Fuzzifying Modal Algebra

Abstract

Fuzzy relations are mappings from pairs of elements into the interval [0, 1]. As a replacement for the complement operation one can use the mapping that sends x to 1 − x. Together with the concepts of t-norm and t-conorm a weak form of Boolean algebra can be defined. However, to our knowledge so far no notion of domain or codomain has been investigated for fuzzy relations. These might, however, be useful, since fuzzy relations can, e.g., be used to model flow problems and many other things. We give a new axiomatisation of two variants of domain and codomain in the more general setting of idempotent left semirings that avoids complementation and hence is applicable to fuzzy relations. Some applications are sketched as well.

A Hierarchy of Algebras for Boolean Subsets

“…Abstracting from the relational IL-semiring to a general one, we arrive at the following definitions [ 8 , 29 ]. A left prepredomain semiring is an IL-semiring S with an additional prepredomain operation satisfying for all .…”

Generalising KAT to Verify Weighted Computations

“…M. Winter [33,34,35] follows this route through a categorical perspective. The work of J. Desharnais et al [7] continued along distinct paths: one [10] proposes a new axiomatisation for domain and codomain operators, leading to algebras of domain elements of which Boolean and Heyting algebras are special cases; another [6] investigates notions of domain and codomain operators to provide applications in fuzzy relations and matrices, by using an idempotent left semiring as the base algebraic structure. This paper builds on such motivations to introduce two generalisations of KAT able to express programs as weighted computations and tests as predicates evaluated in a graded truth space -the graded Kleene algebra with tests (GKAT) and the idempotent graded Kleene algebra with tests (I-GKAT).…”

“…The formalisation and the proof of the soundness of the Hoare logic deductive system using the structure based on Heyting domain semirings, by comparing with the our approach, seems also an appropriate discussion to be made in future work. Additionally, a more recent work [6] investigates a generalisation of these domain algebras to support fuzzy relations, taken as functions from pairs of elements to the interval [0, 1]. Different from our approach, the authors study an axiomatisation of domain and codomain operators in the setting of idempotent left semirings, which do not require left distributivity of multiplication over addition and right annihilation of 0.…”

“…p + (q + r) = (p + q) + r (1) p + q = q + p (2) p; (q; r) = (p; q); r(3)p; 1 = 1; p = p (4) p; (q + r) = (p; q) + (p; r) (5) (p + q); r = (p; r) + (q; r)(6) …”