15 research outputs found
Non-Zero Sum Games for Reactive Synthesis
In this invited contribution, we summarize new solution concepts useful for
the synthesis of reactive systems that we have introduced in several recent
publications. These solution concepts are developed in the context of non-zero
sum games played on graphs. They are part of the contributions obtained in the
inVEST project funded by the European Research Council.Comment: LATA'16 invited pape
Computer aided synthesis: a game theoretic approach
In this invited contribution, we propose a comprehensive introduction to game
theory applied in computer aided synthesis. In this context, we give some
classical results on two-player zero-sum games and then on multi-player non
zero-sum games. The simple case of one-player games is strongly related to
automata theory on infinite words. All along the article, we focus on general
approaches to solve the studied problems, and we provide several illustrative
examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language
Theory" (DLT 2017
Infinite-Duration Bidding Games
Two-player games on graphs are widely studied in formal methods as they model
the interaction between a system and its environment. The game is played by
moving a token throughout a graph to produce an infinite path. There are
several common modes to determine how the players move the token through the
graph; e.g., in turn-based games the players alternate turns in moving the
token. We study the {\em bidding} mode of moving the token, which, to the best
of our knowledge, has never been studied in infinite-duration games. The
following bidding rule was previously defined and called Richman bidding. Both
players have separate {\em budgets}, which sum up to . In each turn, a
bidding takes place: Both players submit bids simultaneously, where a bid is
legal if it does not exceed the available budget, and the higher bidder pays
his bid to the other player and moves the token. The central question studied
in bidding games is a necessary and sufficient initial budget for winning the
game: a {\em threshold} budget in a vertex is a value such that
if Player 's budget exceeds , he can win the game, and if Player 's
budget exceeds , he can win the game. Threshold budgets were previously
shown to exist in every vertex of a reachability game, which have an
interesting connection with {\em random-turn} games -- a sub-class of simple
stochastic games in which the player who moves is chosen randomly. We show the
existence of threshold budgets for a qualitative class of infinite-duration
games, namely parity games, and a quantitative class, namely mean-payoff games.
The key component of the proof is a quantitative solution to strongly-connected
mean-payoff bidding games in which we extend the connection with random-turn
games to these games, and construct explicit optimal strategies for both
players.Comment: A short version appeared in CONCUR 2017. The paper is accepted to
JAC
Finite-Valued Weighted Automata
Any weighted automaton (WA) defines a relation from finite words to values: given an input word, its set of values is obtained as the set of values computed by each accepting run on that word. A WA is k-valued if the relation it defines has degree at most k, i.e., every set of values associated with an input word has cardinality at most k. We investigate the class of quantitative languages defined by k-valued automata, for all parameters k. We consider several measures to associate values with runs: sum, discounted-sum, and more generally values in groups.
We define a general procedure which decides, given a bound k and a WA over a group, whether this automaton is k-valued. We also show that any k-valued WA over a group, under some general conditions, can be decomposed as a union of k unambiguous WA. While inclusion and equivalence are undecidable problems for arbitrary sum-automata, we show, based on this decomposition, that they are decidable for k-valued sum-automata, and k-valued discounted sum-automata over inverted integer discount factors. We finally show that the quantitative Church problem is undecidable for k-valued sum-automata, even given as finite unions of deterministic sum-automata
The target discounted-sum problem
The target discounted-sum problem is the following: Given a rational discount factor 0 < λ < 1 and three rational values a, b, and t, does there exist a finite or an infinite sequence w ε(a, b)∗ or w ε(a, b)w, such that Σ|w| i=0 w(i)λi equals t? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: β-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that λ λ 1/2 or λ = 1/n, for every n ε N. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value problem for nondeterministic automata over finite words and the universality and inclusion problems for functional automata
IST Austria Technical Report
The target discounted-sum problem is the following: Given a rational discount factor 0 < λ < 1 and three rational values a, b, and t, does there exist a finite or an infinite sequence w ε(a, b)∗ or w ε(a, b)w, such that Σ|w| i=0 w(i)λi equals t? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: β-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that λ λ 1/2 or λ = 1/n, for every n ε N. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value problem for nondeterministic automata over finite words and the universality and inclusion problems for functional automata