Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic automata that associate a weight drawn from a semiring $K$ with every transition and every state. Their behaviours can be formalized either as weighted language equivalence or weighted bisimulation. In this paper we explore the properties of weighted automata in the framework of coalgebras over (i) the category $\mathsf{SMod}$ of semimodules over a semiring $K$ and $K$-linear maps, and (ii) the category $\mathsf{Set}$ of sets and maps. We show that the behavioural equivalences defined by the corresponding final coalgebras in these two cases characterize weighted language equivalence and weighted bisimulation, respectively. These results extend earlier work by Bonchi et al. using the category $\mathsf{Vect}$ of vector spaces and linear maps as the underlying model for weighted automata with weights drawn from a field $K$. The key step in our work is generalizing the notions of linear relation and linear bisimulation of Boreale from vector spaces to semimodules using the concept of the kernel of a $K$-linear map in the sense of universal algebra. We also provide an abstract procedure for forward partition refinement for computing weighted language equivalence. Since for weighted automata defined over semirings the problem is undecidable in general, it is guaranteed to halt only in special cases. We provide sufficient conditions for the termination of our procedure. Although the results are similar to those of Bonchi et al., many of our proofs are new, especially those about the coalgebra in $\mathsf{SMod}$ characterizing weighted language equivalence

Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic automata that associate a weight drawn from a semiring $K$ with every transition and every state. Their behaviours can be formalized either as weighted language equivalence or weighted bisimulation. In this paper we explore the properties of weighted automata in the framework of coalgebras over (i) the category $\mathsf{SMod}$ of semimodules over a semiring $K$ and $K$-linear maps, and (ii) the category $\mathsf{Set}$ of sets and maps. We show that the behavioural equivalences defined by the corresponding final coalgebras in these two cases characterize weighted language equivalence and weighted bisimulation, respectively. These results extend earlier work by Bonchi et al. using the category $\mathsf{Vect}$ of vector spaces and linear maps as the underlying model for weighted automata with weights drawn from a field $K$. The key step in our work is generalizing the notions of linear relation and linear bisimulation of Boreale from vector spaces to semimodules using the concept of the kernel of a $K$-linear map in the sense of universal algebra. We also provide an abstract procedure for forward partition refinement for computing weighted language equivalence. Since for weighted automata defined over semirings the problem is undecidable in general, it is guaranteed to halt only in special cases. We provide sufficient conditions for the termination of our procedure. Although the results are similar to those of Bonchi et al., many of our proofs are new, especially those about the coalgebra in $\mathsf{SMod}$ characterizing weighted language equivalence

SCHEDULE VERIFICATION AND SYNTHESIS FOR EMBEDDED REAL-TIME COMPONENTS ∗

Abstract In this paper we address the problems of schedule synthesis and timing verification for component based architectures in embedded systems. We consider a component to be a set of tasks with response times that lie within specified intervals. When a set of components is deployed to implement a desired functionality, we want to guarantee that the components can achieve the timing constraints of the application. We solve the associated synthesis and verification problems using the framework of timed interface automata and timed games. Keywords: Component-based embedded real-time systems, real-time scheduling, timed interfaces, timed games, schedule synthesis. 1

Performance analysis of FlexRay-based systems using real-time calculus, revisited

The FlexRay protocol [4] is likely to be the de facto standard for automotive communication systems. Hence, there is a need to provide hard performance guarantees on properties like worst case response times of messages, their buffer re-quirements, end-to-end latency (for example, from sensor to actuator), etc., for FlexRay based systems. The paper [11] provides an analysis for finding worst case response times of the messages transmitted on the FlexRay bus, but the analysis is done using ILP formulation and is thus compu-tationally expensive. The paper [5] models the FlexRay in the analytic framework of Real-Time Calculus [12, 3] and is compositional as well as scalable. In this paper, we show that the analysis of [5] may lead to results that are over op-timistic; in particular, we show that obtaining the “upper service curves ” is not trivial and does not follow the reason-ing of the “lower service curves ” which the authors obtain. We also provide tighter “lower service curves ” than that of [5]. Finally we show that our model allows the messages to be of variable size which is not the case with [5]. Categories and Subject Descriptors C.3 [Special-purpose and application-based systems]: Real-time and embedded systems; C.4 [Performance of systems]: Design studies and modeling technique

Performance analysis of FlexRay-based systems using real-time calculus, revisited

The FlexRay protocol [4] is likely to be the de facto standard for automotive communication systems. Hence, there is a need to provide hard performance guarantees on properties like worst case response times of messages, their buffer re-quirements, end-to-end latency (for example, from sensor to actuator), etc., for FlexRay based systems. The paper [11] provides an analysis for finding worst case response times of the messages transmitted on the FlexRay bus, but the analysis is done using ILP formulation and is thus compu-tationally expensive. The paper [5] models the FlexRay in the analytic framework of Real-Time Calculus [12, 3] and is compositional as well as scalable. In this paper, we show that the analysis of [5] may lead to results that are over op-timistic; in particular, we show that obtaining the “upper service curves ” is not trivial and does not follow the reason-ing of the “lower service curves ” which the authors obtain. We also provide tighter “lower service curves ” than that of [5]. Finally we show that our model allows the messages to be of variable size which is not the case with [5]. Categories and Subject Descriptors C.3 [Special-purpose and application-based systems]: Real-time and embedded systems; C.4 [Performance of systems]: Design studies and modeling technique