On Kleene Algebras for Weighted Computation

Generalising KAT to Verify Weighted Computations

“…Extending KAT to the domain of weighted computation is the main motivation of this work. The paper extends some preliminary results documented in our previous work [13] in distinct directions. First, we propose three algebraic constructions that represent models for both GKAT and I-GKAT: the set of all fuzzy sets, the set of all fuzzy relations and the set of all fuzzy languages, provided with the appropriate operators over the elements for each case.…”

“…We have just shown that axiom (11) holds for any a, b, c ∈ T in M . Since axioms (1)-(10), (12), (13) are axioms of A, M is indeed a GKAT. 2 Example 1 (2 -the Boolean lattice).…”

“…We denote by + the supremum of K, and operators ; and → satisfying the axioms (1)- (7) and (11) of Figure 2. We use the same notation for operators of T satisfying (1)-(7) plus (11)- (13). Since + and ; are associative, we can generalise them to n-ary operators and use the notation and to represent their iterated versions, respectively.…”

“…Considering the way that the elements of T X and the operators ∪, ⊗ and → are defined, it is straightforward to verify that axioms (1) to (7) and (11) to (13), plus (19) for I-GKAT, are satisfied. We prove that axioms dealing with operator * (8), (9) hold as well.…”

“…{ definition of and zi ∈ X, 1 ≤ i ≤ n} σ(x, z 1 ); η(z 1 , y) + · · · + σ(x, z n ); η(z n , y), for all σ(x, z i ), η(z i , y) = 0, with 1 ≤ i ≤ n Clearly x = z i = y, using the definition of σ(x, y), for all σ ∈ T X×X . Thus, the proof follows directly from (13) for elements of T , as shown below.…”