Kleene Star Research Articles

Conjunctive Regular Path Queries with Capture Groups

In practice, regular expressions are usually extended by so-called capture groups or capture variables, which allow to capture a subexpression by a variable that can be referenced in the regular expression in order to describe repetitions of subwords. We investigate how this concept could be used for pattern-based graph querying; i.e., we investigate conjunctive regular path queries (CRPQs) that are extended by capture variables. If capture variables are added to CRPQs in a completely unrestricted way, then Boolean evaluation becomes PSPACE-hard in data complexity, even for single-edge graph patterns. On the other hand, if capture variables do not occur under a Kleene star, then the data complexity drops to NL-completeness. Combined complexity is in EXPSPACE but drops to PSPACE-completeness if the depth (i.e., the nesting depth of capture variables) is bounded, and it drops to NP-completeness if the size of the images of capture variables is bounded by a constant (regardless of the depth or of whether capture variables occur under a Kleene star). In the application of regular expressions as string searching tools, references to capture variables only describe exact repetitions of subwords (i.e., they implement the equality relation on strings). Following recent developments in graph database research, we also study CRPQs with capture variables that describe arbitrary regular relations. We show that if the expressions have depth 0, or if the size of the images of capture variables is bounded by a constant, then we can allow arbitrary regular relations while staying in the same complexity bounds. We also investigate the problems of checking whether a given tuple is in the solution set and computing the whole solution set. On the conceptual side, we add capture variables to CRPQs in such a way that they can be defined in an expression on one arc of the graph pattern but also referenced in expressions on other arcs. Hence, they add to CRPQs the possibility to define inter-dependencies between different paths, which is a relevant feature of pattern-based graph querying.

The Hodge star operator and the Beltrami equation

An essentially unique homeomorphic solution to the Beltrami equation with measurable coefficients was found in the 1930s by Morrey. The most well-known proof from the 1960s uses the theory of Calderón–Zygmund and singular integral operators in $$L^p(\mathbb {C})$$ . We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar.

Vanishing theorems for p-harmonic forms on Riemannian manifolds

In this paper, we establish several vanishing type theorems for a p-harmonic ℓ-form ω on a Riemannian manifold M. We give some vanishing theorems on a manifold with a weighted Poincaré inequality. To do so, we need to employ a Bochner formula on both ω and ⋆ω, where ⋆ is the Hodge star operator. Then, we need to control the nonlinearity of the curvature terms and inner product of forms. Moreover, using these techniques we can show that if a pure curvature tensor condition is satisfied, then we still obtain another vanishing theorem. Finally, by using Sobolev inequality in terms of Yamabe invariant, a vanishing property for p-harmonic ℓ-forms on Riemannian manifolds is also given.

Moduli stabilization in asymptotic flux compactifications

We present a novel strategy to systematically study complex-structure moduli stabilization in Type IIB and F-theory flux compactifications. In particular, we determine vacua in any asymptotic regime of the complex-structure moduli space by exploiting powerful tools of asymptotic Hodge theory. In a leading approximation the moduli dependence of the vacuum conditions are shown to be polynomial with a dependence given by sl(2)-weights of the fluxes. This simple algebraic dependence can be extracted in any asymptotic regime, even though in nearly all asymptotic regimes essential exponential corrections have to be present for consistency. We give a pedagogical introduction to the sl(2)-approximation as well as a detailed step-by-step procedure for constructing the corresponding Hodge star operator. To exemplify the construction, we present a detailed analysis of several Calabi-Yau three- and fourfold examples. For these examples we illustrate that the vacua in the sl(2)-approximation match the vacua obtained with all polynomial and essential exponential corrections rather well, and we determine the behaviour of the tadpole contribution of the fluxes. Finally, we discuss the structure of vacuum loci and their relations to several swampland conjectures. In particular, we comment on the realization of the so-called linear scenario in view of the tadpole conjecture.

Intuitionistic fuzzy (⊗,N)-general regular languages and their minimization implementation

We firstly give the notion of intuitionistic fuzzy (⊗,N)-general regular languages (IF(⊗,N)-GRLs) under a t-norm ⊗ and a fuzzy negation N and study their properties. The similarities and differences between IF(⊗,N)-GRLs and intuitionistic fuzzy regular languages (IFRLs) are investigated for explaining the existence of intuitionistic fuzzy automata (IFAs) (⊗,N)-recognizing IF(⊗,N)-GRLs. Next, we discuss whether or not the set of IF(⊗,N)-GRLs is closed under the generalized operations shown here. Concretely, the set of IF(⊗,N)-GRLs is closed under the generalized intersection, but not closed under the generalized complement, and the necessary and sufficient condition of closure of that set under the generalized union, the generalized concatenation and the generalized Kleene closure is thought as well. We also deliberate on the Pumping lemma and the Myhill-Nerode theorem in the framework of IF(⊗,N)-GRLs. Finally, we study the minimization implementation of IF(⊗,N)-GRLs, that is, for an IF(⊗,N)-GRL f, if it is (⊗,N)-recognized by a given accessible deterministic intuitionistic fuzzy automaton (DIFA) A, then a polynomial-time algorithm is presented to realize the construction of a minimal DIFA (⊗,N)-recognizing f by using A.

Solving string constraints with Regex-dependent functions through transducers with priorities and variables

Regular expressions are a classical concept in formal language theory. Regular expressions in programming languages (RegEx) such as JavaScript, feature non-standard semantics of operators (e.g. greedy/lazy Kleene star), as well as additional features such as capturing groups and references. While symbolic execution of programs containing RegExes appeals to string solvers natively supporting important features of RegEx, such a string solver is hitherto missing. In this paper, we propose the first string theory and string solver that natively provides such support. The key idea of our string solver is to introduce a new automata model, called prioritized streaming string transducers (PSST), to formalize the semantics of RegEx-dependent string functions. PSSTs combine priorities, which have previously been introduced in prioritized finite-state automata to capture greedy/lazy semantics, with string variables as in streaming string transducers to model capturing groups. We validate the consistency of the formal semantics with the actual JavaScript semantics by extensive experiments. Furthermore, to solve the string constraints, we show that PSSTs enjoy nice closure and algorithmic properties, in particular, the regularity-preserving property (i.e., pre-images of regular constraints under PSSTs are regular), and introduce a sound sequent calculus that exploits these properties and performs propagation of regular constraints by means of taking post-images or pre-images. Although the satisfiability of the string constraint language is generally undecidable, we show that our approach is complete for the so-called straight-line fragment. We evaluate the performance of our string solver on over 195000 string constraints generated from an open-source RegEx library. The experimental results show the efficacy of our approach, drastically improving the existing methods (via symbolic execution) in both precision and efficiency.

Compton Imaging and Machine-Learning techniques for an enhanced sensitivity in key stellar (n,γ) measurements

Neutron capture cross-section measurements are fundamental in the study of astrophysical phenomena, such as the slow neutron capture (s-) process of nucleosynthesis operating in red-giant stars. To enhance the sensitivity of such measurements we have developed the i-TED detector. i-TED is an innovative detection system which exploits the Compton imaging technique with the aim of obtaining information about the incoming direction of the detected γ-rays. The imaging capability allows one to reject a large fraction of the dominant γ-ray background, hence enhancing the (n,γ) detection sensitivity. This work summarizes the main results of the first experimental proof-of-concept of the background rejection with i-TED carried out at CERN n_TOF using an early i-TED demonstrator. Two state-of-the-art C6D6 detectors were also used to benchmark the performance of i-TED. The i-TED prototype built for this study shows a factor of ~3 higher detection sensitivity than C6D6 detectors in the ~10 keV neutron-energy range of astrophysical interest. This works also introduces the perspectives of further enhancement in performance attainable with the final i-TED array and new analysis methodologies based on Machine-Learning techniques. The latter provide higher (n,γ) detection efficiency and similar enhancement in the sensitivity than the analytical method based on the Compton scattering law. Finally, we present our proposal to use this detection system for the first time on key astrophysical (n,γ) measurements, in particular on the s-process branching-point 79Se, which is especially well suited to constrain the thermal conditions of Red Giant and Massive Stars.

On some classes of integral domains with finitely many star operations of finite type

Let R be an integral domain. We investigate when R has only finitely many star operations of finite type. We start by investigating some classes of integral domains with at most two star operations of finite type. Our aim is to generalize or find analogues of several characterizations of well-known classes of integral domains with at most two star operations. Next we concentrate on Mori domains with only finitely many star operations of finite type in a similar study done for Noetherian domains. We end by a section on pullbacks to enrich the literature with new and original examples and also to illustrate several results in the paper.

Infinitary action logic with exponentiation

We introduce infinitary action logic with exponentiation—that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allow some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is Π11-complete. Second, we show that the closure ordinal of its derivability operator is ω1CK. In the case where no subexponential allows contraction, we show that complexity is the same as for infinitary action logic itself. Namely, the derivability problem in this case is Π10-complete and the closure ordinal is not greater than ωω.

Kleene star, subexponentials without contraction, and infinite computations

Kleene star, subexponentials without contraction, and infinite computations

Asymptotic for the number of star operations on one-dimensional Noetherian domains

We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.

Generalization of the coordinates non-commutativity to a general manifold

In the framework of quantum mechanics and based on the non-commutativity between the ‎coordinates in Minkowski space-time, we generalize the geometric non-commutative relation ‎to a space-time other than Minkowski. Using the authority of inserting the unit operator, we ‎exploit the translation operator to derive the Wyle-Moyal star product operator. Up to the first ‎order of translation parameters and by employing the Wyle-Moyal star operator, we find the ‎modified non-commutativity of coordinates relation in terms of geometric structure. The basic ‎premise of this article is that pseudo-Riemannian local homomorphic with Minkowski space-‎time‎‎ are equivalent.

Linear preservers for matrices over a class of semirings

We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator.

Symbolic and automatic differentiation of languages

Formal languages are usually defined in terms of set theory. Choosing type theory instead gives us languages as type-level predicates over strings. Applying a language to a string yields a type whose elements are language membership proofs describing how a string parses in the language. The usual building blocks of languages (including union, concatenation, and Kleene closure) have precise and compelling specifications uncomplicated by operational strategies and are easily generalized to a few general domain-transforming and codomain-transforming operations on predicates. A simple characterization of languages (and indeed functions from lists to any type) captures the essential idea behind language “differentiation” as used for recognizing languages, leading to a collection of lemmas about type-level predicates. These lemmas are the heart of two dual parsing implementations—using (inductive) regular expressions and (coinductive) tries—each containing the same code but in dual arrangements (with representation and primitive operations trading places). The regular expression version corresponds to symbolic differentiation, while the trie version corresponds to automatic differentiation. The relatively easy-to-prove properties of type-level languages transfer almost effortlessly to the decidable implementations. In particular, despite the inductive and coinductive nature of regular expressions and tries respectively, we need neither inductive nor coinductive/bisimulation arguments to prove algebraic properties.

On the integral domains characterized by a Bezout property on intersections of principal ideals

In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of containment of one of the principal ideals in all the others. We relate these classes to many well-studied classes of integral domains, to star operations and to classical and new ring constructions.

A simple and complete discrete exterior calculus on general polygonal meshes

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

Action Logic is Undecidable

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.

Regular Expressions with Lookahead

This paper investigates regular expressions which in addition to the standard operators of union, concatenation, and Kleene star, have lookaheads. We show how to translate regular expressions with lookaheads (REwLA) to equivalent Boolean automata having at most 3 states more than the length of the REwLA. We also investigate the state complexity when translating REwLA to equivalent deterministic finite automata (DFA).

Editorial

I am pleased to announce the fourth issue of 2021. As always, I would like to express my sincere appreciation for the great support that makes the continued publication of novel and high quality articles possible. Thus, I would like to thank all authors for their sound research contributions, the reviewers for their very helpful suggestions and the consortium members for their financial support. I would also like to report on further achievements regarding our new platform. We have successfully migrated all the information of the Board of Editors and we have also started to use the new review module. Due to the cooperation with Pensoft Inc., our new platform provider, we will also be able to offer review acknowledgment on the Publons portal in the future. In this regular issue, I am very pleased to introduce four accepted papers from three different countries and 14 involved authors. Martin Berglund, Brink van der Merwe, and Steyn van Litsenborgh from South Africa investigate in their article regular expressions which contain lookaheads in addition to the standard operators of union, concatenation, and Kleene star. Fairouz Fakhfakh, Slim Kallel and Saoussen Cheikhrouhou from Tunisia research and discuss in their work a crucial issue in modern distributed information systems, i.e. how to verify the correctness of Cloud and Fog systems based on formal verification. Marcia Henke, Eulanda Santos, Eduardo Souto, and Altair O. Santin from Brazil introduce their enhanced spam detection system which is based on analyzing the evolution of features. And finally, also from Brazil, Marcelo Aires Vieira, Elivaldo Lozer Fracalossi Ribeiro, Daniela Barreiro Claro, and Babacar Mane investigate the challenging problem of integrating heterogeneous DaaS and DBaaS sources and explore the Data Join (DJ) method for integrating heterogeneous data.

Union-Freeness Revisited — Between Deterministic and Nondeterministic Union-Free Languages

Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free (d-union-free, for short) languages. In this paper [Formula: see text]-free nondeterministic variants of 1CFPAs are used to define n-union-free languages. The defined language class is shown to be properly between the classes of (nondeterministic) union-free and d-union-free languages (in case of at least binary alphabet). In case of unary alphabet the class of n-union-free languages coincides with the class of union-free languages. Some properties of the new subregular class of languages are discussed, e.g., closure properties. On the other hand, a regular expression is in union normal form if it is a finite union of union-free expressions. It is well known that every regular expression can be written in union normal form, i.e., all regular languages can be described as finite unions of (nondeterministic) union-free languages. It is also known that the same fact does not hold for deterministic union-free languages, that is, there are regular languages that cannot be written as finite unions of d-union-free languages. As an important result here we show that every regular language can be defined by a finite union of n-union-free languages. This fact also allows to define n-union-complexity of regular languages.