Kleene Star Research Articles

The Frobenius and Factor Universality Problems of the Kleene Star of a Finite Set of Words

We solve open problems concerning the Kleene star of a finite set of words over an alphabet . The Frobenius monoid problem is the question for a given finite set of words , whether the language is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets such that the length of the longest words not in (when is cofinite) is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . The factor universality problem is the question for a given finite set of words , whether every word over is a factor (substring) of some word from . We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets such that the length of the shortest words not being a factor of any word in is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . This essentially settles in the negative the longstanding Restivo’s conjecture (1981) and its weak variations. All our solutions are based on one shared construction, and as an auxiliary general tool, we introduce the concept of set rewriting systems . Finally, we complement the results with upper bounds.

V_c-Noetherian domains and Krull domains

Let D be an integrally closed domain, {V_{alpha }} be the set of t-linked valuation overrings of D, and v_c be the star operation on D defined by I^{v_c} = bigcap _{alpha } IV_{alpha } for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a v_c-Noetherian domain if and only if D is a Krull domain, if and only if v_c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then v_c = v if and only if D is a Dedekind domain.

Random Regular Expression Over Huge Alphabets

In this article, we study some properties of random regular expressions of size [Formula: see text], when the cardinality of the alphabet also depends on [Formula: see text]. For this, we revisit and improve the classical Transfer Theorem from the field of analytic combinatorics. This provides precise estimations for the number of regular expressions, the probability of recognizing the empty word and the expected number of Kleene stars in a random expression. For all these statistics, we show that there is a threshold when the size of the alphabet approaches [Formula: see text], at which point the leading term in the asymptotics starts oscillating.

Recursive max-linear models with propagating noise

Recursive max-linear vectors model causal dependence between node variables by a structural equation model, expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph (DAG) and some exogenous innovation. For such a model, there exists a unique minimum DAG, represented by the Kleene star matrix of its edge weight matrix, which identifies the model and can be estimated. For a more realistic statistical modeling we introduce some random observational noise. A probabilistic analysis of this new noisy model reveals that the unique minimum DAG representing the distribution of the non-noisy model remains unchanged and identifiable. Moreover, the distribution of the minimum ratio estimators of the model parameters at their left limits are completely determined by the distribution of the noise variables up to a positive constant. Under a regular variation condition on the noise variables we prove that the estimated Kleene star matrix converges to a matrix of independent Weibull entries after proper centering and scaling.

Geometrical modeling and numerical analysis of edge dislocation

In this study, we conduct modeling and numerical analysis of a straight edge dislocation in a three-dimensional geometrically nonlinear elastic medium. Based on the fundamental framework of geometrical elasto-plasticity, kinematics of the continuum is represented by the reference R, intermediate B and current S states and local deformations are described by the multiplicative decomposition of the deformation gradient; F = Fe·Fp. Here, the reference and current states are Euclidean submanifolds while the intermediate state B is represented by Weitzenbock manifold with non-zero torsion in the affine connection. Following to the equivalence of torsion and dislocation density tensor through Hodge star operation, a plastic deformation gradient Fp is obtained from the Cartan first structure equation using the homotopy operator. The current state S is determined so that it minimizes the strain energy functional. We solve the variational problem by using isogeometric analysis; Galerkin method with non-uniform B-spline basis function. The present analysis shows that all stress components agree quantitatively well with those of the Volterra dislocation model. Around the dislocation core, stress fields are non-singular on the contrary to the Volterra dislocation. In addition, we found that all stress components distribute asymmetrically due to the geometrical nonlinearity of the model.

Two Extensions of a Star Operation on D to the Polynomial Ring D[X

Two Extensions of a Star Operation on D to the Polynomial Ring D[X

ON THE STRUCTURE OF COHOMOLOGICAL MODELS OF ELECTRODYNAMICS AND GENERAL RELATIVITY

In the present paper, we take case of a complex scalar field on a Riemannian manifold and study diff erential geometry and cohomological way to construct field theory Lagrangians. The total Lagrangian of the model is proposed as 4-form on Riemannian manifold. To this end, we use inner product of differential (p, q)-forms and Hodge star operators. It is shown that actions, including that for gravity, can be represented in quadratic forms of fields of matter and basic tetrad fields. Our study is limited to the case of the Levi-Civita metric. We stress some features arisen within the approach regarding nil potency property. Within the model, Klein-Gordon, Maxwell and general relativity actions have been reproduced.

Quantum critical phase transition between two topologically ordered phases in the Ising toric code bilayer

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled $\mathbb{Z}_2\times\mathbb{Z}_2$ and the strongly-coupled $\mathbb{Z}_2$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising$^*$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local $\mathbb{Z}_2$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter.

Regular expressions for some Petri net languages for the producer/consumer problem

We consider a Petri net for the producer/consumer problem (one of the classical synchronization problems) with the bounded buffer of size n and the regular formal languages Ln, generated by the net. We propose regular expressions denoting these languages in the recursive form, in case of the bounded buffer of size from 1 to 3 the explicit formulas are proposed. We transform a reachability graph into a finite automaton and use the state elimination method. We give an upper estimate for the star-height of the mentioned languages, in case of the bounded buffer of size 1 and 2 the exact values are calculated. We also consider union, intersection, Kleene closure, concatenation and set difference operations on mentioned languages. We give the finite automaton and propose regular expressions denoting the set difference of languages Ln \ L1 in the recursive form, for L2 \ L1 the explicit formula is proposed.

The sets of star and semistar operations on semilocal Prüfer domains

We study the sets of semistar and star operations on a semilocal Prüfer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is h-local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.

$\star$-super potent domains

For a finite-type star operation ⋆ on a domain R, we say that R is ⋆-super potent if each maximal ⋆-ideal of R contains a finitely generated ideal I such that (1) I is contained in no other maximal ⋆-ideal of R and (2) J is ⋆-invertible for every finitely generated ideal J⊇I. Examples of t-super potent domains include domains each of whose maximal t-ideals is t-invertible (e.g., Krull domains). We show that if the domain R is ⋆-super potent for some finite-type star operation ⋆, then R is t-super potent, we study t-super potency in polynomial rings and pullbacks, and we prove that a domain R is a generalized Krull domain if and only if it is t-super potent and has t-dimension one.

On bounded linear codes and the commutative equivalence

The problem of the commutative equivalence of semigroups generated by semi-linear languages is studied. In particular conditions ensuring that the Kleene closure of a bounded semi-linear code is commutatively equivalent to a regular language are investigated.

Multiplicative closure operations on ring extensions

Let A ⊆ B be a ring extension and G be a set of A -submodules of B . We introduce a class of closure operations on G (which we call multiplicative operations on ( A , B , G ) ) that generalizes the classes of star, semistar and semiprime operations. We study how the set Mult ( A , B , G ) of these closure operations varies when A , B or G vary, and how Mult ( A , B , G ) behaves under ring homomorphisms . As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.

On Theory of Regular Languages with the Kleene Star Operation

This paper is dedicated to studying model-theoretic and complexity properties of the theory of regular languages with the Kleene star operation. We construct a ‘‘natural’’ infinite set of axioms for this theory, and we prove that it is not finitely axiomatizable. We establish that this theory is countably categorical, but it is not categorical in any uncountable cardinality. Also, we find the exact amount of non-isomorphic models of given cardinality. Finally, we prove that the theory of regular languages with the Kleene star is PSPACE-complete.

Divisibility Properties of the Semiring of Ideals of an Integral Domain

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

On Decidability of Theories of Regular Languages

This paper is dedicated to studying decidability properties of theories of regular languages with classical operations: union, concatenation, and the Kleene star. The theory with union only is a theory of some Boolean algebra, so it is decidable. We prove that the theory of regular languages with the Kleene star only is decidable. If we use union and concatenation simultaneously, then the theory becomes both Σ1- and π1-hard over the one-symbol alphabet. Using methods from the proof of this theorem we establish that the theory of regular languages over one-symbol alphabet with union and the Kleene star is as hard as arithmetic. Then we establish that the theory with all three operations is reducible to arithmetic also, hence, it is equivalent to arithmetic. Finally, we prove that the theory of regular languages over any alphabet with concatenation only is equivalent to arithmetic also. The last result is based on our previous work where an analogous theorem was proved for one-symbol languages.

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].

COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR

AbstractWe consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).

Closure operations on MV-algebras

Closure operations on posets have been investigated and applied extensively to lattices of congruences of various algebraic structures. In this paper, we initiate the study of closure operations on MV-algebras and star operations on MV-modules. The connections between closure (star) operations studied here and those of semirings as well as those of lattice-ordered rings are also investigated. We provide several examples of these closure operations and star operations and also study some of their main properties.

Adaptive estimation of Hodge star operator on simplicial surfaces

The Hodge star operator is a fundamental component of the second-order differential operators that bridges constitutive physical laws, as a matter of fact, it plays a central role in geometry processing and physical simulation. To be admissible, the discrete Hodge operators should be regular, symmetric, sparse and positive definite. Unfortunately, the last criteria is rarely met in the literature, which leads to inconsistent physical simulation behavior. In this paper, we exploit the intrinsic relationship between Hodge operator and the physical Fourier’s constitutive laws, to construct an adaptive discrete Hodge operator by expressing the Fourier’s laws on a surface mesh. As by-product, the new discrete Hodge operator is diagonal, regular and positive definite. Various comparative examples are presented to demonstrate the performance of our approach. The results show that the proposed operator performs better compared with the standard discrete Hodge.