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

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 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 -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.

Energy Büchi Problems

“…Even without Büchi conditions, the approach laid out here would not work for weighted edges. This was already noted in [7]; instead it requires different methods which are developed in [6] (see also [12,13]). There, one-clock weighted timed automata (with edge weights) are translated to finite automata weighted with so-called energy functions instead of integers.…”

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

“…The notions presented in Sections 2 and 3 can be seen as direct extensions of existing algebraic approaches related to lattices and Kleene algebra. In this section, we introduce automorphism-based Kleene algebras [4], which will allow us to make a first bridge between algebraic tools and invariant sets of nonlinear dynamical systems.…”

“…In particular, we will show the link between problems that can be expressed in terms of invariant sets and the Kleene algebra, the elements of which are automorphisms of a lattice [4]. We will take advantage of this algebraic structure to derive new efficient algorithms that are able to solve problems involving invariant sets that were not possible to compute with existing methods.…”

Qualitative Controller Synthesis for Consumption Markov Decision Processes

“…The paper [29] presents polynomial-time algorithms for non-stochastic energy games with special weight structures. Recently, an abstract algebraic perspective on energy models was presented in [22,35,36].…”