140 research outputs found
Hyperplane Separation Technique for Multidimensional Mean-Payoff Games
We consider both finite-state game graphs and recursive game graphs (or
pushdown game graphs), that can model the control flow of sequential programs
with recursion, with multi-dimensional mean-payoff objectives. In pushdown
games two types of strategies are relevant: global strategies, that depend on
the entire global history; and modular strategies, that have only local memory
and thus do not depend on the context of invocation. We present solutions to
several fundamental algorithmic questions and our main contributions are as
follows: (1) We show that finite-state multi-dimensional mean-payoff games can
be solved in polynomial time if the number of dimensions and the maximal
absolute value of the weight is fixed; whereas if the number of dimensions is
arbitrary, then problem is already known to be coNP-complete. (2) We show that
pushdown graphs with multi-dimensional mean-payoff objectives can be solved in
polynomial time. (3) For pushdown games under global strategies both single and
multi-dimensional mean-payoff objectives problems are known to be undecidable,
and we show that under modular strategies the multi-dimensional problem is also
undecidable (whereas under modular strategies the single dimensional problem is
NP-complete). We show that if the number of modules, the number of exits, and
the maximal absolute value of the weight is fixed, then pushdown games under
modular strategies with single dimensional mean-payoff objectives can be solved
in polynomial time, and if either of the number of exits or the number of
modules is not bounded, then the problem is NP-hard. (4) Finally we show that a
fixed parameter tractable algorithm for finite-state multi-dimensional
mean-payoff games or pushdown games under modular strategies with
single-dimensional mean-payoff objectives would imply the solution of the
long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282
Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time
We generalise the hyperplane separation technique (Chatterjee and Velner,
2013) from multi-dimensional mean-payoff to energy games, and achieve an
algorithm for solving the latter whose running time is exponential only in the
dimension, but not in the number of vertices of the game graph. This answers an
open question whether energy games with arbitrary initial credit can be solved
in pseudo-polynomial time for fixed dimensions 3 or larger (Chaloupka, 2013).
It also improves the complexity of solving multi-dimensional energy games with
given initial credit from non-elementary (Br\'azdil, Jan\v{c}ar, and
Ku\v{c}era, 2010) to 2EXPTIME, thus establishing their 2EXPTIME-completeness.Comment: Corrected proof of Lemma 6.2 (thanks to Dmitry Chistikov for spotting
an error in the previous proof
Robust Multidimensional Mean-Payoff Games are Undecidable
Mean-payoff games play a central role in quantitative synthesis and
verification. In a single-dimensional game a weight is assigned to every
transition and the objective of the protagonist is to assure a non-negative
limit-average weight. In the multidimensional setting, a weight vector is
assigned to every transition and the objective of the protagonist is to satisfy
a boolean condition over the limit-average weight of each dimension, e.g.,
\LimAvg(x_1) \leq 0 \vee \LimAvg(x_2)\geq 0 \wedge \LimAvg(x_3) \geq 0. We
recently proved that when one of the players is restricted to finite-memory
strategies then the decidability of determining the winner is inter-reducible
with Hilbert's Tenth problem over rationals (a fundamental long-standing open
problem). In this work we allow arbitrary (infinite-memory) strategies for both
players and we show that the problem is undecidable
Attainability in Repeated Games with Vector Payoffs
We introduce the concept of attainable sets of payoffs in two-player repeated
games with vector payoffs. A set of payoff vectors is called {\em attainable}
if player 1 can ensure that there is a finite horizon such that after time
the distance between the set and the cumulative payoff is arbitrarily
small, regardless of what strategy player 2 is using. This paper focuses on the
case where the attainable set consists of one payoff vector. In this case the
vector is called an attainable vector. We study properties of the set of
attainable vectors, and characterize when a specific vector is attainable and
when every vector is attainable.Comment: 28 pages, 2 figures, conference version at NetGCoop 201
From Security Enforcement to Supervisory Control in Discrete Event Systems: Qualitative and Quantitative Analyses
Cyber-physical systems are technological systems that involve physical components that are monitored and controlled by multiple computational units that exchange information through a communication network.
Examples of cyber-physical systems arise in transportation, power, smart manufacturing, and other classes of systems that have a large degree of automation.
Analysis and control of cyber-physical systems is an active area of research.
The increasing demands for safety, security and performance improvement of cyber-physical systems put stringent constraints on their design and necessitate the use of formal model-based methods to synthesize control strategies that provably enforce required properties.
This dissertation focuses on the higher level control logic in cyber-physical systems using the framework of discrete event systems.
It tackles two classes of problems for discrete event systems.
The first class of problems is related to system security.
This problem is formulated in terms of the information flow property of opacity.
In this part of the dissertation, an interface-based approach called insertion/edit function is developed to enforce opacity under the potential inference of malicious intruders that may or may not know the implementation of the insertion/edit function.
The focus is the synthesis of insertion/edit functions that solve the opacity enforcement problem in the framework of qualitative and quantitative games on finite graphs.
The second problem treated in the dissertation is that of performance optimization in the context of supervisory control under partial observation.
This problem is transformed to a two-player quantitative game and an information structure where the game is played is constructed.
A novel approach to synthesize supervisors by solving the game is developed.
The main contributions of this dissertation are grouped into the following five categories.
(i) The transformation of the formulated opacity enforcement and supervisory control problems to games on finite graphs provides a systematic way of performing worst case analysis in design of discrete event systems.
(ii) These games have state spaces that are as compact as possible using the notion of information states in each corresponding problem.
(iii) A formal model-based approach is employed in the entire dissertation, which results in provably correct solutions.
(iv) The approaches developed in this dissertation reveal the interconnection between control theory and formal methods.
(v) The results in this dissertation are applicable to many types of cyber-physical systems with security-critical and performance-aware requirements.PHDElectrical and Computer EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150002/1/jiyiding_1.pd
The Theory of Universal Graphs for Infinite Duration Games
We introduce the notion of universal graphs as a tool for constructing
algorithms solving games of infinite duration such as parity games and mean
payoff games. In the first part we develop the theory of universal graphs, with
two goals: showing an equivalence and normalisation result between different
recently introduced related models, and constructing generic value iteration
algorithms for any positionally determined objective. In the second part we
give four applications: to parity games, to mean payoff games, and to
combinations of them (in the form of disjunctions of objectives). For each of
these four cases we construct algorithms achieving or improving over the best
known time and space complexity.Comment: 43 pages, 10 figure
IST Austria Technical Report
Simulation is an attractive alternative for language inclusion for automata as it is an under-approximation of language inclusion, but usually has much lower complexity. For non-deterministic automata, while language inclusion is PSPACE-complete, simulation can be computed in polynomial time. Simulation has also been extended in two orthogonal directions, namely, (1) fair simulation, for simulation over specified set of infinite runs; and (2) quantitative simulation, for simulation between weighted automata. Again, while fair trace inclusion is PSPACE-complete, fair simulation can be computed in polynomial time. For weighted automata, the (quantitative) language inclusion problem is undecidable for mean-payoff automata and the decidability is open for discounted-sum automata, whereas the (quantitative) simulation reduce to mean-payoff games and discounted-sum games, which admit pseudo-polynomial time algorithms.
In this work, we study (quantitative) simulation for weighted automata with Büchi acceptance conditions, i.e., we generalize fair simulation from non-weighted automata to weighted automata. We show that imposing Büchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games. For example, whereas for mean-payoff and discounted-sum games, the players do not need memory to play optimally; we show in contrast that for simulation games with Büchi acceptance conditions, (i) for mean-payoff objectives, optimal strategies for both players require infinite memory in general, and (ii) for discounted-sum objectives, optimal strategies need not exist for both players. While the simulation games with Büchi acceptance conditions are more complicated (e.g., due to infinite-memory requirements for mean-payoff objectives) as compared to their counterpart without Büchi acceptance conditions, we still present pseudo-polynomial time algorithms to solve simulation games with Büchi acceptance conditions for both weighted mean-payoff and weighted discounted-sum automata
Quantum computational finance: martingale asset pricing for incomplete markets
A derivative is a financial security whose value is a function of underlying
traded assets and market outcomes. Pricing a financial derivative involves
setting up a market model, finding a martingale (``fair game") probability
measure for the model from the given asset prices, and using that probability
measure to price the derivative. When the number of underlying assets and/or
the number of market outcomes in the model is large, pricing can be
computationally demanding. We show that a variety of quantum techniques can be
applied to the pricing problem in finance, with a particular focus on
incomplete markets. We discuss three different methods that are distinct from
previous works: they do not use the quantum algorithms for Monte Carlo
estimation and they extract the martingale measure from market variables akin
to bootstrapping, a common practice among financial institutions. The first two
methods are based on a formulation of the pricing problem into a linear program
and are using respectively the quantum zero-sum game algorithm and the quantum
simplex algorithm as subroutines. For the last algorithm, we formalize a new
market assumption milder than market completeness for which quantum linear
systems solvers can be applied with the associated potential for large
speedups. As a prototype use case, we conduct numerical experiments in the
framework of the Black-Scholes-Merton model.Comment: 31 pages, 6 figure
Evolutionary Game Theory Squared: Evolving Agents in Endogenously Evolving Zero-Sum Games
The predominant paradigm in evolutionary game theory and more generally
online learning in games is based on a clear distinction between a population
of dynamic agents that interact given a fixed, static game. In this paper, we
move away from the artificial divide between dynamic agents and static games,
to introduce and analyze a large class of competitive settings where both the
agents and the games they play evolve strategically over time. We focus on
arguably the most archetypal game-theoretic setting -- zero-sum games (as well
as network generalizations) -- and the most studied evolutionary learning
dynamic -- replicator, the continuous-time analogue of multiplicative weights.
Populations of agents compete against each other in a zero-sum competition that
itself evolves adversarially to the current population mixture. Remarkably,
despite the chaotic coevolution of agents and games, we prove that the system
exhibits a number of regularities. First, the system has conservation laws of
an information-theoretic flavor that couple the behavior of all agents and
games. Secondly, the system is Poincar\'{e} recurrent, with effectively all
possible initializations of agents and games lying on recurrent orbits that
come arbitrarily close to their initial conditions infinitely often. Thirdly,
the time-average agent behavior and utility converge to the Nash equilibrium
values of the time-average game. Finally, we provide a polynomial time
algorithm to efficiently predict this time-average behavior for any such
coevolving network game.Comment: To appear in AAAI 202
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