Contract Syntax syntactically defined as follows: Definition 5 The set of contract formulae denoted by C (with variable ϕ P C to range over the contracts) is The reparation operator, written ϕ1 § ϕ2, is the contract which starts off as ϕ1, but when violated trig- gers contract ϕ2, e.g., O1 a 2 § P2 b 5 is the con- tract which obliges agent 1 to perform action a in 2 time units, but if she does not, permits agent 2 to per- form action b in 5 time units. p qr sp q p qr s p qr s The formula condk a d ϕ1, ϕ2 is a conditional contract where (i) if party k performs action a within d time units it proceeds to behave like ϕ1; otherwise (ii) if d time units elapse without a being performed by k, it then proceeds to behave like ϕ2. Note that we can generalize to more general conditions on the system, but we limit it to the ability of a party to perform an action for the scope of this paper. p q p qr s p qr s ^ Finally, rec x.ϕ and x handles recursive contracts, e.g., rec x.Op a d ; x is the contract which obliges agent p to repeatedly perform action a within d time units of each other. In contrast, rec x.Or pr 10 wait 30 ; x, is the contract in which agent r is repeat- edly obliged to pay rent (action pr) during the first 10 days of the month. Using these basic contract combinators, we can define more complex ones, for example, a prohibi- tion which persists until a particular action is per- formed — a prohibition on agent k from perform- ing action a until party l performs action b, written F pra, ks U rb, lsq, and defined as follows: , , = rec . k , F pra ks U rb lsq df x `cond paqr8spK Jq^ x x k ϕ ::= J | K | P paqrds | O paqrds | F paqrds condlpbqr8spJ, xq˘ | waitpdq | condkpaqrdspϕ1, ϕ2q | ϕ1; ϕ2 J K | ϕ1 ^ ϕ2 | ϕ1 vϕ2 | ϕ1 § ϕ2 | rec x.ϕ | x where a P Act, x P fvars, k P I and d P T Yt8u. The basic formulae and indicate, respectively, the contracts that are trivially satisfied and violated. The key modalities we use from deontic logic to spec- Example 1 The contract of the plane boarding sys- tem from Section 2, can be formalised using our con- tract calculus as follows: ϕ0 ::= Pppcheckinqrt0 − 120s ϕ1 ::= OppPBPqr5s ϕ2 ::= OppShPqr5s ϕ3 ::= pF prweapon, ps U rlanding, csqq^ ify contracts are permissions, obligations, and prohi- p qr s bitions. The formula Pk a d indicates the permis- sion of agent k to perform action a within d time units, ϕ4 :: ppF prlq, ps U rboarding, psqq § Oppdlqqr10sq = pPppbrdqr90s; Ppphlqr10sq§ pOcpbrdqr90s; Ocphlhldqr20sq p q p qr s p qr s while Ok ...
Contract Syntax. Definition 5. The set of contract formulae denoted by C (with variable ϕ P C to range over the contracts) is contract where (i) if party k performs action a within d time units it proceeds to behave like ϕ1; otherwise (ii) if d time units elapse without a being performed by k, it then proceeds to behave like ϕ2. Note that we can generalize to more general conditions on the system, but we limit it to the ability of a party to perform an action for the scope of this paper. p q p qr s p qr s ^ Finally, rec x.ϕ and x handles recursive contracts, e.g., rec x.Op a d ; x is the contract which obliges agent p to repeatedly perform action a within d time units of each other. In contrast, rec x.Or pr 10 wait 30 ; x, is the contract in which agent r is repeat- edly obliged to pay rent (action pr) during the first 10 days of the month. Using these basic contract combinators, we can define more complex ones, for example, a prohibi- tion which persists until a particular action is per- formed — a prohibition on agent k from perform- ing action a until party l performs action b, written F pra, ks U rb, lsq, and defined as follows: ` ˘ df syntactically defined as follows: ϕ ::= J | K | P paqrds | O paqrds | F paqrds F pra, ks U rb, lsq = rec x. condkpaqr8spK, Jq^ condl pbqr8spJ, xq
