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Zoltan Esik

Publications and source records attributed to Zoltan Esik.

At least 19 recordsLinked to original sources
arXiv2019

On Free $ω$-Continuous and Regular Ordered Algebras

We study varieties of certain ordered $Σ$-algebras with restricted completeness and continuity properties. We give a general characterization of their free algebras in terms of submonads of the monad of $Σ$-coterms. Varieties of this form are called \emph{quasi-regular}. For example, we show that if $E$ is a set of inequalities between finite $Σ$-terms, and if $\mathcal{V}_ω$ and $\mathcal{V}_\mathrm{reg}$ denote the varieties of all $ω$-continuous ordered $Σ$-algebras and regular ordered $Σ$-algebras satisfying $E$, respectively, then the free $\mathcal{V}_\mathrm{reg}$-algebra $F_\mathrm{reg}(X)$ on generators $X$ is the subalgebra of the corresponding free $\mathcal{V}_ω$-algebra $F_ω(X)$ determined by those elements of $F_ω(X)$ denoted by the regular $Σ$-coterms. This is a special case of a more general construction that applies to any quasi-regular family. Examples include the *-continuous Kleene algebras, context-free languages, $ω$-continuous semirings and $ω$-continuous idempotent semirings, OI-macro languages, and iteration theories.

cs.LO↗
arXiv2016

A representation theorem for stratified complete lattices

We consider complete lattices equipped with preorderings indexed by the ordinals less than a given (limit) ordinal subject to certain axioms. These structures, called stratified complete lattices, and weakly monotone functions over them, provide a framework for solving fixed point equations involving non-monotone operations such as negation or complement, and have been used to give semantics to logic programs with negation. More precisely, we consider stratified complete lattices subject to two slightly different systems of axioms defining `models' and `strong models'. We prove that a stratified complete lattice is a model iff it is isomorphic to the stratified complete lattice determined by the limit of an inverse system of complete lattices with `locally completely additive' projections. Moreover, we prove that a stratified complete lattice is a strong model iff it is isomorphic to the stratified complete lattice determined by the limit of an inverse system of complete lattices with completely additive projections. We use the inverse limit representation to give alternative proofs of some recent results and to derive some new ones for models and strong models. In particular, we use the representation theorem to prove that every model gives rise to another complete lattice structure, which in limit models corresponds to the lexicographic order. Moreover, we prove that the set of all fixed points of a weakly monotone function over a model, equipped with the new ordering, is a complete lattice. We also consider symmetric models that satisfy, together with each axiom, the dual axiom, and use the inverse limit representation to prove that every strong model is symmetric.

cs.LO↗
arXiv2016

Some Remarks on Conway and Iteration Theories

We present an axiomatization of Conway theories which yields,as a corollary, a very concise axiomatization of iteration theories satisfying the functorial implication for base morphisms.

cs.LO↗
arXiv2015

Equational properties of stratified least fixed points

Recently, a novel fixed point operation has been introduced over certain non-monotonic functions between stratified complete lattices and used to give semantics to logic programs with negation and boolean context-free grammars. We prove that this new operation satisfies `the standard' identities of fixed point operations as described by the axioms of iteration theories. We also study this new fixed point operation in connection with lambda-abstraction.

cs.LO↗
arXiv2015

An analysis of the equational properties of the well-founded fixed point

Well-founded fixed points have been used in several areas of knowledge representation and reasoning and to give semantics to logic programs involving negation. They are an important ingredient of approximation fixed point theory. We study the logical properties of the (parametric) well-founded fixed point operation. We show that the operation satisfies several, but not all of the equational properties of fixed point operations described by the axioms of iteration theories.

cs.DM↗
arXiv2015

Equational axioms associated with finite automata for fixed point operations in cartesian categories

The axioms of iteration theories, or iteration categories, capture the equational properties of fixed point operations in several computationally significant categories. Iteration categories may be axiomatized by the Conway identities and identities associated with finite automata. We show that in conjunction with the Conway identities, each identity associated with a finite automaton implies the identity associated with any input extension of the automaton. We conclude that the Conway identities and the identities associated with the members of a subclass $\cQ$ of finite automata is complete for iteration categories iff for every finite simple group $G$ there is an automaton $\bQ \in \cQ$ such that $G$ is a quotient of a group in the monoid $M(\bQ)$ of the automaton $\bQ$. We also prove a stronger result that concerns identities associated with finite automata with a distinguished initial state.

cs.LO↗
arXiv2013

A connection between concurrency and language theory

We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and simulation equivalence classes of (regular) synchronization trees (or processes). The results reveal a close relationship between classical language theory and process algebra.

cs.LO↗
arXiv2011

On context-free languages of scattered words

It is known that if a Büchi context-free language (BCFL) consists of scattered words, then there is an integer $n$, depending only on the language, such that the Hausdorff rank of each word in the language is bounded by $n$. Every BCFL is a Müller context-free language (MCFL). In the first part of the paper, we prove that an MCFL of scattered words is a BCFL iff the rank of every word in the language is bounded by an integer depending only on the language. Then we establish operational characterizations of the BCFLs of well-ordered and scattered words. We prove that a language is a BCFL consisting of well-ordered words iff it can be generated from the singleton languages containing the letters of the alphabet by substitution into ordinary context-free languages and the $ω$-power operation. We also establish a corresponding result for BCFLs of scattered words and define expressions denoting BCFLs of well-ordered and scattered words. In the final part of the paper we give some applications.

cs.FL↗
arXiv2011

Free iterative and iteration K-semialgebras

We consider algebras of rational power series over an alphabet $Σ$ with coefficients in a commutative semiring $K$ and characterize them as the free algebras in various classes of algebraic structures.

cs.FL↗
arXiv2011

Multi-linear iterative K-Sigma-semialgebras

We consider $K$-semialgebras for a commutative semiring $K$ that are at the same time $Σ$-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear $K$-$Σ$-semialgebra. Examples of such algebras include the algebras of $Σ$-tree series over an alphabet $A$ with coefficients in $K$, and the algebra of all rational tree series. We show that for many commutative semirings $K$, the rational $Σ$-tree series over $A$ with coefficients in $K$ form the free multi-linear iterative $K$-$Σ$-semialgebra on $A$.

cs.DM↗
arXiv2011

Context-free ordinals

We consider context-free languages equipped with the lexicographic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than $ω^ω$. As a corollary of this result we obtain that an ordinal is the order type of a well-ordered context-free language iff it is less than $ω^{ω^ω}$.

cs.FL↗
arXiv2011

Free inductive K-semialgebras

We consider rational power series over an alphabet $Σ$ with coefficients in a ordered commutative semiring $K$ and characterize them as the free ordered $K$-semialgebras in various classes of ordered $K$-semialgebras equipped with a star operation satisfying the least pre-fixed point rule and/or its dual. The results are generalizations of Kozen's axiomatization of regular languages.

cs.FL↗
arXiv2011

Residuated Park Theories

When $L$ is a complete lattice, the collection $\Mon_L$ of all monotone functions $L^p \to L^n$, $n,p \geq 0$, forms a Lawvere theory. We enrich this Lawvere theory with the binary supremum operation $\vee$, an operation of (left) residuation $\res$ and the parameterized least fixed point operation $^\dagger$. We exhibit a system of \emph{equational} axioms which is sound and proves all valid equations of the theories $\Mon_L$ involving only the theory operations, $\vee$ and $^\dagger$, i.e., all valid equations not involving residuation. We also present an alternative axiomatization, where $^\dagger$ is replaced by a star operation, and provide an application to regular tree languages.

cs.LO↗
arXiv2010

Simulation vs. Equivalence

For several semirings S, two weighted finite automata with multiplicities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called "proper". It is known that the Boolean semiring, the semiring of natural numbers, the ring of integers, all finite commutative positively ordered semirings and all fields are proper. The semiring S is Noetherian if every subsemimodule of a finitely generated S-semimodule is finitely generated. First, it is shown that all Noetherian semirings and thus all commutative rings and all finite semirings are proper. Second, the tropical semiring is shown not to be proper. So far there has not been any example of a semiring that is not proper.

cs.FL↗
arXiv2010

An undecidable property of context-free languages

We prove that there exists no algorithm to decide whether the language generated by a context-free grammar is dense with respect to the lexicographic ordering. As a corollary to this result, we show that it is undecidable whether the lexicographic orderings of the languages generated by two context-free grammars have the same order type.

cs.FL↗