9 research outputs found
On G\'acs' quantum algorithmic entropy
We define an infinite dimensional modification of lower-semicomputability of
density operators by G\'acs with an attempt to fix some problem in the paper.
Our attempt is partly achieved by showing the existence of universal operator
under some additional assumption. It is left as a future task to eliminate this
assumption. We also see some properties and examples which stimulate further
research. In particular, we show that universal operator has certain nontrivial
form if it exists.Comment: In Proceedings QPL 2014, arXiv:1412.810
Connectivity in the presence of an opponent
The paper introduces two player connectivity games played on finite bipartite
graphs. Algorithms that solve these connectivity games can be used as
subroutines for solving M\"uller games. M\"uller games constitute a well
established class of games in model checking and verification. In connectivity
games, the objective of one of the players is to visit every node of the game
graph infinitely often. The first contribution of this paper is our proof that
solving connectivity games can be reduced to the incremental strongly connected
component maintenance (ISCCM) problem, an important problem in graph algorithms
and data structures. The second contribution is that we non-trivially adapt two
known algorithms for the ISCCM problem to provide two efficient algorithms that
solve the connectivity games problem. Finally, based on the techniques
developed, we recast Horn's polynomial time algorithm that solves explicitly
given M\"uller games and provide an alternative proof of its correctness. Our
algorithms are more efficient than that of Horn's algorithm. Our solution for
connectivity games is used as a subroutine in the algorithm
Learning Density-Based Correlated Equilibria for Markov Games
Correlated Equilibrium (CE) is a well-established solution concept that
captures coordination among agents and enjoys good algorithmic properties. In
real-world multi-agent systems, in addition to being in an equilibrium, agents'
policies are often expected to meet requirements with respect to safety, and
fairness. Such additional requirements can often be expressed in terms of the
state density which measures the state-visitation frequencies during the course
of a game. However, existing CE notions or CE-finding approaches cannot
explicitly specify a CE with particular properties concerning state density;
they do so implicitly by either modifying reward functions or using value
functions as the selection criteria. The resulting CE may thus not fully fulfil
the state-density requirements. In this paper, we propose Density-Based
Correlated Equilibria (DBCE), a new notion of CE that explicitly takes state
density as selection criterion. Concretely, we instantiate DBCE by specifying
different state-density requirements motivated by real-world applications. To
compute DBCE, we put forward the Density Based Correlated Policy Iteration
algorithm for the underlying control problem. We perform experiments on various
games where results demonstrate the advantage of our CE-finding approach over
existing methods in scenarios with state-density concerns
無限文字列の大規模幾何
京都大学0048新制・課程博士博士(理学)甲第20886号理博第4338号新制||理||1623(附属図書館)京都大学大学院理学研究科数学・数理解析専攻(主査)教授 長谷川 真人, 教授 向井 茂, 准教授 照井 一成学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDFA
Moment Propagation Through Carleman Linearization with Application to Probabilistic Safety Analysis
We develop a method to approximate the moments of a discrete-time stochastic
polynomial system. Our method is built upon Carleman linearization with
truncation. Specifically, we take a stochastic polynomial system with finitely
many states and transform it into an infinite-dimensional system with linear
deterministic dynamics, which describe the exact evolution of the moments of
the original polynomial system. We then truncate this deterministic system to
obtain a finite-dimensional linear system, and use it for moment approximation
by iteratively propagating the moments along the finite-dimensional linear
dynamics across time. We provide efficient online computation methods for this
propagation scheme with several error bounds for the approximation. Our result
also shows that precise values of certain moments can be obtained when the
truncated system is sufficiently large. Furthermore, we investigate techniques
to reduce the offline computation load using reduced Kronecker power. Based on
the obtained approximate moments and their errors, we also provide probability
bounds for the state to be outside of given hyperellipsoidal regions. Those
bounds allow us to conduct probabilistic safety analysis online through convex
optimization. We demonstrate our results on a logistic map with stochastic
dynamics and a vehicle dynamics subject to stochastic disturbance.Comment: Preprint submitted to Automatica. arXiv admin note: substantial text
overlap with arXiv:1911.1268