113 research outputs found
THE ADDITION OF TEMPORAL NEIGHBORHOOD MAKES THE LOGIC OF PREFIXES AND SUB-INTERVALS EXPSPACE-COMPLETE
A classic result by Stockmeyer [Sto74] gives a non-elementary lower bound to the emptiness problem for generalized ∗-free regular expressions. This result is intimately connected to the satisfiability problem for the interval temporal logic of the chop modality under the homogeneity assumption [HMM83]. The chop modality can indeed be viewed as the inverse of the concatenation operator of regular languages, and such a correspondence enables reductions between the two problems. 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 introduce the logic BDhom featuring modalities B (for begins), corresponding to the prefix relation on pairs of intervals, and D (for during), corresponding to the infix relation, whose satisfiability problem, under the homogeneity assumption, has been recently shown to be PSpace-complete [BMPS21b]. The homogeneous models of BDhom naturally correspond to languages defined by restricted forms of generalized *-free regular expressions, that feature operators for union, complementation, and the inverses of the prefix and infix relations. Then, we study the extension of BDhom with the temporal neighborhood modality A, corresponding to the Allen relation Meets, and prove that such an addition increases both the expressiveness and the complexity of the logic. In particular, we show that the resulting logic BDAhom is ExpSpace-complete
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
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
PSPACE-completeness of the temporal logic of sub-intervals and suffixes
In this paper, we prove PSPACE-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality 〈E〉, for the “suffix” relation on pairs of intervals, and modality 〈D〉, for the “sub-interval” relation, under the homogeneity assumption. The result significantly improves the EXPSPACE upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (〈E〉) or, symmetrically, the modality for prefixes (〈B〉) to the logic of sub-intervals (featuring only 〈D〉)
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
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
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