14 research outputs found
Decisive Markov Chains
We consider qualitative and quantitative verification problems for
infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given
set of target states F if it almost certainly eventually reaches either F or a
state from which F can no longer be reached. While all finite Markov chains are
trivially decisive (for every set F), this also holds for many classes of
infinite Markov chains. Infinite Markov chains which contain a finite attractor
are decisive w.r.t. every set F. In particular, this holds for probabilistic
lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains
are decisive. This class includes probabilistic vector addition systems (PVASS)
and probabilistic noisy Turing machines (PNTM). We consider both safety and
liveness problems for decisive Markov chains, i.e., the probabilities that a
given set of states F is eventually reached or reached infinitely often,
respectively. 1. We express the qualitative problems in abstract terms for
decisive Markov chains, and show an almost complete picture of its decidability
for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm
of Iyer and Narasimha terminates for decisive Markov chains and can thus be
used to solve the approximate quantitative safety problem. A modified variant
of this algorithm solves the approximate quantitative liveness problem. 3.
Finally, we show that the exact probability of (repeatedly) reaching F cannot
be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS,
PVASS or (P)NTM.Comment: 32 pages, 0 figure
Eager Markov Chains
Abstract. We consider infinite-state discrete Markov chains which are eager: the probability of avoiding a defined set of final states for more thanÒsteps is bounded by some exponentially decreasing function�(Ò). We prove that eager Markov chains include those induced by Probabilistic Lossy Channel Systems, Probabilistic Vector Addition Systems with States, and Noisy Turing Machines, and that the bounding function�(Ò) can be effectively constructed for them. Furthermore, we study the problem of computing the expected reward (or cost) of runs until reaching the final states, where rewards are assigned to individual runs by computable reward functions. For eager Markov chains, an effective path exploration scheme, based on forward reachability analysis, can be used to approximate the expected reward up-to an arbitrarily small error.
Does Church-Turing thesis apply outside computer science?
We analyze whether Church-Turing thesis can be applied to mathematical and physical systems. We find the factors that allow to a class of systems to reach a Turing or a super-Turing computational power. We illustrate our general statements by some more concrete theorems on hybrid and stochastic systems
Robust computations with dynamical systems
In this paper we discuss the computational power of Lipschitz
dynamical systems which are robust to in nitesimal perturbations.
Whereas the study in [1] was done only for not-so-natural systems from
a classical mathematical point of view (discontinuous di erential equation
systems, discontinuous piecewise a ne maps, or perturbed Turing
machines), we prove that the results presented there can be generalized
to Lipschitz and computable dynamical systems.
In other words, we prove that the perturbed reachability problem (i.e. the
reachability problem for systems which are subjected to in nitesimal perturbations)
is co-recursively enumerable for this kind of systems. Using
this result we show that if robustness to in nitesimal perturbations is
also required, the reachability problem becomes decidable. This result
can be interpreted in the following manner: undecidability of veri cation
doesn't hold for Lipschitz, computable and robust systems.
We also show that the perturbed reachability problem is co-r.e. complete
even for C1-systems
Computation with perturbed dynamical systems
This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability. (C) 2013 Elsevier Inc. All rights reserved.INRIA program "Equipe Associee" ComputR; Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT project [PEst-OE/EEI/LA0008/2011]info:eu-repo/semantics/publishedVersio
Stochastic Games with Lossy Channels
We consider turn-based stochastic games on infinite graphs induced by game probabilistic lossy channel systems (GPLCS), the game version of probabilistic lossy channel systems (PLCS). We study games with Büchi (repeated reachability) objectives and almost-sure winning conditions. These games are pure memoryless determined and, under the assumption that the target set is regular, a symbolic representation of the set of winning states for each player can be effectively constructed. Thus, turn-based stochastic games on GPLCS are decidable. This generalizes the decidability result for PLCS-induced Markov decision processes in [10]
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Sensitive finite state computations using a distributed network with a noisy network attractor
This is the author accepted manuscript. The final version is available from IEEE via the DOI in this record.We exhibit a class of smooth continuous-state
neural-inspired networks composed of simple nonlinear elements
that can be made to function as a finite state computational
machine. We give an explicit construction of arbitrary finitestate
virtual machines in the spatio-temporal dynamics of the
network. The dynamics of the functional network can be completely
characterised as a “noisy network attractor” in phase
space operating in either an “excitable” or a “free-running”
regime, respectively corresponding to excitable or heteroclinic
connections between states. The regime depends on the sign of
an “excitability parameter”. Viewing the network as a nonlinear
stochastic differential equation where deterministic (signal)
and/or stochastic (noise) input are applied to any element, we
explore the influence of signal to noise ratio on the error rate of
the computations. The free-running regime is extremely sensitive
to inputs: arbitrarily small amplitude perturbations can be used
to perform computations with the system as long as the input
dominates the noise. We find a counter-intuitive regime where
increasing noise amplitude can lead to more, rather than less,
accurate computation. We suggest that noisy network attractors
will be useful for understanding neural networks that reliably
and sensitively perform finite-state computations in a noisy
environment.PA gratefully acknowledges the financial support of the
EPSRC via grant EP/N014391/1. CMP acknowledges travel
funding from the University of Auckland and support from the
London Mathematical Laboratory