Notions of Bisimulation for Heyting-Valued Modal Languages

Abstract

We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy-Milner type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.

Fuzzy Bisimulation for Gödel Modal Logic

“…Hence, there is a dearth of works on the investigation of bisimulation and the resultant modal invariance results. The only exception is the notion of weak bisimulation for Heyting-valued modal logics L H ♦ introduced in [13], which is defined as a family of binary relations between two Heytingvalued possible world models. In the particular case that the underlying Heyting algebra H is ([0, 1], ≤), L H ♦ is the same as G( ♦).…”

“…However, it can be regarded as a cutbased definition of fuzzy bisimulation by using the results in [13] and this paper. In [13], it is proved that for any two image-finite models M = (W, R, V ), M = (W , R , V ), w ∈ W, w ∈ W , and c ∈ (0, 1], w and w are weakly c-bisimilar iff min(c, V (w, ϕ)) = min(c, V (w , ϕ)) for any ϕ ∈ G( ♦), which is equivalent to inf ϕ∈G( ♦) (V (w, ϕ) ⇔ V (w , ϕ)) ≥ c. Thus, by Theorem 3, w and w are weakly cbisimilar iff (w, w ) belongs to the c-cut of the maximum fuzzy bisimulation between M and M . As the result in [13] holds for any Heyting-valued modal logics, its application scope is indeed more general than ours.…”

How True It Is = Who Says It’s True

“…We noted above that the Boolean algebra approach generalizes naturally to a Heyting algebra version, with an intuition involving dependent agents. We also noted that there has been work on bisimulations in the Heyting algebra setting, [3]. It is not clear that matrix methods carry over to Heyting algebras, particularly in the light of the independence of and ♦.…”

“…If one considers dependent agents, bringing Heyting algebras into the picture, propositional connectives have an intuitionistic flavor, and and ♦ are no longer dual operators, as noted above, so it would seem that the notion of a bisimulation must become more complex than in Section 8. This is investigated in [3], where two different notions of bisimulation are introduced, strong bisimulation and weak t bisimulation. Although it was developed independently of [9], the work in [3] seems to show that weak t bisimulation is the Heyting algebra counterpart of the Boolean version considered here, though this has not been fully checked yet.…”

“…This is investigated in [3], where two different notions of bisimulation are introduced, strong bisimulation and weak t bisimulation. Although it was developed independently of [9], the work in [3] seems to show that weak t bisimulation is the Heyting algebra counterpart of the Boolean version considered here, though this has not been fully checked yet.…”