9 research outputs found
Polynomial Time Algorithms for Multi-Type Branching Processes and Stochastic Context-Free Grammars
We show that one can approximate the least fixed point solution for a
multivariate system of monotone probabilistic polynomial equations in time
polynomial in both the encoding size of the system of equations and in
log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the
solution. (The model of computation is the standard Turing machine model.)
We use this result to resolve several open problems regarding the
computational complexity of computing key quantities associated with some
classic and heavily studied stochastic processes, including multi-type
branching processes and stochastic context-free grammars
Diagnosis in Infinite-State Probabilistic Systems
In a recent work, we introduced four variants of diagnosability
(FA, IA, FF, IF) in (finite) probabilistic
systems (pLTS) depending whether one considers (1) finite or
infinite runs and (2) faulty or all runs. We studied their
relationship and established that the corresponding decision
problems are PSPACE-complete. A key ingredient of the decision
procedures was a characterisation of diagnosability by the fact that
a random run almost surely lies in an open set whose specification
only depends on the qualitative behaviour of the pLTS. Here we
investigate similar issues for infinite pLTS. We first show that
this characterisation still holds for FF-diagnosability but
with a G-delta set instead of an open set and also for IF-
and IA-diagnosability when pLTS are finitely branching. We also
prove that surprisingly FA-diagnosability cannot be
characterised in this way even in the finitely branching case. Then
we apply our characterisations for a partially observable
probabilistic extension of visibly pushdown automata (POpVPA),
yielding EXPSPACE procedures for solving diagnosability problems.
In addition, we establish some computational lower bounds and show
that slight extensions of POpVPA lead to undecidability
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
In the last two decades, there has been much progress on model checking of
both probabilistic systems and higher-order programs. In spite of the emergence
of higher-order probabilistic programming languages, not much has been done to
combine those two approaches. In this paper, we initiate a study on the
probabilistic higher-order model checking problem, by giving some first
theoretical and experimental results. As a first step towards our goal, we
introduce PHORS, a probabilistic extension of higher-order recursion schemes
(HORS), as a model of probabilistic higher-order programs. The model of PHORS
may alternatively be viewed as a higher-order extension of recursive Markov
chains. We then investigate the probabilistic termination problem -- or,
equivalently, the probabilistic reachability problem. We prove that almost sure
termination of order-2 PHORS is undecidable. We also provide a fixpoint
characterization of the termination probability of PHORS, and develop a sound
(but possibly incomplete) procedure for approximately computing the termination
probability. We have implemented the procedure for order-2 PHORSs, and
confirmed that the procedure works well through preliminary experiments that
are reported at the end of the article
Linear-Time Model Checking Branching Processes
(Multi-type) branching processes are a natural and well-studied model for generating random infinite trees. Branching processes feature both nondeterministic and probabilistic branching, generalizing both transition systems and Markov chains (but not generally Markov decision processes). We study the complexity of model checking branching processes against linear-time omega-regular specifications: is it the case almost surely that every branch of a tree randomly generated by the branching process satisfies the omega-regular specification? The main result is that for LTL specifications this problem is in PSPACE, subsuming classical results for transition systems and Markov chains, respectively. The underlying general model-checking algorithm is based on the automata-theoretic approach, using unambiguous BĂŒchi automata