159 research outputs found
Clustering of spectra and fractals of regular graphs
We exhibit a characteristic structure of the class of all regular graphs of
degree d that stems from the spectra of their adjacency matrices. The structure
has a fractal threadlike appearance. Points with coordinates given by the mean
and variance of the exponentials of graph eigenvalues cluster around a line
segment that we call a filar. Zooming-in reveals that this cluster splits into
smaller segments (filars) labeled by the number of triangles in graphs. Further
zooming-in shows that the smaller filars split into subfilars labelled by the
number of quadrangles in graphs, etc. We call this fractal structure,
discovered in a numerical experiment, a multifilar structure. We also provide a
mathematical explanation of this phenomenon based on the Ihara-Selberg trace
formula, and compute the coordinates and slopes of all filars in terms of
Bessel functions of the first kind.Comment: 10 pages, 5 figure
Discounting in Games across Time Scales
We introduce two-level discounted games played by two players on a
perfect-information stochastic game graph. The upper level game is a discounted
game and the lower level game is an undiscounted reachability game. Two-level
games model hierarchical and sequential decision making under uncertainty
across different time scales. We show the existence of pure memoryless optimal
strategies for both players and an ordered field property for such games. We
show that if there is only one player (Markov decision processes), then the
values can be computed in polynomial time. It follows that whether the value of
a player is equal to a given rational constant in two-level discounted games
can be decided in NP intersected coNP. We also give an alternate strategy
improvement algorithm to compute the value
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
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
Probabilistic Model Checking for Energy Analysis in Software Product Lines
In a software product line (SPL), a collection of software products is
defined by their commonalities in terms of features rather than explicitly
specifying all products one-by-one. Several verification techniques were
adapted to establish temporal properties of SPLs. Symbolic and family-based
model checking have been proven to be successful for tackling the combinatorial
blow-up arising when reasoning about several feature combinations. However,
most formal verification approaches for SPLs presented in the literature focus
on the static SPLs, where the features of a product are fixed and cannot be
changed during runtime. This is in contrast to dynamic SPLs, allowing to adapt
feature combinations of a product dynamically after deployment. The main
contribution of the paper is a compositional modeling framework for dynamic
SPLs, which supports probabilistic and nondeterministic choices and allows for
quantitative analysis. We specify the feature changes during runtime within an
automata-based coordination component, enabling to reason over strategies how
to trigger dynamic feature changes for optimizing various quantitative
objectives, e.g., energy or monetary costs and reliability. For our framework
there is a natural and conceptually simple translation into the input language
of the prominent probabilistic model checker PRISM. This facilitates the
application of PRISM's powerful symbolic engine to the operational behavior of
dynamic SPLs and their family-based analysis against various quantitative
queries. We demonstrate feasibility of our approach by a case study issuing an
energy-aware bonding network device.Comment: 14 pages, 11 figure
The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games
We analyse the computational complexity of finding Nash equilibria in simple
stochastic multiplayer games. We show that restricting the search space to
equilibria whose payoffs fall into a certain interval may lead to
undecidability. In particular, we prove that the following problem is
undecidable: Given a game G, does there exist a pure-strategy Nash equilibrium
of G where player 0 wins with probability 1. Moreover, this problem remains
undecidable if it is restricted to strategies with (unbounded) finite memory.
However, if mixed strategies are allowed, decidability remains an open problem.
One way to obtain a provably decidable variant of the problem is restricting
the strategies to be positional or stationary. For the complexity of these two
problems, we obtain a common lower bound of NP and upper bounds of NP and
PSPACE respectively.Comment: 23 pages; revised versio
Value Iteration for Long-run Average Reward in Markov Decision Processes
Markov decision processes (MDPs) are standard models for probabilistic
systems with non-deterministic behaviours. Long-run average rewards provide a
mathematically elegant formalism for expressing long term performance. Value
iteration (VI) is one of the simplest and most efficient algorithmic approaches
to MDPs with other properties, such as reachability objectives. Unfortunately,
a naive extension of VI does not work for MDPs with long-run average rewards,
as there is no known stopping criterion. In this work our contributions are
threefold. (1) We refute a conjecture related to stopping criteria for MDPs
with long-run average rewards. (2) We present two practical algorithms for MDPs
with long-run average rewards based on VI. First, we show that a combination of
applying VI locally for each maximal end-component (MEC) and VI for
reachability objectives can provide approximation guarantees. Second, extending
the above approach with a simulation-guided on-demand variant of VI, we present
an anytime algorithm that is able to deal with very large models. (3) Finally,
we present experimental results showing that our methods significantly
outperform the standard approaches on several benchmarks
Simulation-Based Graph Similarity
We present symmetric and asymmetric similarity measures for labeled directed rooted graphs that are inspired by the simulation and bisimulation relations on labeled transition systems. Computation of the similarity measures has close connections to discounted Markov decision processes in the asymmetric case and to perfect-information stochastic games in the symmetric case. For the symmetric case, we also give a polynomial-time algorithm that approximates the similarity to any desired precision
Synchronizing Objectives for Markov Decision Processes
We introduce synchronizing objectives for Markov decision processes (MDP).
Intuitively, a synchronizing objective requires that eventually, at every step
there is a state which concentrates almost all the probability mass. In
particular, it implies that the probabilistic system behaves in the long run
like a deterministic system: eventually, the current state of the MDP can be
identified with almost certainty.
We study the problem of deciding the existence of a strategy to enforce a
synchronizing objective in MDPs. We show that the problem is decidable for
general strategies, as well as for blind strategies where the player cannot
observe the current state of the MDP. We also show that pure strategies are
sufficient, but memory may be necessary.Comment: In Proceedings iWIGP 2011, arXiv:1102.374
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