2,532 research outputs found
Nonapproximability Results for Partially Observable Markov Decision Processes
We show that for several variations of partially observable Markov decision
processes, polynomial-time algorithms for finding control policies are unlikely
to or simply don't have guarantees of finding policies within a constant factor
or a constant summand of optimal. Here "unlikely" means "unless some complexity
classes collapse," where the collapses considered are P=NP, P=PSPACE, or P=EXP.
Until or unless these collapses are shown to hold, any control-policy designer
must choose between such performance guarantees and efficient computation
Non-causal computation
Computation models such as circuits describe sequences of computation steps
that are carried out one after the other. In other words, algorithm design is
traditionally subject to the restriction imposed by a fixed causal order. We
address a novel computing paradigm beyond quantum computing, replacing this
assumption by mere logical consistency: We study non-causal circuits, where a
fixed time structure within a gate is locally assumed whilst the global causal
structure between the gates is dropped. We present examples of logically
consistent non- causal circuits outperforming all causal ones; they imply that
suppressing loops entirely is more restrictive than just avoiding the
contradictions they can give rise to. That fact is already known for
correlations as well as for communication, and we here extend it to
computation.Comment: 6 pages, 4 figure
The Complexity of Relating Quantum Channels to Master Equations
Completely positive, trace preserving (CPT) maps and Lindblad master
equations are both widely used to describe the dynamics of open quantum
systems. The connection between these two descriptions is a classic topic in
mathematical physics. One direction was solved by the now famous result due to
Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete
characterisation of the master equations that generate completely positive
semi-groups. However, the other direction has remained open: given a CPT map,
is there a Lindblad master equation that generates it (and if so, can we find
it's form)? This is sometimes known as the Markovianity problem. Physically, it
is asking how one can deduce underlying physical processes from experimental
observations.
We give a complexity theoretic answer to this problem: it is NP-hard. We also
give an explicit algorithm that reduces the problem to integer semi-definite
programming, a well-known NP problem. Together, these results imply that
resolving the question of which CPT maps can be generated by master equations
is tantamount to solving P=NP: any efficiently computable criterion for
Markovianity would imply P=NP; whereas a proof that P=NP would imply that our
algorithm already gives an efficiently computable criterion. Thus, unless P
does equal NP, there cannot exist any simple criterion for determining when a
CPT map has a master equation description.
However, we also show that if the system dimension is fixed (relevant for
current quantum process tomography experiments), then our algorithm scales
efficiently in the required precision, allowing an underlying Lindblad master
equation to be determined efficiently from even a single snapshot in this case.
Our work also leads to similar complexity-theoretic answers to a related
long-standing open problem in probability theory.Comment: V1: 43 pages, single column, 8 figures. V2: titled changed; added
proof-overview and accompanying figure; 50 pages, single column, 9 figure
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
Tree Stochastic Processes
Stochastic processes play a vital role in understanding the development of many natural and computational systems over time. In this thesis, we will study two settings where stochastic processes on trees play a significant role. The first setting is in the reconstruction of evolutionary trees from biological sequence data. Most previous work done in this area has assumed that different positions in a sequence evolve independently. This independence however is a strong assumption that has been shown to possibly cause inaccuracies in the reconstructed trees \cite{schoniger1994stochastic,tillier1995neighbor}. In our work, we provide a first step toward realizing the effects of dependency in such situations by creating a model in which two positions may evolve dependently. For two characters with transition matrices and , their joint transition matrix is the tensor product . Our dependence model modifies the joint transition matrix by adding an `error matrix,\u27 a matrix with rows summing to 0. We show when such dependence can be detected.
The second setting concerns computing in the presence of faults. In pushing the limits of computing hardware, there is tradeoff between the reliability of components and their cost (e.g. \cite{kadric2014energy}). We first examine a method of identifying faulty gates in a read-once formula when our access is limited to providing an input and reading its output. We show that determining \emph{whether} a fault exists can always be done, and that locating these faults can be done efficiently as long as the read-once formula satisfies a certain balance condition. Finally for a fixed topology, we provide a dynamic program which allows us to optimize how to allocate resources to individual gates so as to optimize the reliability of the whole system under a known input product distribution
Non deterministic Repairable Fault Trees for computing optimal repair strategy
In this paper, the Non deterministic Repairable Fault Tree (NdRFT) formalism is proposed: it allows to model failure modes of complex systems as well as their repair processes. The originality of this formalism
with respect to other Fault Tree extensions is that it allows to face repair strategies optimization problems: in an NdRFT model, the decision on whether to start or not a given repair action is non deterministic, so
that all the possibilities are left open. The formalism is rather powerful allowing to specify which failure events are observable, whether local repair or global repair can be applied, and the resources needed to start
a repair action. The optimal repair strategy can then be computed by solving an optimization problem on a Markov Decision Process (MDP) derived from the NdRFT. A software framework is proposed in order to perform in automatic way the derivation of an MDP from a NdRFT model, and to deal with the solution of the MDP
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