Kleene Star Research Articles

Expressiveness modulo bisimilarity of regular expressions with parallel composition

The languages accepted by finite automata are precisely the languages denoted by regular expressions. In contrast, finite automata may exhibit behaviours that cannot be described by regular expressions up to bisimilarity. In this paper, we consider extensions of the theory of regular expressions with various forms of parallel composition and study the effect on expressiveness. First we prove that adding pure interleaving to the theory of regular expressions strictly increases its expressiveness modulo bisimilarity. Then, we prove that replacing the operation for pure interleaving by ACP-style parallel composition gives a further increase in expressiveness, still insufficient, however, to facilitate the expression of all finite automata up to bisimilarity. Finally, we prove that the theory of regular expressions with ACP-style parallel composition and encapsulation is expressive enough to express all finite automata up to bisimilarity. Our results extend the expressiveness results obtained by Bergstra, Bethke and Ponse for process algebras with (the binary variant of) Kleene's star operation.

Note on characterizations of semistar operations and star operations on an integral domain, II

Note on characterizations of semistar operations and star operations on an integral domain, II

The construction of all the star operations and all the semistar operations on 1-dimensional Prüfer domains

The construction of all the star operations and all the semistar operations on 1-dimensional Prüfer domains

Bad Worlds

The idea of relevant logic—that irrelevant inferences are invalid—is appealing. But the standard semantics for relevant logics involve baroque metaphysics: a three-place accessibility relation, a star operator, and ‘bad’ (impossible/non-normal) worlds. In this article we propose that these oddities express a mismatch between non-classical object theory and classical metatheory. A uniformly relevant semantics for relevant logic is a better fit.

State Complexity of Kleene-Star Operations on Regular Tree Languages

The concatenation of trees can be defined either as a sequential or a parallel operation, and the corresponding iterated operation gives an extension of Kleene-star to tree languages. Since the sequential tree concatenation is not associative, we get two essentially different iterated sequential concatenation operations that we call the bottom-up star and top-down star operation, respectively. We establish that the worst-case state complexity of bottom-up star is (n + 3/2) · 2 n−1. The bound differs by an order of magnitude from the corresponding result for string languages. The state complexity of top-down star is similar as in the string case. We consider also the state complexity of the star of the concatenation of a regular tree language with the set of all trees.

On the Power of SPARQL in Expressing Navigational Queries

Navigational queries on graph databases return binary relations over the nodes of the graph. The calculus of relations, popularized by Tarski, serves as a natural benchmark for first-order navigational querying. Recently, nested regular expressions (nre's) have been proposed to extend navigational querying to resource description framework graphs, i.e. ternary relations. This paper investigates the expressiveness of nre's and their relationship to basic SPARQL queries. An elegant proof is given to the effect that nre queries are already expressible as basic SPARQL queries. This result takes exception of nre's involving Kleene star (transitive closure), but on the other hand it holds even when extending nre's with negation (complementation). The resulting language of ‘star-free nre's with negation (sfnre¬)’ can in fact be captured by a precisely delineated fragment of SPARQL, called Tarski-SPARQL. The resulting language is also compared with an alternative extension that adds negation in the form of the difference operator. While sfnre¬ queries are subsumed by first-order logic with three variables (FO3), it is shown that some natural FO3 queries are not expressible in nre¬, even when allowing transitive closure.

Integral domains in which any two v-coprime elements are comaximal

Domains in which the star operations d (the trivial star operation) and w coincide have received a good deal of attention recently. These are exactly the domains D in which I=D whenever I is a finitely generated ideal of D with Iv=D. In this work, we study what happens when “finitely generated” is replaced by “two-generated.” It turns out that these are precisely the domains in which d=F, where F is a certain star operation closely connected to, but more complicated than, the w-operation.

Robustness and closure properties of recognizable languages in adhesive categories

We consider recognizable languages of cospans in adhesive categories defined via automaton functors, of which recognizable graph languages are a special case.There are three contributions in this paper: we first show that the notion of recognizable languages is robust in the sense that also semi-functors, i.e., functors that do not necessarily preserve identities, characterize recognizable languages. This is done by converting semi-functors to functors with a procedure akin to epsilon elimination for non-deterministic finite automata.Second, relying on this result, we show that recognizable languages are closed under concatenation, i.e. under cospan composition, by providing a concrete construction that creates a concatenation automaton from two given automata. The construction is considerably more complex than the corresponding construction for finite automata. We conclude by showing negative closure properties for Kleene star and substitution.

Star operations on numerical semigroups: the multiplicity 3 case

We prove an explicit formula for the number of star operations on numerical semigroups of multiplicity 3 in terms of the generators of the semigroup. We also estimate the number of semigroups of multiplicity 3 with exactly \(n\) star operations.

2D quantum double models from a 3D perspective

In this paper we look at three dimensional (3D) lattice models that are generalizations of the state sum model used to define the Kuperberg invariant of 3-manifolds. The partition function is a scalar constructed as a tensor network where the building blocks are tensors given by the structure constants of an involutory Hopf algebra . These models are very general and are hard to solve in its entire parameter space. One can obtain familiar models, such as ordinary gauge theories, by letting be the group algebra of a discrete group G and staying on a certain region of the parameter space. We consider the transfer matrix of the model and show that quantum double Hamiltonians are derived from a particular choice of the parameters. Such a construction naturally leads to the star and plaquette operators of the quantum double Hamiltonians, of which the toric code is a special case when . This formulation is convenient to study ground states of these generalized quantum double models where they can naturally be interpreted as tensor network states. For a surface Σ, the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of . It is also possible to obtain extra models by simply enlarging the allowed parameter space but keeping the solubility of the model. While some of these extra models have appeared before in the literature, our 3D perspective allows for an uniform description of them.

Oscillation of the Measurement Accuracy of the Center Location of an Ellipse on a Digital Image

Abstract. Circular targets are often utilized in photogrammetry and a circle on a plane is projected as an ellipse onto an oblique image. This paper reports an experiment conducted to investigate whether the measurement accuracy of the center location of an ellipse on a digital image oscillates as its dimension increases. The experiment was executed by the Monte Carlo simulation using 1024 synthesized images of which the centers were randomly distributed in one pixel for each ellipse. We investigated four typical measurement methods: intensity-weighted centroid method, non-iterative ellipse fitting, iterative ellipse fitting with the star operator, and least-squares matching. Three flattenings 0.00, 0.25, 0.50, and three rotation angles 0.0°, 22.5°, 45.0° were investigated in the experiment. The experiment results clearly show that the measurement accuracy by all the investigated methods would oscillate as the dimension of an ellipse increases. The measurement accuracy by the intensity-weighted centroid method and the non-iterative ellipse fitting would oscillate smoothly, while that by the least-squares matching would oscillate considerably roughly. It would be impossible to determine the cycle of the oscillation except the measurement accuracy by the intensity-weighted centroid method and the non-iterative ellipse fitting when the rotation angle of an ellipse is 0.0° and 45.0°. The experiment results indicate that the flattening and the rotation angle of an ellipse would affect the cycle of the oscillation as well.

An Improved Algorithm for the Metamorphosis of Semi-Extended Regular Expressions to Deterministic Finite Automata

Semi-extended regular expressions are regular expressions with intersection, union, concatenation and Kleene closure operators. The aim of this paper is the metamorphosis of semi-extended regular expressions to deterministic finite automata. The proposed algorithm is based on the follow-positions of symbols present in the semi-extended regular expression. Comparison demonstrates that the deterministic finite automaton generated using the proposed algorithm is smaller than the earlier existing approaches in the literature. Finally, to expound upon this metamorphosis, we provide the reader with numerical examples.

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.

Star Almost Schreier Domains

Let D be an integral domain and * a star operation on D. We study the following property: for every *-invertible ideals I, H 1, H 2 of D such that I ⊇ H 1 H 2, there exist an integer k ≥ 1 and ideals I 1, I 2 such that (I k )* = (I 1 I 2)* and , i = 1, 2. The k = 1 variant of the property above is also studied.

Modeling and Control of Weight-Balanced Timed Event Graphs in Dioids

The class of timed event graphs (TEGs) has widely been studied thanks to an approach known as the theory of max-plus linear systems. In particular, the modeling of TEGs via formal power series in a dioid called Min <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ax</sup> [[ γ,δ]] has led to input-output representations on which some model matching control problems have been solved. Our work attempts to extend the class of systems for which a similar control synthesis is possible. To this end, a subclass of timed Petri nets that we call weight-balanced timed event graphs (WBTEGs) will be first defined. They can model synchronization and delays (WBTEGs contain TEGs) and can also describe dynamic phenomena such as batching and event duplications (unbatching). Their behavior is described by rational compositions (sum, product and Kleene star) of four elementary operators γ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> , μm, and βb on a dioid of formal power series denoted ε* [[ δ]]. The main feature is that the transfer series of WBTEGs have a property of ultimate periodicity (such as rational series in M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ax</sup> [[ γ,δ]]). Finally, the existing results on control synthesis for max-plus linear systems find a natural application in this framework.

Suslin forms and the Hodge star operator

We show that the Suslin forms are themselves a product of Suslin matrices (over the integers) and that their corresponding linear transformation is the Hodge star operator. The Fossum–Foxby–Iversen theory of acyclic complexes, as developed by Suslin, plays a vital role in this analysis.

A characterization of local Noetherian domains which admit only finitely many star operations: The infinite residue field case

We characterize local Noetherian domains R with infinite residue field which admit only finitely many star (prime) operations. We further show that a positive integer n is equal to the number of star operations on such an R if and only if n=2,4 or an odd positive integer.

Line bundles and the Thom construction in noncommutative geometry

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct \mathbb{Z} - and \mathbb{N} -graded algebras, the \mathbb{Z} -graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the \mathbb{N} -graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

A matrix equation on triangulated Riemann surfaces

In~[1], Wilson defined holomorphic 1-cochains and combinatrial period matrices of triangulated Riemann surfaces by using the combinatorial Hodge star operator, introduced in~[2]. In this paper, we define a matrix and call this matrix the associate matrix. Then, we prove that among the three matrices, which are a period matrix, a combinatorial period matrix which is introduced by Wilson~[2] and an associate matrix, there exists a matrix equation. Then we also show that an associate matrix is an element of the Siegel upper half space, so this means that a trianguted Riemann surface gives three elements of the Siegel upper half space.

NetKAT

Recent years have seen growing interest in high-level languages for programming networks. But the design of these languages has been largely ad hoc, driven more by the needs of applications and the capabilities of network hardware than by foundational principles. The lack of a semantic foundation has left language designers with little guidance in determining how to incorporate new features, and programmers without a means to reason precisely about their code. This paper presents NetKAT, a new network programming language that is based on a solid mathematical foundation and comes equipped with a sound and complete equational theory. We describe the design of NetKAT, including primitives for filtering, modifying, and transmitting packets; union and sequential composition operators; and a Kleene star operator that iterates programs. We show that NetKAT is an instance of a canonical and well-studied mathematical structure called a Kleene algebra with tests (KAT) and prove that its equational theory is sound and complete with respect to its denotational semantics. Finally, we present practical applications of the equational theory including syntactic techniques for checking reachability, proving non-interference properties that ensure isolation between programs, and establishing the correctness of compilation algorithms.