7 research outputs found
A Perfect Model for Bounded Verification
A class of languages C is perfect if it is closed under Boolean operations
and the emptiness problem is decidable. Perfect language classes are the basis
for the automata-theoretic approach to model checking: a system is correct if
the language generated by the system is disjoint from the language of bad
traces. Regular languages are perfect, but because the disjointness problem for
CFLs is undecidable, no class containing the CFLs can be perfect.
In practice, verification problems for language classes that are not perfect
are often under-approximated by checking if the property holds for all
behaviors of the system belonging to a fixed subset. A general way to specify a
subset of behaviors is by using bounded languages (languages of the form w1*
... wk* for fixed words w1,...,wk). A class of languages C is perfect modulo
bounded languages if it is closed under Boolean operations relative to every
bounded language, and if the emptiness problem is decidable relative to every
bounded language.
We consider finding perfect classes of languages modulo bounded languages. We
show that the class of languages accepted by multi-head pushdown automata are
perfect modulo bounded languages, and characterize the complexities of decision
problems. We also show that bounded languages form a maximal class for which
perfection is obtained. We show that computations of several known models of
systems, such as recursive multi-threaded programs, recursive counter machines,
and communicating finite-state machines can be encoded as multi-head pushdown
automata, giving uniform and optimal underapproximation algorithms modulo
bounded languages.Comment: 14 pages, 6 figure
On the Path-Width of Integer Linear Programming
We consider the feasibility problem of integer linear programming (ILP). We
show that solutions of any ILP instance can be naturally represented by an
FO-definable class of graphs. For each solution there may be many graphs
representing it. However, one of these graphs is of path-width at most 2n,
where n is the number of variables in the instance. Since FO is decidable on
graphs of bounded path- width, we obtain an alternative decidability result for
ILP. The technique we use underlines a common principle to prove decidability
which has previously been employed for automata with auxiliary storage. We also
show how this new result links to automata theory and program verification.Comment: In Proceedings GandALF 2014, arXiv:1408.556
The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems
International audienceBounded languages have recently proved to be an important class of languages for the analysis of Turing-powerful models. For instance, bounded context-free languages are used to under-approximate the behav-iors of recursive programs. Ginsburg and Spanier have shown in 1966 that a bounded language L â a * 1 · · · a * d is context-free if, and only if, its Parikh image is a stratifiable semilinear set. However, the question whether a semilinear set is stratifiable, hereafter called the stratifiability problem, was left open, and remains so. In this paper, we give a partial answer to this problem. We focus on semilinear sets that are given as finite systems of linear inequalities, and we show that stratifiability is coNP-complete in this case. Then, we apply our techniques to the context-freeness problem for flat counter systems, that asks whether the trace language of a counter system intersected with a bounded regular language is context-free. As main result of the paper, we show that this problem is coNP-complete
Bounded Context Switching for Valence Systems
We study valence systems, finite-control programs over infinite-state memories modeled in terms of graph monoids. Our contribution is a notion of bounded context switching (BCS). Valence systems generalize pushdowns, concurrent pushdowns, and Petri nets. In these settings, our definition conservatively generalizes existing notions. The main finding is that reachability within a bounded number of context switches is in NPTIME, independent of the memory (the graph monoid). Our proof is genuinely algebraic, and therefore contributes a new way to think about BCS. In addition, we exhibit a class of storage mechanisms for which BCS reachability belongs to PTIME
Reachability analysis of reversal-bounded automata on seriesâparallel 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 seriesâparallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on seriesâparallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of seriesâparallel graphs described by a context-free graph transformation system (GTS), and a concurrent automaton over seriesâparallel 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 seriesâparallel 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