*-Continuous Kleene ω-Algebras for Energy Problems

Abstract

Energy problems are important in the formal analysis of embedded or autonomous systems. Using recent results on * -continuous Kleene ω-algebras, we show here that energy problems can be solved by algebraic manipulations on the transition matrix of energy automata. To this end, we prove general results about certain classes of finitely additive functions on complete lattices which should be of a more general interest. IntroductionEnergy problems are concerned with the question whether a given system admits infinite schedules during which (1) certain tasks can be repeatedly accomplished and (2) the system never runs out of energy (or other specified resources). These are important in areas such as embedded systems or autonomous systems and, starting with [4], have attracted some attention in recent years, for example in [3, 5-8, 16, 19, 23, 24].With the purpose of generalizing some of the above approaches, we have in [12,17] introduced energy automata. These are finite automata whose transitions are labeled with energy functions which specify how energy values change from one system state to another. Using the theory of semiring-weighted automata [9], we have shown in [12] that energy problems in such automata can be solved in a simple static way which only involves manipulations of energy functions.In order to put the work of [12] on a more solid theoretical footing and with an eye to future generalizations, we have recently introduced a new algebraic structure of * -continuous Kleene ω-algebras [10] (see also [11] for the long version). We show here that energy functions form such a * -continuous Kleene ω-algebra. Using the fact, proven in [10], that for automata with transition weights in * -continuous Kleene ω-algebras, reachability and Büchi acceptance can be computed by algebraic manipulations on the transition matrix of the automaton, the results from [12] follow. Energy AutomataThe transition labels on the energy automata which we consider in the paper, will be functions which model transformations of energy levels between system states. Such transformations have the (natural) properties that below a certain energy level, the transition might be disabled (not enough energy is available to perform the transition), and an increase in input energy always yields at least the same increase in output energy.

Featured Weighted Automata

“…Recall that E denotes the set of energy functions: functions f : It can be shown [13] that n f n is ∞-continuous for any infinite sequence f 0 , f 1 , . .…”

“…is provides a uniform framework to treat problems such as minimum reachability, maximum flow, energy problems [13], and others. Here we extend techniques from weighted automata to featured weighted automata, i.e., weighted automata in which the transitions are annotated with feature expressions.…”

“…5, we extend our results to develop an application to featured energy problems. is is based on the recent result in [13] that energy problems can be stated as Büchi problems in automata weighted in * -continuous Kleene ω-algebras, which are certain types of semimodules over * -continuous Kleene algebras; hence we need to extend our results to such semimodules. e paper is followed by a separate appendix which contains some of the proofs of our results.…”

“…[13]). ( , Î) is a * -continuous Kleene ω-algebra.Hence the energy problems stated at the end of Sect.…”

“…( , Î) is a * -continuous Kleene ω-algebra.Hence the energy problems stated at the end of Sect. 5.1 can be solved by computing reachability and Büchi values of energy automata:P 5.4 ( [13]). Let S = (S, I , F ,T ) be an energy automaton and x 0 ∈ Ê ≥0 .…”