79 research outputs found
On a temporal logic of prefixes and infixes
A classic result by Stockmeyer [16] gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of the chop operator under the homogeneity assumption [5]. In this paper, we study the complexity of the satisfiability problem for a suitable weakening of the chop interval temporal logic, that can be equivalently viewed as a fragment of Halpern and Shoham interval logic featuring the operators B, for \u201cbegins\u201d, corresponding to the prefix relation on pairs of intervals, and D, for \u201cduring\u201d, corresponding to the infix relation. The homogeneous models of the considered logic naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations
The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete
A classic result by Stockmeyer gives a non-elementary lower bound to the
emptiness problem for star-free generalized regular expressions. This result is
intimately connected to the satisfiability problem for interval temporal logic,
notably for formulas that make use of the so-called chop operator. Such an
operator can indeed be interpreted as the inverse of the concatenation
operation on regular languages, and this correspondence enables reductions
between non-emptiness of star-free generalized regular expressions and
satisfiability of formulas of the interval temporal logic of chop under the
homogeneity assumption. In this paper, we study the complexity of the
satisfiability problem for suitable weakenings of the chop interval temporal
logic, that can be equivalently viewed as fragments of Halpern and Shoham
interval logic. We first consider the logic featuring
modalities , for \emph{begins}, corresponding to the prefix relation on
pairs of intervals, and , for \emph{during}, corresponding to the infix
relation. The homogeneous models of naturally correspond to
languages defined by restricted forms of regular expressions, that use union,
complementation, and the inverses of the prefix and infix relations. Such a
fragment has been recently shown to be PSPACE-complete . In this paper, we
study the extension with the temporal neighborhood modality
(corresponding to the Allen relation \emph{Meets}), and prove that it
increases both its expressiveness and complexity. In particular, we show that
the resulting logic is EXPSPACE-complete.Comment: arXiv admin note: substantial text overlap with arXiv:2109.0832
Zero-Reachability in Probabilistic Multi-Counter Automata
We study the qualitative and quantitative zero-reachability problem in
probabilistic multi-counter systems. We identify the undecidable variants of
the problems, and then we concentrate on the remaining two cases. In the first
case, when we are interested in the probability of all runs that visit zero in
some counter, we show that the qualitative zero-reachability is decidable in
time which is polynomial in the size of a given pMC and doubly exponential in
the number of counters. Further, we show that the probability of all
zero-reaching runs can be effectively approximated up to an arbitrarily small
given error epsilon > 0 in time which is polynomial in log(epsilon),
exponential in the size of a given pMC, and doubly exponential in the number of
counters. In the second case, we are interested in the probability of all runs
that visit zero in some counter different from the last counter. Here we show
that the qualitative zero-reachability is decidable and SquareRootSum-hard, and
the probability of all zero-reaching runs can be effectively approximated up to
an arbitrarily small given error epsilon > 0 (these result applies to pMC
satisfying a suitable technical condition that can be verified in polynomial
time). The proof techniques invented in the second case allow to construct
counterexamples for some classical results about ergodicity in stochastic Petri
nets.Comment: 20 page
LTL Parameter Synthesis of Parametric Timed Automata
The parameter synthesis problem for parametric timed automata is undecidable
in general even for very simple reachability properties. In this paper we
introduce restrictions on parameter valuations under which the parameter
synthesis problem is decidable for LTL properties. The investigated bounded
integer parameter synthesis problem could be solved using an explicit
enumeration of all possible parameter valuations. We propose an alternative
symbolic zone-based method for this problem which results in a faster
computation. Our technique extends the ideas of the automata-based approach to
LTL model checking of timed automata. To justify the usefulness of our
approach, we provide experimental evaluation and compare our method with
explicit enumeration technique.Comment: 23 pages, extended versio
Language Emptiness of Continuous-Time Parametric Timed Automata
Parametric timed automata extend the standard timed automata with the
possibility to use parameters in the clock guards. In general, if the
parameters are real-valued, the problem of language emptiness of such automata
is undecidable even for various restricted subclasses. We thus focus on the
case where parameters are assumed to be integer-valued, while the time still
remains continuous. On the one hand, we show that the problem remains
undecidable for parametric timed automata with three clocks and one parameter.
On the other hand, for the case with arbitrary many clocks where only one of
these clocks is compared with (an arbitrary number of) parameters, we show that
the parametric language emptiness is decidable. The undecidability result
tightens the bounds of a previous result which assumed six parameters, while
the decidability result extends the existing approaches that deal with
discrete-time semantics only. To the best of our knowledge, this is the first
positive result in the case of continuous-time and unbounded integer
parameters, except for the rather simple case of single-clock automata
The ideal view on Rackoff's coverability technique
Rackoff’s small witness property for the coverability problem is the standard means to prove tight upper bounds in vector addition systems (VAS) and many extensions. We show how to derive the same bounds directly on the computations of the VAS instantiation of the generic backward coverability algorithm. This relies on a dual view of the algorithm using ideal decompositions of downwards-closed sets, which exhibits a key structural invariant in the VAS case. The same reasoning readily generalises to several VAS extensions
Interprocedural Reachability for Flat Integer Programs
We study programs with integer data, procedure calls and arbitrary call
graphs. We show that, whenever the guards and updates are given by octagonal
relations, the reachability problem along control flow paths within some
language w1* ... wd* over program statements is decidable in Nexptime. To
achieve this upper bound, we combine a program transformation into the same
class of programs but without procedures, with an Np-completeness result for
the reachability problem of procedure-less programs. Besides the program, the
expression w1* ... wd* is also mapped onto an expression of a similar form but
this time over the transformed program statements. Several arguments involving
context-free grammars and their generative process enable us to give tight
bounds on the size of the resulting expression. The currently existing gap
between Np-hard and Nexptime can be closed to Np-complete when a certain
parameter of the analysis is assumed to be constant.Comment: 38 pages, 1 figur
An in-depth investigation of interval temporal logic model checking with regular expressions
In the last years, the model checking (MC) problem for interval temporal logic (ITL) has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained by imposing suitable restrictions on interval labeling. In this paper, we overcome such limitations by using regular expressions to define the behavior of proposition letters over intervals in terms of the component states. We first prove that MC for Halpern and Shoham’s ITL (HS), extended with regular expressions, is decidable. Then, we show that formulas of a large class of HS fragments, namely, all fragments featuring (a subset of) HS modalities for Allen’s relations meets, met-by, starts, and started-by, can be model checked in polynomial working space (MC for all these fragments turns out to be PSPACE-complete)
Approaching the Coverability Problem Continuously
The coverability problem for Petri nets plays a central role in the
verification of concurrent shared-memory programs. However, its high
EXPSPACE-complete complexity poses a challenge when encountered in real-world
instances. In this paper, we develop a new approach to this problem which is
primarily based on applying forward coverability in continuous Petri nets as a
pruning criterion inside a backward coverability framework. A cornerstone of
our approach is the efficient encoding of a recently developed polynomial-time
algorithm for reachability in continuous Petri nets into SMT. We demonstrate
the effectiveness of our approach on standard benchmarks from the literature,
which shows that our approach decides significantly more instances than any
existing tool and is in addition often much faster, in particular on large
instances.Comment: 18 pages, 4 figure
Realizing Omega-regular Hyperproperties
We studied the hyperlogic HyperQPTL, which combines the concepts of trace
relations and -regularity. We showed that HyperQPTL is very expressive,
it can express properties like promptness, bounded waiting for a grant,
epistemic properties, and, in particular, any -regular property. Those
properties are not expressible in previously studied hyperlogics like HyperLTL.
At the same time, we argued that the expressiveness of HyperQPTL is optimal in
a sense that a more expressive logic for -regular hyperproperties would
have an undecidable model checking problem. We furthermore studied the
realizability problem of HyperQPTL. We showed that realizability is decidable
for HyperQPTL fragments that contain properties like promptness. But still, in
contrast to the satisfiability problem, propositional quantification does make
the realizability problem of hyperlogics harder. More specifically, the
HyperQPTL fragment of formulas with a universal-existential propositional
quantifier alternation followed by a single trace quantifier is undecidable in
general, even though the projection of the fragment to HyperLTL has a decidable
realizability problem. Lastly, we implemented the bounded synthesis problem for
HyperQPTL in the prototype tool BoSy. Using BoSy with HyperQPTL specifications,
we have been able to synthesize several resource arbiters. The synthesis
problem of non-linear-time hyperlogics is still open. For example, it is not
yet known how to synthesize systems from specifications given in branching-time
hyperlogics like HyperCTL.Comment: International Conference on Computer Aided Verification (CAV 2020
- …