Model Checking
Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K.
While the model checking problems for LTL and to a lesser extent ATL have been
very active research areas for the past decades, the model checking problem for
the more basic Multi-agent Modal Logic K (MMLK) has important applications as a
formal framework for perfect information multi-player games on its own.
We present Minimal Proof Search (MPS), an effort number based algorithm
solving the model checking problem for MMLK. We prove two important properties
for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal
cost for a general definition of cost, and MPS is an optimal algorithm for
finding (dis)proofs of minimal cost. Optimality means that any comparable
algorithm either needs to explore a bigger or equal state space than MPS, or is
not guaranteed to find a (dis)proof of minimal cost on every input.
As such, our work relates to A* and AO* in heuristic search, to Proof Number
Search and DFPN+ in two-player games, and to counterexample minimization in
software model checking.Comment: Extended version of the JELIA 2012 paper with the same titl
Positional Games and QBF: The Corrective Encoding
Positional games are a mathematical class of two-player games comprising
Tic-tac-toe and its generalizations. We propose a novel encoding of these games
into Quantified Boolean Formulas (QBF) such that a game instance admits a
winning strategy for first player if and only if the corresponding formula is
true. Our approach improves over previous QBF encodings of games in multiple
ways. First, it is generic and lets us encode other positional games, such as
Hex. Second, structural properties of positional games together with a careful
treatment of illegal moves let us generate more compact instances that can be
solved faster by state-of-the-art QBF solvers. We establish the latter fact
through extensive experiments. Finally, the compactness of our new encoding
makes it feasible to translate realistic game problems. We identify a few such
problems of historical significance and put them forward to the QBF community
as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on
Theory and Applications of Satisfiability Testing (SAT2020
Vision Transformers for Computer Go
Motivated by the success of transformers in various fields, such as language
understanding and image analysis, this investigation explores their application
in the context of the game of Go. In particular, our study focuses on the
analysis of the Transformer in Vision. Through a detailed analysis of numerous
points such as prediction accuracy, win rates, memory, speed, size, or even
learning rate, we have been able to highlight the substantial role that
transformers can play in the game of Go. This study was carried out by
comparing them to the usual Residual Networks
The parameterized complexity of positional games
We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterized by formula size. We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise. Short Maker-Maker, Short Maker-Breaker, and Short Enforcer-Avoider are respectively AW[*]-, W[1]-, and co-W[1]-complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the W-hierarchy when the winning condition only depends on which vertices one player has been able to pick, but AW[*]-complete when it depends on which vertices both players have picked. However, some positional games with highly structured board and winning configurations are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when parameterized by the number of moves
On Bellman's Optimality Principle for zs-POSGs
Many non-trivial sequential decision-making problems are efficiently solved
by relying on Bellman's optimality principle, i.e., exploiting the fact that
sub-problems are nested recursively within the original problem. Here we show
how it can apply to (infinite horizon) 2-player zero-sum partially observable
stochastic games (zs-POSGs) by (i) taking a central planner's viewpoint, which
can only reason on a sufficient statistic called occupancy state, and (ii)
turning such problems into zero-sum occupancy Markov games (zs-OMGs). Then,
exploiting the Lipschitz-continuity of the value function in occupancy space,
one can derive a version of the HSVI algorithm (Heuristic Search Value
Iteration) that provably finds an -Nash equilibrium in finite time.Comment: 18 pages, 0 figures, 1 algorith
Recherche heuristique pour jeux stochastiques (Ă somme nulle)
National audienceIn various types of problems, such as sequential decision-making, heuristic search algorithms allow exploiting the knowledge of the initial situation and of an admissible heuristic to efficiently search for an optimal solution. Such algorithms exist including in case of uncertain dynamics, of partial observability, of multiple criteria, or of multiple collaborating agents. Here we propose a heuristic search algorithm for two-player zero-sum stochastic games with discounted criterion. This algorithm relies on HSVI—hence on generating trajectories. We demonstrate that, each player acting in an optimistic manner, and employing simple heuristic initializations, the resulting algorithm converges in finite time to an-optimal solution.Dans divers types de problèmes, par exemple de prise de décision séquentielle, les algorithmes de recherche heuristique permettent d'exploiter la connaissance d'une situation initiale et d'une heuristique admissible pour rechercher efficacement une solution optimale. De tels algorithmes existent y compris en cas de dynamique incertaine, d'observabilité partielle, de critères multiples, ou d'agents multiples collaborant. Nous proposons ici un algorithme de recherche heuristique pour jeux stochastiques à deux joueurs et à somme nulle, et avec critère décompté, algorithme reposant sur HSVI—donc sur la génération de trajectoires. Nous démontrons que, chaque joueur agissant de manière optimiste, et en employant des initialisations heuristiques simples, l'algorithme obtenu converge vers une solution-optimale en temps fini
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