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

Zoltán Ésik, Uli Fahrenberg, Axel Legay, Karin Quaas
2017 Acta Cybernetica
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
more » ... ues 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 -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. We recall the energy automata introduced in  and the decision problems we are interested in. Let [0, ∞] ⊥ = {⊥} ∪ [0, ∞] denote the complete lattice of nonnegative real numbers together with extra elements ⊥ and ∞, with the standard order on Definition 1. An (extended) energy function is a mapping f : [0, ∞] ⊥ → [0, ∞] ⊥ , for which ⊥f = ⊥ and yf ≥ xf + y − x ( * )