Positive Fork Graph Calculus

On Graph Refutation for Relational Inclusions

“…Relational terms, slices and graphs are labels and every label is equivalent to a basic graph and to a slice. 8 Our goal-oriented calculus is simpler than some of the available graph relational calculi [7,8,9,10,11,12]. It is conceptually simpler as it proceeds by eliminating relational operations and its rules require only the concept of (draft) morphism (rather than slice homomorphism -a draft morphism that respects input and output nodes -and graph cover [9]).…”

“…On the other hand, an assertion like "r ⊑ s follows from r⊓s ⊑ I ⊥", which is trivial in our approach, will require using the expansion rule in the direct approach. 10 8 Also, any Boolean combination of inclusions is equivalent to an inclusion L ⊑ I ⊥ [16]. 9 The basic forms of these 2 terms are single-slice graphs, with the following slices S and T, respectively:…”

“…So, one can reason about relations by manipulating their representations. This is a key idea underlying graph methods for reasoning about relations [5,6,7,8,9,10,11,12]. Some relational operations (like complementation) are not so easy to handle.…”

A Graph Calculus for Predicate Logic

“…] M = / 0, then some assignment satisfies S. 5 For instance, for arc p/w of D , we have arc p/w of D ; for arcs q/v 1 w 1 and q/v 2 w 2 of D , we have arc q/v w of D . 6 Take δ(D ) as q(v, w) ∧ p(w ) ∧ r(v, w ) ∧ s(v, u) ∧ t(u, w) ∧ a(u, v ) ∧ b(v , w) and δ(D ) as the conjunction of q(v 1 , w 1 ), q(v 2 , w 2 ), p(w ), r(v, w ), r(v 1 , w ), r(v 2 , w ), s(v 2 , u 3 ), t(u 2 , w 1 ), a(u 1 , v ), a(u 3 , v ), b(v , w), b(v , w 1 ) and b(v , w 2 ). 7 The extension of slice T can be described by the formula ∃y (r(u, y) ∧ s(y, w)).…”

“…So, there appear to be advantages and disadvantages on both sides. Some further work on our calculus would be: add function symbols (for this purpose, some ideas used for structured nodes [6] seem promising); provide a detailed comparison between it and [9] (such a comparison between Peirce's and Rensink's approaches is reported difficult [9], p. 333); develop a "middle-ground" between our approach and Dau's [4], with the best features from each one.…”