30 research outputs found
On Frequency LTL in Probabilistic Systems
We study frequency linear-time temporal logic (fLTL) which extends the
linear-time temporal logic (LTL) with a path operator expressing that on
a path, certain formula holds with at least a given frequency p, thus relaxing
the semantics of the usual G operator of LTL. Such logic is particularly useful
in probabilistic systems, where some undesirable events such as random failures
may occur and are acceptable if they are rare enough.
Frequency-related extensions of LTL have been previously studied by several
authors, where mostly the logic is equipped with an extended "until" and
"globally" operator, leading to undecidability of most interesting problems.
For the variant we study, we are able to establish fundamental decidability
results. We show that for Markov chains, the problem of computing the
probability with which a given fLTL formula holds has the same complexity as
the analogous problem for LTL. We also show that for Markov decision processes
the problem becomes more delicate, but when restricting the frequency bound
to be 1 and negations not to be outside any operator, we can compute the
maximum probability of satisfying the fLTL formula. This can be again performed
with the same time complexity as for the ordinary LTL formulas.Comment: A paper presented at CONCUR 2015, with appendi
Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata
We study the bisimilarity problem for probabilistic pushdown automata (pPDA)
and subclasses thereof. Our definition of pPDA allows both probabilistic and
non-deterministic branching, generalising the classical notion of pushdown
automata (without epsilon-transitions). We first show a general
characterization of probabilistic bisimilarity in terms of two-player games,
which naturally reduces checking bisimilarity of probabilistic labelled
transition systems to checking bisimilarity of standard (non-deterministic)
labelled transition systems. This reduction can be easily implemented in the
framework of pPDA, allowing to use known results for standard
(non-probabilistic) PDA and their subclasses. A direct use of the reduction
incurs an exponential increase of complexity, which does not matter in deriving
decidability of bisimilarity for pPDA due to the non-elementary complexity of
the problem. In the cases of probabilistic one-counter automata (pOCA), of
probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic
process algebras (i.e., single-state pPDA) we show that an implicit use of the
reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and
2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic
versions. The bisimilarity problems for OCA and vPDA are known to have matching
lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively);
we show that these lower bounds also hold for fully probabilistic versions that
do not use non-determinism
Trading Performance for Stability in Markov Decision Processes
We study the complexity of central controller synthesis problems for
finite-state Markov decision processes, where the objective is to optimize both
the expected mean-payoff performance of the system and its stability.
We argue that the basic theoretical notion of expressing the stability in
terms of the variance of the mean-payoff (called global variance in our paper)
is not always sufficient, since it ignores possible instabilities on respective
runs. For this reason we propose alernative definitions of stability, which we
call local and hybrid variance, and which express how rewards on each run
deviate from the run's own mean-payoff and from the expected mean-payoff,
respectively.
We show that a strategy ensuring both the expected mean-payoff and the
variance below given bounds requires randomization and memory, under all the
above semantics of variance. We then look at the problem of determining whether
there is a such a strategy. For the global variance, we show that the problem
is in PSPACE, and that the answer can be approximated in pseudo-polynomial
time. For the hybrid variance, the analogous decision problem is in NP, and a
polynomial-time approximating algorithm also exists. For local variance, we
show that the decision problem is in NP. Since the overall performance can be
traded for stability (and vice versa), we also present algorithms for
approximating the associated Pareto curve in all the three cases.
Finally, we study a special case of the decision problems, where we require a
given expected mean-payoff together with zero variance. Here we show that the
problems can be all solved in polynomial time.Comment: Extended version of a paper presented at LICS 201
Markov Decision Processes with Multiple Long-run Average Objectives
We study Markov decision processes (MDPs) with multiple limit-average (or
mean-payoff) functions. We consider two different objectives, namely,
expectation and satisfaction objectives. Given an MDP with k limit-average
functions, in the expectation objective the goal is to maximize the expected
limit-average value, and in the satisfaction objective the goal is to maximize
the probability of runs such that the limit-average value stays above a given
vector. We show that under the expectation objective, in contrast to the case
of one limit-average function, both randomization and memory are necessary for
strategies even for epsilon-approximation, and that finite-memory randomized
strategies are sufficient for achieving Pareto optimal values. Under the
satisfaction objective, in contrast to the case of one limit-average function,
infinite memory is necessary for strategies achieving a specific value (i.e.
randomized finite-memory strategies are not sufficient), whereas memoryless
randomized strategies are sufficient for epsilon-approximation, for all
epsilon>0. We further prove that the decision problems for both expectation and
satisfaction objectives can be solved in polynomial time and the trade-off
curve (Pareto curve) can be epsilon-approximated in time polynomial in the size
of the MDP and 1/epsilon, and exponential in the number of limit-average
functions, for all epsilon>0. Our analysis also reveals flaws in previous work
for MDPs with multiple mean-payoff functions under the expectation objective,
corrects the flaws, and allows us to obtain improved results