53 research outputs found
A Unifying Survey on Weighted Logics and Weighted Automata: Core Weighted Logic: Minimal and Versatile Specification of Quantitative Properties
International audienceLogical formalisms equivalent to weighted automata have been the topic of numerous research papers in the recent years. It started with the seminal result by Droste and Gastin on weighted logics over semir-ings for words. It has been extended in two dimensions by many authors. First, the weight domain has been extended to valuation monoids, valuation structures, etc., to capture more quantitative properties. Along another dimension, different structures such as ranked or unranked trees, nested words, Mazurkiewiz traces, etc., have been considered. The long and involved proofs of equivalences in all these papers are implicitely based on the same core arguments. This article provides a meta-theorem which unifies these different approaches. Towards this, we first introduce a core weighted logic with a minimal number of features and a simplified syntax. Then, we define a new semantics for weighted automata and weighted logics in two phases—an abstract semantics based on multisets of weight structures (independent of particular weight domains) followed by a concrete semantics. We show at the level of the abstract semantics that weighted automata and core weighted logic have the same expressive power. We show how previous results can be recovered from our result by logical reasoning. In this paper, we prove the meta-theorem for words, ranked and unranked trees, showing the robustness of our approach
Determinisation of Finitely-Ambiguous Copyless Cost Register Automata
Cost register automata (CRA) are machines reading an input word while computing values using write-only registers: values from registers are combined using the two operations, as well as the constants, of a semiring. Particularly interesting is the subclass of copyless CRAs where the content of a register cannot be used twice for updating the registers. Originally deterministic, non-deterministic variant of CRA may also be defined: the semantics is then obtained by combining the values of all accepting runs with the additive operation of the semiring (as for weighted automata). We show that finitely-ambiguous copyless non-deterministic CRAs (i.e. the ones that admit a bounded number of accepting runs on every input word) can be effectively transformed into an equivalent copyless (deterministic) CRA, without requiring any specific property on the semiring. As a corollary, this also shows that regular look-ahead can effectively be removed from copyless CRAs
Decidability of One-Clock Weighted Timed Games with Arbitrary Weights
Weighted Timed Games (WTG for short) are the most widely used model to
describe controller synthesis problems involving real-time issues.
Unfortunately, they are notoriously difficult, and undecidable in general. As a
consequence, one-clock WTGs have attracted a lot of attention, especially
because they are known to be decidable when only non-negative weights are
allowed. However, when arbitrary weights are considered, despite several recent
works, their decidability status was still unknown. In this paper, we solve
this problem positively and show that the value function can be computed in
exponential time (if weights are encoded in unary)
Simple Priced Timed Games Are Not That Simple
Priced timed games are two-player zero-sum games played on priced timed
automata (whose locations and transitions are labeled by weights modeling the
costs of spending time in a state and executing an action, respectively). The
goals of the players are to minimise and maximise the cost to reach a target
location, respectively. We consider priced timed games with one clock and
arbitrary (positive and negative) weights and show that, for an important
subclass of theirs (the so-called simple priced timed games), one can compute,
in exponential time, the optimal values that the players can achieve, with
their associated optimal strategies. As side results, we also show that
one-clock priced timed games are determined and that we can use our result on
simple priced timed games to solve the more general class of so-called
reset-acyclic priced timed games (with arbitrary weights and one-clock)
Efficient Energy Distribution in a Smart Grid using Multi-Player Games
Algorithms and models based on game theory have nowadays become prominent
techniques for the design of digital controllers for critical systems. Indeed,
such techniques enable automatic synthesis: given a model of the environment
and a property that the controller must enforce, those techniques automatically
produce a correct controller, when it exists. In the present paper, we consider
a class of concurrent, weighted, multi-player games that are well-suited to
model and study the interactions of several agents who are competing for some
measurable resources like energy. We prove that a subclass of those games
always admit a Nash equilibrium, i.e. a situation in which all players play in
such a way that they have no incentive to deviate. Moreover, the strategies
yielding those Nash equilibria have a special structure: when one of the agents
deviate from the equilibrium, all the others form a coalition that will enforce
a retaliation mechanism that punishes the deviant agent. We apply those results
to a real-life case study in which several smart houses that produce their own
energy with solar panels, and can share this energy among them in micro-grid,
must distribute the use of this energy along the day in order to avoid
consuming electricity that must be bought from the global grid. We demonstrate
that our theory allows one to synthesise an efficient controller for these
houses: using penalties to be paid in the utility bill as an incentive, we
force the houses to follow a pre-computed schedule that maximises the
proportion of the locally produced energy that is consumed.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017
Decidability of One-Clock Weighted Timed Games with Arbitrary Weights
Weighted Timed Games (WTG for short) are the most widely used model to describe controller synthesis problems involving real-time issues. Unfortunately, they are notoriously difficult, and undecidable in general. As a consequence, one-clock WTG has attracted a lot of attention, especially because they are known to be decidable when only non-negative weights are allowed. However, when arbitrary weights are considered, despite several recent works, their decidability status was still unknown. In this paper, we solve this problem positively and show that the value function can be computed in exponential time (if weights are encoded in unary)
Reaching Your Goal Optimally by Playing at Random with No Memory
Shortest-path games are two-player zero-sum games played on a graph equipped with integer weights. One player, that we call Min, wants to reach a target set of states while minimising the total weight, and the other one has an antagonistic objective. This combination of a qualitative reachability objective and a quantitative total-payoff objective is one of the simplest settings where Min needs memory (pseudo-polynomial in the weights) to play optimally. In this article, we aim at studying a tradeoff allowing Min to play at random, but using no memory. We show that Min can achieve the same optimal value in both cases. In particular, we compute a randomised memoryless ?-optimal strategy when it exists, where probabilities are parametrised by ?. We also show that for some games, no optimal randomised strategies exist. We then characterise, and decide in polynomial time, the class of games admitting an optimal randomised memoryless strategy
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
Playing Stochastically in Weighted Timed Games to Emulate Memory
Weighted timed games are two-player zero-sum games played in a timed automaton equipped with integer weights. We consider optimal reachability objectives, in which one of the players, that we call Min, wants to reach a target location while minimising the cumulated weight. While knowing if Min has a strategy to guarantee a value lower than a given threshold is known to be undecidable (with two or more clocks), several conditions, one of them being the divergence, have been given to recover decidability. In such weighted timed games (like in untimed weighted games in the presence of negative weights), Min may need finite memory to play (close to) optimally. This is thus tempting to try to emulate this finite memory with other strategic capabilities. In this work, we allow the players to use stochastic decisions, both in the choice of transitions and of timing delays. We give for the first time a definition of the expected value in weighted timed games, overcoming several theoretical challenges. We then show that, in divergent weighted timed games, the stochastic value is indeed equal to the classical (deterministic) value, thus proving that Min can guarantee the same value while only using stochastic choices, and no memory
- …