Kleene Algebras and Semimodules for Energy Problems

With the purpose of unifying a number of approaches to energy problems found in the literature, we introduce generalized energy automata. These are finite automata whose edges are labeled with energy functions that define how energy levels evolve during transitions. Uncovering a close connection between energy problems and reachability and B\"uchi acceptance for semiring-weighted automata, we show that these generalized energy problems are decidable. We also provide complexity results for important special cases

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

-Continuous Kleene ω-Algebras

We define and study basic properties of *-continuous Kleene ω-algebras that involve a *-continuous Kleene algebra with a *-continuous action on a semimodule and an infinite product operation that is also *-continuous. We show that *-continuous Kleene ω-algebras give rise to iteration semiring-semimodule pairs. We show how our work can be applied to solve certain energy problems for hybrid systems

Continuous semiring-semimodule pairs and mixed algebraic systems

We associate with every commutative continuous semiring S and alphabet Σ a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series S⟪(Y⊎Σ)*⟫ of finite words over Y⊎Σ, an X × Y -matrix over S⟪(Y⊎Σ)*⟫, and a function from into the continuous S⟪(Y⊎Σ)*⟫-semimodule S⟪(Y⊎Σ)ω⟫ of series of ω-words over Y⊎Σ. When S is also an ω-semiring (equipped with an infinite product operation), then we define a fixed point operation over our category and show that it satisfies all identities of iteration categories. We then use this fixed point operation to give semantics to recursion schemes defining series of finite and infinite words. In the particular case when the semiring is the Boolean semiring, we obtain the context-free languages of finite and ω-words

A Kleene theorem for weighted ω-pushdown automata

Weighted ω-pushdown automata were introduced as generalization of the classical pushdown automata accepting infinite words by Büchi acceptance. The main result in the proof of the Kleene Theorem is the construction of a weighted ω-pushdown automaton for the ω-algebraic closure of subsets of a continuous star-omega semiring