An algebraic approach to energy problems II - the algebra of energy functions

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 certain 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

Energiautomater, energifunktioner og Kleene-algebra

Forfatterne til denne artikel har, sammen med mange gode kolleger, i en del år arbejdet med såkaldte energiproblemer. Disse handler om, at man i en formel model ønsker at bestemme, om der findes en endelig eller uendelig eksekvering under hvilken en given energivariabel aldrig bliver negativ. Den formelle model kan være en vægtet tidsautomat, en endelig automat som er annoteret med energifunktioner eller lignende. Fælles for alle disse modeller er, at det har vist sig ualmindeligt svært at løse sådanne energiproblemer og at teknikker fra Kleene-algebra har været en stor hjælp.  Formålet med denne artikel er at give et overblik over nylig forskning i energiproblemer (for første gang på dansk) samt at udvide anvendelsen af Kleene-algebra i et forsøg på at lukke et åbent problem fra artiklen som startede hele dette område.&nbsp

An algebraic approach to energy problems I - continuous Kleene ω-algebras ‡

Energy problems are important in the formal analysis of embedded or autonomous systems. With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Motivated by this application and in order to compute with energy functions, we introduce a new algebraic structure of *-continuous Kleene ω-algebras. These involve a *-continuous Kleene algebra with a *-continuous action on a semimodule and an infinite product operation that is also *-continuous. We define both a finitary and a non-finitary version of *-continuous Kleene ω-algebras. We then establish some of their properties, including a characterization of the free finitary *-continuous Kleene ω-algebras. We also show that every *-continuous Kleene ω-algebra gives rise to an iteration semiring-semimodule pair

An Algebraic Approach to Energy Problems II — The Algebra of Energy Functions

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 certain 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