### An Algebraic Approach to Energy Problems I — *-Continuous Kleene ω-Algebras

Zoltán Ésik, Uli Fahrenberg, Axel Legay, Karin Quaas
2017 Acta Cybernetica
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.
more » ... 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 semiringsemimodule pair. Let [0, ∞] ⊥ = {⊥}∪[0, ∞] denote the complete lattice of non-negative real numbers together with extra elements ⊥ and ∞, with the standard order on Definition 3. An extended energy function is a mapping f : [0, ∞] ⊥ → [0, ∞] ⊥ , for which ⊥f = ⊥ and yf ≥ xf + y − x for all x ≤ y. Moreover, ∞f = ∞, unless xf = ⊥ for all x ∈ [0, ∞] ⊥ . The class of all extended energy functions is denoted E. This means, in particular, that xf = ⊥ implies yf = ⊥ for all y ≤ x, and xf = ∞ implies yf = ∞ for all y ≥ x. Hence, except for the extension to ∞, these