Encoding partial constraint satisfaction in the semiring-based framework for soft constraints

Abstract

Abstract

A semiring-based framework for the deduction/abduction reasoning in access control with weighted credentials

“…Semirings. An absorptive semiring [4] S can be represented as a A, +, ×, 0, 1 tuple such that: i) A is a set and 0, 1 ∈ A; ii) + is commutative, associative and 0 is its unit element; iii) × is associative, distributes over +, 1 is its unit element and 0 is its absorbing element. Moreover, + is idempotent, 1 is its absorbing element and × is commutative.…”

“…In [4] the authors extended the semiring structure by adding the notion of division, i.e., ÷, as a weak inverse operation of ×. An absorptive semiring S is invertible if, for all the elements a, b ∈ A such that a ≤ b, there exists an element c ∈ A such that b × c = a [4].…”

“…In [4] the authors extended the semiring structure by adding the notion of division, i.e., ÷, as a weak inverse operation of ×. An absorptive semiring S is invertible if, for all the elements a, b ∈ A such that a ≤ b, there exists an element c ∈ A such that b × c = a [4]. If S is absorptive and invertible, then, S is invertible by residuation if the set {x ∈ A | b × x = a} admits a maximum for all elements a, b ∈ A such that a ≤ b [4].…”

“…An absorptive semiring S is invertible if, for all the elements a, b ∈ A such that a ≤ b, there exists an element c ∈ A such that b × c = a [4]. If S is absorptive and invertible, then, S is invertible by residuation if the set {x ∈ A | b × x = a} admits a maximum for all elements a, b ∈ A such that a ≤ b [4]. Moreover, an absorptive S is residuated if the set {x ∈ A | b × x ≤ a} admits a maximum for all elements a, b ∈ A: with an abuse of notation, the maximal element among solutions is denoted a ÷ b.…”

“…The notion of semiring division can be applied to all the classical semirings instances [8], as the Boolean {false, true}, ∨, ∧, false, true , Fuzzy [0, 1], max, min, 0, 1 , Probabilistic [0, 1], max, ×, 0, 1 ( × is the arithmetic multiplication) and Weighted R + ∪ {+∞}, min, +, ∞, 0 semiring ( + is the arithmetic plus operation): the reason is that all these semirings are complete and consequently residuated [4]. Therefore, for all these semirings it is possible to use the ÷ operation as a "particular" inverse of ×.…”