118 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
Verifying nondeterministic probabilistic channel systems against -regular linear-time properties
Lossy channel systems (LCSs) are systems of finite state automata that
communicate via unreliable unbounded fifo channels. In order to circumvent the
undecidability of model checking for nondeterministic
LCSs, probabilistic models have been introduced, where it can be decided
whether a linear-time property holds almost surely. However, such fully
probabilistic systems are not a faithful model of nondeterministic protocols.
We study a hybrid model for LCSs where losses of messages are seen as faults
occurring with some given probability, and where the internal behavior of the
system remains nondeterministic. Thus the semantics is in terms of
infinite-state Markov decision processes. The purpose of this article is to
discuss the decidability of linear-time properties formalized by formulas of
linear temporal logic (LTL). Our focus is on the qualitative setting where one
asks, e.g., whether a LTL-formula holds almost surely or with zero probability
(in case the formula describes the bad behaviors). Surprisingly, it turns out
that -- in contrast to finite-state Markov decision processes -- the
satisfaction relation for LTL formulas depends on the chosen type of schedulers
that resolve the nondeterminism. While all variants of the qualitative LTL
model checking problem for the full class of history-dependent schedulers are
undecidable, the same questions for finite-memory scheduler can be solved
algorithmically. However, the restriction to reachability properties and
special kinds of recurrent reachability properties yields decidable
verification problems for the full class of schedulers, which -- for this
restricted class of properties -- are as powerful as finite-memory schedulers,
or even a subclass of them.Comment: 39 page
Model Checking Probabilistic Pushdown Automata
We consider the model checking problem for probabilistic pushdown automata
(pPDA) and properties expressible in various probabilistic logics. We start
with properties that can be formulated as instances of a generalized random
walk problem. We prove that both qualitative and quantitative model checking
for this class of properties and pPDA is decidable. Then we show that model
checking for the qualitative fragment of the logic PCTL and pPDA is also
decidable. Moreover, we develop an error-tolerant model checking algorithm for
PCTL and the subclass of stateless pPDA. Finally, we consider the class of
omega-regular properties and show that both qualitative and quantitative model
checking for pPDA is decidable
Reachability Analysis of Communicating Pushdown Systems
The reachability analysis of recursive programs that communicate
asynchronously over reliable FIFO channels calls for restrictions to ensure
decidability. Our first result characterizes communication topologies with a
decidable reachability problem restricted to eager runs (i.e., runs where
messages are either received immediately after being sent, or never received).
The problem is EXPTIME-complete in the decidable case. The second result is a
doubly exponential time algorithm for bounded context analysis in this setting,
together with a matching lower bound. Both results extend and improve previous
work from La Torre et al
Equivalence-Checking on Infinite-State Systems: Techniques and Results
The paper presents a selection of recently developed and/or used techniques
for equivalence-checking on infinite-state systems, and an up-to-date overview
of existing results (as of September 2004)
Simulating perfect channels with probabilistic lossy channels
AbstractWe consider the problem of deciding whether an infinite-state system (expressed as a Markov chain) satisfies a correctness property with probability 1. This problem is, of course, undecidable for general infinite-state systems. We focus our attention on the model of probabilistic lossy channel systems consisting of finite-state processes that communicate over unbounded lossy FIFO channels. Abdulla and Jonsson have shown that safety properties are decidable while progress properties are undecidable for non-probabilistic lossy channel systems. Under assumptions of “sufficiently high” probability of loss, Baier and Engelen have shown how to check whether a property holds of probabilistic lossy channel system with probability 1. In this paper, we consider a model of probabilistic lossy channel systems, where messages can be lost only during send transitions. In contrast to the model of Baier and Engelen, once a message is successfully sent to channel, it can only be removed through a transition which receives the message. We show that checking whether safety properties hold with probability 1 is undecidable for this model. Our proof depends upon simulating a perfect channel, with a high degree of confidence, using lossy channels
Reachability of Communicating Timed Processes
We study the reachability problem for communicating timed processes, both in
discrete and dense time. Our model comprises automata with local timing
constraints communicating over unbounded FIFO channels. Each automaton can only
access its set of local clocks; all clocks evolve at the same rate. Our main
contribution is a complete characterization of decidable and undecidable
communication topologies, for both discrete and dense time. We also obtain
complexity results, by showing that communicating timed processes are at least
as hard as Petri nets; in the discrete time, we also show equivalence with
Petri nets. Our results follow from mutual topology-preserving reductions
between timed automata and (untimed) counter automata.Comment: Extended versio
LIPIcs
Fault-tolerant distributed algorithms play an important role in many critical/high-availability applications. These algorithms are notoriously difficult to implement correctly, due to asynchronous communication and the occurrence of faults, such as the network dropping messages or computers crashing. Nonetheless there is surprisingly little language and verification support to build distributed systems based on fault-tolerant algorithms. In this paper, we present some of the challenges that a designer has to overcome to implement a fault-tolerant distributed system. Then we review different models that have been proposed to reason about distributed algorithms and sketch how such a model can form the basis for a domain-specific programming language. Adopting a high-level programming model can simplify the programmer's life and make the code amenable to automated verification, while still compiling to efficiently executable code. We conclude by summarizing the current status of an ongoing language design and implementation project that is based on this idea
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