1,106 research outputs found
Tableaux for Policy Synthesis for MDPs with PCTL* Constraints
Markov decision processes (MDPs) are the standard formalism for modelling
sequential decision making in stochastic environments. Policy synthesis
addresses the problem of how to control or limit the decisions an agent makes
so that a given specification is met. In this paper we consider PCTL*, the
probabilistic counterpart of CTL*, as the specification language. Because in
general the policy synthesis problem for PCTL* is undecidable, we restrict to
policies whose execution history memory is finitely bounded a priori.
Surprisingly, no algorithm for policy synthesis for this natural and
expressive framework has been developed so far. We close this gap and describe
a tableau-based algorithm that, given an MDP and a PCTL* specification, derives
in a non-deterministic way a system of (possibly nonlinear) equalities and
inequalities. The solutions of this system, if any, describe the desired
(stochastic) policies.
Our main result in this paper is the correctness of our method, i.e.,
soundness, completeness and termination.Comment: This is a long version of a conference paper published at TABLEAUX
2017. It contains proofs of the main results and fixes a bug. See the
footnote on page 1 for detail
The RAM equivalent of P vs. RP
One of the fundamental open questions in computational complexity is whether
the class of problems solvable by use of stochasticity under the Random
Polynomial time (RP) model is larger than the class of those solvable in
deterministic polynomial time (P). However, this question is only open for
Turing Machines, not for Random Access Machines (RAMs).
Simon (1981) was able to show that for a sufficiently equipped Random Access
Machine, the ability to switch states nondeterministically does not entail any
computational advantage. However, in the same paper, Simon describes a
different (and arguably more natural) scenario for stochasticity under the RAM
model. According to Simon's proposal, instead of receiving a new random bit at
each execution step, the RAM program is able to execute the pseudofunction
, which returns a uniformly distributed random integer in the
range . Whether the ability to allot a random integer in this fashion is
more powerful than the ability to allot a random bit remained an open question
for the last 30 years.
In this paper, we close Simon's open problem, by fully characterising the
class of languages recognisable in polynomial time by each of the RAMs
regarding which the question was posed. We show that for some of these,
stochasticity entails no advantage, but, more interestingly, we show that for
others it does.Comment: 23 page
The oriented swap process and last passage percolation
We present new probabilistic and combinatorial identities relating three
random processes: the oriented swap process on particles, the corner growth
process, and the last passage percolation model. We prove one of the
probabilistic identities, relating a random vector of last passage percolation
times to its dual, using the duality between the Robinson-Schensted-Knuth and
Burge correspondences. A second probabilistic identity, relating those two
vectors to a vector of 'last swap times' in the oriented swap process, is
conjectural. We give a computer-assisted proof of this identity for
after first reformulating it as a purely combinatorial identity, and discuss
its relation to the Edelman-Greene correspondence. The conjectural identity
provides precise finite- and asymptotic predictions on the distribution of
the absorbing time of the oriented swap process, thus conditionally solving an
open problem posed by Angel, Holroyd and Romik.Comment: 36 pages, 6 figures. Full version of the FPSAC 2020 extended abstract
arXiv:2003.0333
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