4,673 research outputs found
Regular Separability of Well-Structured Transition Systems
We investigate the languages recognized by well-structured transition systems (WSTS) with upward and downward compatibility. Our first result shows that, under very mild assumptions, every two disjoint WSTS languages are regular separable: There is a regular language containing one of them and being disjoint from the other. As a consequence, if a language as well as its complement are both recognized by WSTS, then they are necessarily regular. In particular, no subclass of WSTS languages beyond the regular languages is closed under complement. Our second result shows that for Petri nets, the complexity of the backwards coverability algorithm yields a bound on the size of the regular separator. We complement it by a lower bound construction
Separability and Non-Determinizability of WSTS
There is a recent separability result for the languages of well-structured
transition systems (WSTS) that is surprisingly general: disjoint WSTS languages
are always separated by a regular language. The result assumes that one of the
languages is accepted by a deterministic WSTS, and it is not known whether this
assumption is needed. There are two ways to get rid of the assumption, none of
which has led to conclusions so far: (i) show that WSTS can be determinized or
(ii) generalize the separability result to non-deterministic WSTS languages.
Our contribution is to show that (i) does not work but (ii) does. As for (i),
we give a non-deterministic WSTS language that we prove cannot be accepted by a
deterministic WSTS. The proof relies on a novel characterization of the
languages accepted by deterministic WSTS. As for (ii), we show how to find
finitely represented inductive invariants without having the tool of ideal
decompositions at hand. Instead, we work with closures under converging
sequences. Our results hold for upward- and downward-compatible WSTS
A Characterization for Decidable Separability by Piecewise Testable Languages
The separability problem for word languages of a class by
languages of a class asks, for two given languages and
from , whether there exists a language from that
includes and excludes , that is, and . In this work, we assume some mild closure properties for
and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
Our characterization involves a result of independent interest, which states
that for any kind of languages and , non-separability by PTL is
equivalent to the existence of common patterns in and
Regular Separability and Intersection Emptiness Are Independent Problems
The problem of regular separability asks, given two languages K and L, whether there exists a regular language S that includes K and is disjoint from L. This problem becomes interesting when the input languages K and L are drawn from language classes beyond the regular languages. For such classes, a mild and useful assumption is that they are full trios, i.e. closed under rational transductions.
All the results on regular separability for full trios obtained so far exhibited a noteworthy correspondence with the intersection emptiness problem: In each case, regular separability is decidable if and only if intersection emptiness is decidable. This raises the question whether for full trios, regular separability can be reduced to intersection emptiness or vice-versa.
We present counterexamples showing that neither of the two problems can be reduced to the other. More specifically, we describe full trios C_1, D_1, C_2, D_2 such that (i) intersection emptiness is decidable for C_1 and D_1, but regular separability is undecidable for C_1 and D_1 and (ii) regular separability is decidable for C_2 and D_2, but intersection emptiness is undecidable for C_2 and D_2
Classical Emergence of Intrinsic Spin-Orbit Interaction of Light at the Nanoscale
Traditionally, in macroscopic geometrical optics intrinsic polarization and
spatial degrees of freedom of light can be treated independently. However, at
the subwavelength scale these properties appear to be coupled together, giving
rise to the spin-orbit interaction (SOI) of light. In this work we address
theoretically the classical emergence of the optical SOI at the nanoscale. By
means of a full-vector analysis involving spherical vector waves we show that
the spin-orbit factorizability condition, accounting the mutual influence
between the amplitude (spin) and phase (orbit), is fulfilled only in the
far-field limit. On the other side, in the near-field region, an additional
relative phase introduces an extra term that hinders the factorization and
reveals an intricate dynamical behavior according to the SOI regime. As a
result, we find a suitable theoretical framework able to capture analytically
the main features of intrinsic SOI of light. Besides allowing for a better
understanding into the mechanism leading to its classical emergence at the
nanoscale, our approach may be useful in order to design experimental setups
that enhance the response of SOI-based effects.Comment: 10 pages, 5 figure
An Approach to Regular Separability in Vector Addition Systems
We study the problem of regular separability of languages of vector addition
systems with states (VASS). It asks whether for two given VASS languages K and
L, there exists a regular language R that includes K and is disjoint from L.
While decidability of the problem in full generality remains an open question,
there are several subclasses for which decidability has been shown: It is
decidable for (i) one-dimensional VASS, (ii) VASS coverability languages, (iii)
languages of integer VASS, and (iv) commutative VASS languages. We propose a
general approach to deciding regular separability. We use it to decide regular
separability of an arbitrary VASS language from any language in the classes
(i), (ii), and (iii). This generalizes all previous results, including (iv)
An Aggregation Technique for Large-Scale PEPA Models with Non-Uniform Populations
Performance analysis based on modelling consists of two major steps: model
construction and model analysis. Formal modelling techniques significantly aid
model construction but can exacerbate model analysis. In particular, here we
consider the analysis of large-scale systems which consist of one or more
entities replicated many times to form large populations. The replication of
entities in such models can cause their state spaces to grow exponentially to
the extent that their exact stochastic analysis becomes computationally
expensive or even infeasible.
In this paper, we propose a new approximate aggregation algorithm for a class
of large-scale PEPA models. For a given model, the method quickly checks if it
satisfies a syntactic condition, indicating that the model may be solved
approximately with high accuracy. If so, an aggregated CTMC is generated
directly from the model description. This CTMC can be used for efficient
derivation of an approximate marginal probability distribution over some of the
model's populations. In the context of a large-scale client-server system, we
demonstrate the usefulness of our method
Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable
We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems
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