1,134 research outputs found
Recurrent Reachability Analysis in Regular Model Checking
Abstract. We consider the problem of recurrent reachability over infinite systems given by regular relations on words and trees, i.e, whether a given regular set of states can be reached infinitely often from a given initial state in the given transition system. Under the condition that the transitive closure of the transition relation is regular, we show that the problem is decidable, and the set of all initial states satisfying the property is regular. Moreover, our algorithm constructs an automaton for this set in polynomial time, assuming that a transducer of the transitive closure can be computed in poly-time. We then demonstrate that transition systems generated by pushdown systems, regular ground tree rewrite systems, and the well-known process algebra PA satisfy our condition and transducers for their transitive closures can be computed in poly-time. Our result also implies that model checking EF-logic extended by recurrent reachability predicate (EGF) over such systems is decidable.
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
On computing fixpoints in well-structured regular model checking, with applications to lossy channel systems
We prove a general finite convergence theorem for "upward-guarded" fixpoint
expressions over a well-quasi-ordered set. This has immediate applications in
regular model checking of well-structured systems, where a main issue is the
eventual convergence of fixpoint computations. In particular, we are able to
directly obtain several new decidability results on lossy channel systems.Comment: 16 page
Liveness of Randomised Parameterised Systems under Arbitrary Schedulers (Technical Report)
We consider the problem of verifying liveness for systems with a finite, but
unbounded, number of processes, commonly known as parameterised systems.
Typical examples of such systems include distributed protocols (e.g. for the
dining philosopher problem). Unlike the case of verifying safety, proving
liveness is still considered extremely challenging, especially in the presence
of randomness in the system. In this paper we consider liveness under arbitrary
(including unfair) schedulers, which is often considered a desirable property
in the literature of self-stabilising systems. We introduce an automatic method
of proving liveness for randomised parameterised systems under arbitrary
schedulers. Viewing liveness as a two-player reachability game (between
Scheduler and Process), our method is a CEGAR approach that synthesises a
progress relation for Process that can be symbolically represented as a
finite-state automaton. The method is incremental and exploits both
Angluin-style L*-learning and SAT-solvers. Our experiments show that our
algorithm is able to prove liveness automatically for well-known randomised
distributed protocols, including Lehmann-Rabin Randomised Dining Philosopher
Protocol and randomised self-stabilising protocols (such as the Israeli-Jalfon
Protocol). To the best of our knowledge, this is the first fully-automatic
method that can prove liveness for randomised protocols.Comment: Full version of CAV'16 pape
Model Checking Synchronized Products of Infinite Transition Systems
Formal verification using the model checking paradigm has to deal with two
aspects: The system models are structured, often as products of components, and
the specification logic has to be expressive enough to allow the formalization
of reachability properties. The present paper is a study on what can be
achieved for infinite transition systems under these premises. As models we
consider products of infinite transition systems with different synchronization
constraints. We introduce finitely synchronized transition systems, i.e.
product systems which contain only finitely many (parameterized) synchronized
transitions, and show that the decidability of FO(R), first-order logic
extended by reachability predicates, of the product system can be reduced to
the decidability of FO(R) of the components. This result is optimal in the
following sense: (1) If we allow semifinite synchronization, i.e. just in one
component infinitely many transitions are synchronized, the FO(R)-theory of the
product system is in general undecidable. (2) We cannot extend the expressive
power of the logic under consideration. Already a weak extension of first-order
logic with transitive closure, where we restrict the transitive closure
operators to arity one and nesting depth two, is undecidable for an
asynchronous (and hence finitely synchronized) product, namely for the infinite
grid.Comment: 18 page
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
Solving Stochastic B\"uchi Games on Infinite Arenas with a Finite Attractor
We consider games played on an infinite probabilistic arena where the first
player aims at satisfying generalized B\"uchi objectives almost surely, i.e.,
with probability one. We provide a fixpoint characterization of the winning
sets and associated winning strategies in the case where the arena satisfies
the finite-attractor property. From this we directly deduce the decidability of
these games on probabilistic lossy channel systems.Comment: In Proceedings QAPL 2013, arXiv:1306.241
The First-Order Theory of Ground Tree Rewrite Graphs
We prove that the complexity of the uniform first-order theory of ground tree
rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower
bound, we show that there is some fixed ground tree rewrite graph whose
first-order theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to
logspace reductions. Finally, we prove that there exists a fixed ground tree
rewrite graph together with a single unary predicate in form of a regular tree
language such that the resulting structure has a non-elementary first-order
theory.Comment: accepted for Logical Methods in Computer Scienc
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