6,517 research outputs found
On reachability in graphs with bounded independence number
Abstract. We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC 0 -circuits, but cannot be computed using AC 0 -circuits. We also study the succinct version of the problem and show that it is Î P 2 -complete for all k
Reachability analysis of reversal-bounded automata on series–parallel graphs
Extensions to finite-state automata on strings, such as multi-head automata or multi-counter automata, have been successfully used to encode many infinite-state non-regular verification problems. In this paper, we consider a generalization of automata-theoretic infinite-state verification from strings to labelled series–parallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on series–parallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of series–parallel graphs described by a context-free graph transformation system (GTS), and a concurrent automaton over series–parallel graphs, is some graph generated by the GTS accepted by the automaton? The general problem is undecidable already for (one-way) multi-head automata over strings. We show that a bounded version, where the automata make a fixed number of reversals along the graph and use a fixed number of shared registers is decidable, even though there is no bound on the sizes of series–parallel graphs generated by the GTS. Our decidability result is based on establishing that the number of context switches can be bounded and on an encoding of the computation of bounded concurrent automata that allows us to reduce the reachability problem to the emptiness problem for pushdown automata
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
Weak Singular Hybrid Automata
The framework of Hybrid automata, introduced by Alur, Courcourbetis,
Henzinger, and Ho, provides a formal modeling and analysis environment to
analyze the interaction between the discrete and the continuous parts of
cyber-physical systems. Hybrid automata can be considered as generalizations of
finite state automata augmented with a finite set of real-valued variables
whose dynamics in each state is governed by a system of ordinary differential
equations. Moreover, the discrete transitions of hybrid automata are guarded by
constraints over the values of these real-valued variables, and enable
discontinuous jumps in the evolution of these variables. Singular hybrid
automata are a subclass of hybrid automata where dynamics is specified by
state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed
that for even very restricted subclasses of singular hybrid automata, the
fundamental verification questions, like reachability and schedulability, are
undecidable. In this paper we present \emph{weak singular hybrid automata}
(WSHA), a previously unexplored subclass of singular hybrid automata, and show
the decidability (and the exact complexity) of various verification questions
for this class including reachability (NP-Complete) and LTL model-checking
(PSPACE-Complete). We further show that extending WSHA with a single
unrestricted clock or extending WSHA with unrestricted variable updates lead to
undecidability of reachability problem
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
Algorithms and Conditional Lower Bounds for Planning Problems
We consider planning problems for graphs, Markov decision processes (MDPs),
and games on graphs. While graphs represent the most basic planning model, MDPs
represent interaction with nature and games on graphs represent interaction
with an adversarial environment. We consider two planning problems where there
are k different target sets, and the problems are as follows: (a) the coverage
problem asks whether there is a plan for each individual target set, and (b)
the sequential target reachability problem asks whether the targets can be
reached in sequence. For the coverage problem, we present a linear-time
algorithm for graphs and quadratic conditional lower bound for MDPs and games
on graphs. For the sequential target problem, we present a linear-time
algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic
conditional lower bound for games on graphs. Our results with conditional lower
bounds establish (i) model-separation results showing that for the coverage
problem MDPs and games on graphs are harder than graphs and for the sequential
reachability problem games on graphs are harder than MDPs and graphs; (ii)
objective-separation results showing that for MDPs the coverage problem is
harder than the sequential target problem.Comment: Accepted at ICAPS'1
Pushdown Control-Flow Analysis of Higher-Order Programs
Context-free approaches to static analysis gain precision over classical
approaches by perfectly matching returns to call sites---a property that
eliminates spurious interprocedural paths. Vardoulakis and Shivers's recent
formulation of CFA2 showed that it is possible (if expensive) to apply
context-free methods to higher-order languages and gain the same boost in
precision achieved over first-order programs.
To this young body of work on context-free analysis of higher-order programs,
we contribute a pushdown control-flow analysis framework, which we derive as an
abstract interpretation of a CESK machine with an unbounded stack. One
instantiation of this framework marks the first polyvariant pushdown analysis
of higher-order programs; another marks the first polynomial-time analysis. In
the end, we arrive at a framework for control-flow analysis that can
efficiently compute pushdown generalizations of classical control-flow
analyses.Comment: The 2010 Workshop on Scheme and Functional Programmin
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