Contracts Acting on Systems. We can now define how contracts evolve alongside a system, and what it means for a system to satisfy a contract. viopJq = ff viopKq = tt df df df df viopPkpaqrdsq = pa, kq viopOkpaqrdsq = ff = = viopFkpaqrdsq pa, kq viopwaitpdqq ff Definition 10 Given a contract ϕ ϵ C with a set of viopϕ; ϕ1q df viopϕq vioprec x.ϕq df viopϕq actions Act1 and a system A, we define the semantics = = = viopcondkpaqrdspϕ, ϕ1qq df ff of ϕ A — the combination of the system with the con- tract — with alphabet Act with Act1 ≤ Act through = viopϕ^ϕ1q df viopϕqvviopϕ1q = = viopϕvϕ1q df viopϕq^viopϕ1q viopϕ§ ϕ1q df viopϕq^viopϕ1q the following rules:
Contracts Acting on Systems. We can now define how contracts evolve along- side a system, and what it means for a system to satisfy a contract. }A P C Definition 10 Given a contract ϕ with a set of actions Act1 and a system , we define the semantics of ϕ — the combination of the sys- tem with the contract — with alphabet Act with Act1 Ď Act through the following rules: (M1) ϕ −−a−,−kÑ ϕ1, A −−−a−,sÑ A1 ϕ } A =ñ ϕ1 } A1 k P s (M2) ϕ −−pa,kqÑ ϕ1, A ( xa, ky −−− ϕ } A =ñ ϕ1 } A (M3) A −−−−Ñ A1 a R Act1 ϕ } A =ñ ϕ } A1 (M4) d d A ✿✿❀ A1, ϕ ✿✿❀ ϕ1, @d1 < d ¨ if A ✿ d1 A2 and ✿❀ ϕ ✿ d1 ϕ2 then A2 ( viopϕ2q ✿❀ ϕ } A =ñ ϕ1 } A1 Rule M1 and M2 handles synchronization be- tween the contract and the system. If an action a performed by the system is of interest to the con- tract, the contract evolves alongside the system (M1), if the contract allows an agent to perform an action but only agent k (and no other agent) is willing to engage in the action, then only the contract evolves (M2). Rule M3 handles actions on the system in which the contract is not in- terested in. Finally, rule M4 ensures that time cannot skip over a violation. A P C Definition 11 Let be a system and ϕ be a contract. ě • A pA q System can breach ϕ, written breach , ϕ , if there exists a computation that leads to a violation of the contract: for some n 0 and contracts ϕ0 till ϕn such that: ϕ } A = ϕ0 } A0 =ñ . . . ϕn−1 } An−1 =ñ ϕn } An, and An ( viopϕnq. ě • A pA q System may fulfil ϕ, written fulfill , ϕ , if there exists a computation of the system that fulfils the contract: for some n 0 and con- tracts ϕ0 till ϕn: ϕ} A = ϕ0 } A0 =ñ . . . ϕn−1 } An−1 =ñ ϕn } An , and A ( viopϕkq for 0 ď k < n, and ϕn ” J. r 8spK 8q Note that there are contracts that may never be fulfilled. An example of such a contract is ϕ = rec x. a, k, , , which may never be fulfilled since there are no transitions from this contract leading to . Nevertheless, if agent k never performs action a, then neither is the con- tract broken. 5 Refinement
Contracts Acting on Systems p q We can now formally define contract violation. First, we define the predicate vio ϕ . This predicate will be used to verify if a contract is currently violated, which enables us to determine how a system can be monitored with respect to a contract. p q
