# *-Continuous Kleene ω-Algebras

@article{sik2015ContinuousK, title={*-Continuous Kleene $\omega$-Algebras}, author={Zolt{\'a}n {\'E}sik and Ulrich Fahrenberg and Axel Legay}, journal={ArXiv}, year={2015}, volume={abs/1501.01118} }

We define and study basic properties of \(^*\)-continuous Kleene \(\omega \)-algebras that involve a \(^*\)-continuous Kleene algebra with a \(^*\)-continuous action on a semimodule and an infinite product operation that is also \(^*\)-continuous. We show that \(^*\)-continuous Kleene \(\omega \)-algebras give rise to iteration semiring-semimodule pairs, and that for Buchi automata over \(^*\)-continuous Kleene \(\omega \)-algebras, one can compute the associated infinitary power series.

## 7 Citations

### *-Continuous Kleene ω-Algebras for Energy Problems

- Mathematics, Computer ScienceFICS
- 2015

It is shown here that energy problems can be solved by algebraic manipulations on the transition matrix of energy automata, and general results about certain classes of finitely additive functions on complete lattices are proved.

### An Algebraic Approach to Energy Problems II - The Algebra of Energy Functions

- Computer Science, MathematicsActa Cybern.
- 2017

It is shown that energy functions form a *-continuous Kleene ω-algebra, as an application of a general result that finitely additive, locally *-closed and T-continuously functions on complete lattices form *- continuous Kleeneπ-algebras.

### Quantitative properties of featured automata

- Computer ScienceInternational Journal on Software Tools for Technology Transfer
- 2019

A model of featured weighted automata is introduced which combines featured transition systems and (semiring-) weighted automation and applications to minimum reachability and to energy properties are shown.

### Polymorphic Iterable Sequential Effect Systems

- MathematicsACM Trans. Program. Lang. Syst.
- 2021

This work presents an abstract polymorphic effect system parameterized by an effect quantale—an algebraic structure with well-defined properties that can model the effects of a range of existing sequential effect systems.

### Energiautomater, energifunktioner og Kleene-algebra

- Mathematics, Computer ScienceNIK
- 2018

Formålet med denne artikel er at give et overblik over nylig forskning i energiproblemer (for første gang p̊a dansk) samt at udvide anvendelsen af Kleenealgebra i et forsøg p̶�a at lukke et åbent problem fra [7].

### Featured Weighted Automata

- Computer Science2017 IEEE/ACM 5th International FME Workshop on Formal Methods in Software Engineering (FormaliSE)
- 2017

A model of featured weighted automata is introduced which combines featured transition systems and (semiring-) weighted automation and applications to minimum reachability and to energy properties are shown.

## References

SHOWING 1-10 OF 18 REFERENCES

### A completeness theorem for Kleene algebras and the algebra of regular events

- Mathematics[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
- 1991

A finitary axiomatization of the algebra of regular events involving only equations and equational implications that is sound for all interpretations over Kleene algebras is given. Axioms for Kleene…

### On Kleene Algebras and Closed Semirings

- Mathematics, Computer ScienceMFCS
- 1990

The literature contains at several inequivalent definitions of Kleene algebras and related algebraic structures: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms.

### An Eilenberg Theorem for Infinity-Languages

- MathematicsICALP
- 1991

A one-to-one correspondence between special classes of regular ∞-languages and pseudovarieties of right binoids according to Eilenberg's theorem for regular sets of finite words is established.

### Two Complete Axiom Systems for the Algebra of Regular Events

- MathematicsJACM
- 1966

Two formal systems for the algebraic transformation of regular expressions are developed, one based upon the uniqueness of the solution of certain regular expression equations, and the other based upon some facts concerning the representation theory of regular events.

### Kleene Algebras and Semimodules for Energy Problems

- Computer ScienceATVA
- 2013

With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce generalized energy automata. These are finite automata whose edges are labeled with energy…

### Iteration Theories: The Equational Logic of Iterative Processes

- Mathematics
- 1993

Written both for graduate students and research scientists in theoretical computer science and mathematics, this book provides a detailed investigation of the properties of the fixed point or…

### Iteration Semirings

- MathematicsDevelopments in Language Theory
- 2008

The role of iteration semirings in the axiomatization of regular languages and rational power series, and in the equational theory of continuous and completeSemirings, is reviewed.

### Rationally Additive Semirings

- MathematicsJ. Univers. Comput. Sci.
- 2002

It is proved that every rationally additive semiring is an iteration semiring, and the semirings of rational power series with coefficients in N∞, the semiring of natural numbers equipped with a top element, are characterized as the free rationally additives.

### On Iteration Semiring-Semimodule Pairs

- Mathematics
- 2007

Conway semiring-module pairs and iteration semiring-semimodule pairs were shown to provide an axiomatic basis to automata on ω -words in [Bloom, Esik: Iteration Theories, Springer, 1993]. In this…