Adding Contracts about bounds on the frequency of task arrivals and time to completions. In addition, they can capture the dependencies As in [5] we equip our real-time interfaces with a notion of contracts. Contracts are pairs (A, G) where A is an assumption about the environment of a component, and G is the guarantee that the component offers to its environment [4]. For real- time interfaces, both assumptions and guarantees will talk The events ai and fi represent the activation and completion of task i. Suppose task i has execution time of ci on the processor. The translation of A into a set of task executions is given by α(A) = (ici 0pi−ci )ω and that of G by α(G) = (0٨(ici 0di−ci ))ω. ||| ||| With this translation, checking whether an interface com- plies to a contract reduces to language inclusion with respect to the mapping α. Indeed, the mapping involves an abstraction of the behaviour, which causes problems for more complex scenarios, as the mapping induces more behaviour in the in- terface than intended. With the revised definition of interfaces, these issues in the relation with contracts disappear: ⬦ | ∩ ⊆ Definition 5: [4] Let C = (P, ΣP , A, G) be a contract. An implementation M of the contract satisfies C, written M ▶ C, if and only if M P A G. Here M , A and G are all sets of traces (sequences). Considering interfaces as implementations of contracts, we get the following relation. An interface IK = (K, ΣK, LK) satisfies a contract C = (P, ΣP , A, G) if LK satisfies C. We define a parallel composition of contracts that is consis- tent with the definition in [4]. However, in order to reason about contracts over different port sets, it is necessary to equalise the alphabets of the involved assertions. This is done exactly as for interfaces: p∈P (ΣP1 →P |p ∪ ΣP2 →P |p), and Q Definition 6: Let C1 = (P1, ΣP1 , A1, G1) and C2 = (P2, ΣP2 , A2, G2) be contracts. The parallel composition C1 C2 is the contract C = (P, ΣP , A, G) where P = P1 ∪ P2, ΣP = between tasks, for example, by stating that “task 2 is triggered whenever task 1 completes”. Both, the assumptions A and the guarantees G, consist of A = (A ∩ A′2) ∪ ¬(G′1 ∩ G′2), G = G′1 ∩ G′2, A′ = proj −1(ΣP , ΣP )(Ai) and G′ = proj −1(ΣP , ΣP )(Gi). task release (or arrival) times as well as task finishing (or i i i i ¬ ⬦ P P completion) times. These are again modelled using ω-regular languages, but now the semantics is about the behaviour observed at the ports P of a component. An ω-language of a contract is defin...
Adding Contracts. As in [5] we equip our real-time interfaces with a notion of contracts. Contracts are pairs (A, G) where A is an assumption about the environment of a component, and G is the guarantee that the component offers to its environment [4]. For real- time interfaces, both assumptions and guarantees will talk The events ai and fi represent the activation and completion ||| |||
