Propositional Gödel Logic and Delannoy Paths

An analysis of Ruspini partitions in Gödel logic

“…Indeed, Theorem 4.6 is the exact counterpart for condition (1.2) of Theorem 3. 10. There are, however, two significant differences.…”

“…Example 2.3 Let µ, ν, ξ be assignments such that One checks that ≤ in Definition 2.2 indeed is a partial order on F n , and (F n , ≤) is in fact a forest [10,Lemma 3.3]. Direct inspection shows that a) the roots of the trees are the equivalence classes of Boolean assignments, b) the equivalence class [µ] ≡n such that µ(X 1 ) = · · · = µ(X n ) = 0 is the only tree having height 1, and c) the leaves are those equivalence classes of assignments in which no variable is set to 1.…”

Best Approximation of Ruspini Partitions in Gödel Logic

“…. , n} such that One checks that ≤ in Definition 3 indeed is a partial order on F n , and (F n , ≤) is in fact a forest [4,Lemma 3.3]. We immediately notice that a) the roots of the trees are the classes of Boolean assignments, b) the class [µ] ≡n such that µ(X 1 ) = • • • = µ(X n ) = 0 is the only tree having height 1, and c) the leaves are those classes of assignments in which no variable is set to 1.…”

“…⊓ ⊔ Figure 2 shows the forest F 2 , whose nodes are labelled by the ordering of variables under a given assignment as in (4). However, for the sake of readability, here and in the following figure we write X i instead of µ(X i ).…”