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...iables, then a constraint is a function that, given an assignment η, returns a truth value indicating if the constraint is satisfied by η. For instance, the assignment {cost 7→ 500} satisfies the first constraint, while {cost 7→ 500, bw 7→ 8000} does not satisfy the second constraint, that is, instead, satisfied by {cost 7→ 400, bw 7→ 8000}. An assignment that satisfies a constraint is called a solution. The constraints we have presented are called crisp in the literature, because they can only be satisfied or violated. In fact, we can also use more general constraints called soft constraints [45]. These constraints, given an assignment for the variables, return an element of an arbitrary constraint semiring (c-semiring, [46]), namely a partially ordered set of ‘preference’ values equipped with two suitable operations for combination (×) and comparison (+) of (tuples of) values and constraints. Formally, a c-semiring is an algebraic structure 〈A,+,×, 0, 1〉 such that: A is a set and 0, 1 ∈ A; + is a binary operation on A that is commutative, associative, idempotent, 0 is its unit element and 1 is its absorbing element; × is a binary operation on A that is commutative, associative, distr...

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...all η. Proposition 3. Csoft is a named c-semiring with idempotent product ×. Note that the support supp(c) of an element of Csoft coincides with the support of a functional constraint c as defined in =-=[3]-=-.sDefinition 4 ((named) constraint). Given a named c-semiring with equalities 〈A,+, ×,ρ, νx.,0,1〉, a (named) constraint c is an element of A. We define here the notions of consistency and entailment o...

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... concurrent constraint programming framework [Sar93], where the fundamental operation of "constraint asking" can be modelled precisely by checking for the presence of a constraint in the com=-=mon store [MR91]-=-. In [MR92b], we model concurrent constraint programming via graph grammars, thus obtaining a trueconcurrency semantics where agents asking for the same constraint are shown to have the possibility of...

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...describing, respectively, the temporal precedence, the possible simultaneity, and the nondeterministic choices among steps of cc computations. This semantics is a conservative extension of the one in =-=[10, 11]-=-, since by restricting the obtained contextual net to subnets without mutual exclusion, we get essentially the partial orders defined in that semantics. What was missing there, and is now present in t...

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...n the SCSP framework a token is simply a soft constraint (that is domain assignment and some associated semiring values); the entailment rule will compute/change new soft constraint (and new levels) (=-=Bistarelli et al. 2002-=-). 3 Constraint Programming for Protocol Analysis This section presents our framework for analysing security protocols. Using soft constraints requires the definition of a c-semiring. Our security sem...

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...ication is defined to satisfy the equation c1 ∗ c2 [υ] = c1 [υ] ∗ c2 [υ], and constraint addition is defined analogously. Then it is easy to prove that the set of constraints forms again a c-semiring =-=[5]-=-. Moreover, we define a projection operator project and (for constraint propagation) a partial evaluation operator peval. For any constraint c = [al | cst] and any valuation υ = [bl ↦→ dl] with bl ⊆ a...