1,047 research outputs found
Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs
In this paper, we consider termination of probabilistic programs with
real-valued variables. The questions concerned are:
1. qualitative ones that ask (i) whether the program terminates with
probability 1 (almost-sure termination) and (ii) whether the expected
termination time is finite (finite termination); 2. quantitative ones that ask
(i) to approximate the expected termination time (expectation problem) and (ii)
to compute a bound B such that the probability to terminate after B steps
decreases exponentially (concentration problem).
To solve these questions, we utilize the notion of ranking supermartingales
which is a powerful approach for proving termination of probabilistic programs.
In detail, we focus on algorithmic synthesis of linear ranking-supermartingales
over affine probabilistic programs (APP's) with both angelic and demonic
non-determinism. An important subclass of APP's is LRAPP which is defined as
the class of all APP's over which a linear ranking-supermartingale exists.
Our main contributions are as follows. Firstly, we show that the membership
problem of LRAPP (i) can be decided in polynomial time for APP's with at most
demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with
angelic non-determinism; moreover, the NP-hardness result holds already for
APP's without probability and demonic non-determinism. Secondly, we show that
the concentration problem over LRAPP can be solved in the same complexity as
for the membership problem of LRAPP. Finally, we show that the expectation
problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's
without probability and non-determinism (i.e., deterministic programs). Our
experimental results demonstrate the effectiveness of our approach to answer
the qualitative and quantitative questions over APP's with at most demonic
non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201
Computational Approaches for Stochastic Shortest Path on Succinct MDPs
We consider the stochastic shortest path (SSP) problem for succinct Markov
decision processes (MDPs), where the MDP consists of a set of variables, and a
set of nondeterministic rules that update the variables. First, we show that
several examples from the AI literature can be modeled as succinct MDPs. Then
we present computational approaches for upper and lower bounds for the SSP
problem: (a)~for computing upper bounds, our method is polynomial-time in the
implicit description of the MDP; (b)~for lower bounds, we present a
polynomial-time (in the size of the implicit description) reduction to
quadratic programming. Our approach is applicable even to infinite-state MDPs.
Finally, we present experimental results to demonstrate the effectiveness of
our approach on several classical examples from the AI literature
Bi-continuous semigroups for flows in infinite networks
We study transport processes on infinite metric graphs with non-constant
velocities and matrix boundary conditions in the -setting.
We apply the theory of bi-continuous operator semigroups to obtain
well-posedness of the problem under different assumptions on the velocities and
for general stochastic matrices appearing in the boundary conditions.Comment: 12 page
Proving Expected Sensitivity of Probabilistic Programs with Randomized Variable-Dependent Termination Time
The notion of program sensitivity (aka Lipschitz continuity) specifies that
changes in the program input result in proportional changes to the program
output. For probabilistic programs the notion is naturally extended to expected
sensitivity. A previous approach develops a relational program logic framework
for proving expected sensitivity of probabilistic while loops, where the number
of iterations is fixed and bounded. In this work, we consider probabilistic
while loops where the number of iterations is not fixed, but randomized and
depends on the initial input values. We present a sound approach for proving
expected sensitivity of such programs. Our sound approach is martingale-based
and can be automated through existing martingale-synthesis algorithms.
Furthermore, our approach is compositional for sequential composition of while
loops under a mild side condition. We demonstrate the effectiveness of our
approach on several classical examples from Gambler's Ruin, stochastic hybrid
systems and stochastic gradient descent. We also present experimental results
showing that our automated approach can handle various probabilistic programs
in the literature
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs
Computing reachability probabilities is a fundamental problem in the analysis
of probabilistic programs. This paper aims at a comprehensive and comparative
account on various martingale-based methods for over- and under-approximating
reachability probabilities. Based on the existing works that stretch across
different communities (formal verification, control theory, etc.), we offer a
unifying account. In particular, we emphasize the role of order-theoretic fixed
points---a classic topic in computer science---in the analysis of probabilistic
programs. This leads us to two new martingale-based techniques, too. We give
rigorous proofs for their soundness and completeness. We also make an
experimental comparison using our implementation of template-based synthesis
algorithms for those martingales
The periodic decomposition problem
If a function can be represented as the sum of
periodic functions as with
(), then it also satisfies a corresponding -order difference
equation . The periodic
decomposition problem asks for the converse implication, which may hold or fail
depending on the context (on the system of periods, on the function class in
which the problem is considered, etc.). The problem has natural extensions and
ramifications in various directions, and is related to several other problems
in real analysis, Fourier and functional analysis. We give a survey about the
available methods and results, and present a number of intriguing open
problems
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