### Abstract

Energy and resource management problems are important in areas such as embedded systems or autonomous systems. They are concerned with the question whether a given system admits infinite schedules during which cer tain tasks can be repeatedly accomplished and the system never runs out of energy (or other resources). In order to develop a general theory of energy problems, we introduce energy automata: Finite automata whose transitions are labeled with energy functions which specify how energy values change from one system state to another. We show that energy functions form a ∗-continuous Kleene ω-algebra, as an application of a general result that finitely additive, locally ∗-closed and T-continuous functions on complete lattices form ∗-continuous Kleene ω-algebras. This permits to solve energy problems in energy automata in a generic, algebraic way. In order to put our work in context, we also review extensions of energy problems to higher dimensions and to games.

Original language | English |
---|---|

Pages (from-to) | 229-268 |

Number of pages | 40 |

Journal | Acta Cybernetica |

Volume | 23 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2017 |

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### Keywords

- Energy problem
- Kleene algebra
- ∗-continuity
- ∗-continuous Kleene ω-algebra

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Acta Cybernetica*,

*23*(1), 229-268. https://doi.org/10.14232/actacyb.23.1.2017.14

**An algebraic approach to Energy Problems II the algebra of energy functions.** / Ésik, Z.; Fahrenberg, Uli; Legay, Axel; Quaas, Karin.

Research output: Contribution to journal › Article

*Acta Cybernetica*, vol. 23, no. 1, pp. 229-268. https://doi.org/10.14232/actacyb.23.1.2017.14

}

TY - JOUR

T1 - An algebraic approach to Energy Problems II the algebra of energy functions

AU - Ésik, Z.

AU - Fahrenberg, Uli

AU - Legay, Axel

AU - Quaas, Karin

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Energy and resource management problems are important in areas such as embedded systems or autonomous systems. They are concerned with the question whether a given system admits infinite schedules during which cer tain tasks can be repeatedly accomplished and the system never runs out of energy (or other resources). In order to develop a general theory of energy problems, we introduce energy automata: Finite automata whose transitions are labeled with energy functions which specify how energy values change from one system state to another. We show that energy functions form a ∗-continuous Kleene ω-algebra, as an application of a general result that finitely additive, locally ∗-closed and T-continuous functions on complete lattices form ∗-continuous Kleene ω-algebras. This permits to solve energy problems in energy automata in a generic, algebraic way. In order to put our work in context, we also review extensions of energy problems to higher dimensions and to games.

AB - Energy and resource management problems are important in areas such as embedded systems or autonomous systems. They are concerned with the question whether a given system admits infinite schedules during which cer tain tasks can be repeatedly accomplished and the system never runs out of energy (or other resources). In order to develop a general theory of energy problems, we introduce energy automata: Finite automata whose transitions are labeled with energy functions which specify how energy values change from one system state to another. We show that energy functions form a ∗-continuous Kleene ω-algebra, as an application of a general result that finitely additive, locally ∗-closed and T-continuous functions on complete lattices form ∗-continuous Kleene ω-algebras. This permits to solve energy problems in energy automata in a generic, algebraic way. In order to put our work in context, we also review extensions of energy problems to higher dimensions and to games.

KW - Energy problem

KW - Kleene algebra

KW - ∗-continuity

KW - ∗-continuous Kleene ω-algebra

UR - http://www.scopus.com/inward/record.url?scp=85020124198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020124198&partnerID=8YFLogxK

U2 - 10.14232/actacyb.23.1.2017.14

DO - 10.14232/actacyb.23.1.2017.14

M3 - Article

AN - SCOPUS:85020124198

VL - 23

SP - 229

EP - 268

JO - Acta Cybernetica

JF - Acta Cybernetica

SN - 0324-721X

IS - 1

ER -