Common use of Operational Semantics Clause in Contracts

Operational Semantics. We can now define an operational semantics for our contract calculus. The rules of the operational ▇▇▇▇▇- tics appear in Figure 1. The semantics take one of three forms: (i) ϕ −Ñ ϕ1 to denote that contract ϕ a,k can evolve (in one s−tep) to ϕ1 when action a is per- formed, which involves party k (and possibly other parties); or (ii) ϕ pa,kÑq ϕ1 indicating that the contract units: OppPBPqr5s. In this case, equivalence rule 13 can be applied after 5 time units: OppPBPqr5s ϕ can evolve to ϕ−1−w−−hen the action a is not offered by any party other than k; or (iii) ϕ d ϕ1 to represent OppPBPqr0s ãÑ K. that contract ϕ can evolve to contract ϕ1 when d time \[ p q p q In order to justify the simplification of contract formulae by applying these rules repeatedly, we will need to prove that the rewriting process is terminat- ing and confluent. To prove confluence of ã , we will first prove local confluence, from which conflu- ence follows using a standard result from computer science. Ñ P Ø Proposition 2. The ã C C relation is: (i) ter- minating: there is no infinite sequence ϕ1, ϕ2 . . . , units pass. We will use variable α to stand for a label of either form: a, k or a, k . The rules of the op- erational semantics are always applied to irreducible terms. The core of any contract reasoning formalism is the rules defining the semantics of the deontic modal- ities.

Operational Semantics. We can now define an operational semantics for our contract calculus. The rules of the operational ▇▇▇▇▇- tics appear in Figure 1. The semantics take one of three forms: (i) ϕ −Ñ −→ ϕ1 to denote that contract ϕ a,k can evolve (in one s−tep) to ϕ1 when action a is per- formed, which involves party k (and possibly other parties); or (ii) ϕ pa,kÑq k→q ϕ1 indicating that the contract units: OppPBPqr5sOp(PBP)|5]. In this case, equivalence rule 13 can be applied after 5 time units: OppPBPqr5s Op(PBP)|5] ϕ can evolve to ϕ−1−w−−hen the action a is not offered by any party other than k; or (iii) ϕ d ϕ1 to represent OppPBPqr0s ãÑ K. Op(PBP)|0] ã→ †. that contract ϕ can evolve to contract ϕ1 when d time \[ p q p q un → ( ) ( ) In order to justify the simplification of contract formulae by applying these rules repeatedly, we will need to prove that the rewriting process is terminat- ing and confluent. To prove confluence of ã , we will first prove local confluence, from which conflu- ence follows using a standard result from computer science. Ñ P Ø Proposition 2. The ã C C relation is: (i) ter- minating: there is no infinite sequence ϕ1, ϕ2 . . . , units pass. We will use variable α to stand for a label of either form: a, k or a, k . The rules of the op- erational semantics are always applied to irreducible terms. The core of any contract reasoning formalism is the rules defining the semantics of the deontic modal- ities.→ ϵ m