On a Graph Calculus for Algebras of Relations

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]).…”

“…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

“…The mapping v → v ; w → w ; v, v 1 , v 2 → v; w, w 1 , w 2 → w and u 1 , u 2 , u 3 → u preserves arcs. 5 So, we have a morphism η : D D . We also have formulas δ(D ) and δ(D ) such that g : A morphism transfers satisfying assignments by composition.…”

“…] 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 ).…”

“…Graph calculi rely on two-dimensional representations providing better visualization [2]. 1 In the realm of binary relations, a simple calculus (with linear derivations) [2,3] was extended for handling complement: direct calculi [5,7] and refutation calculi [13]. Our new calculus is a further extension, inheriting much of the earlier terminology (such as 'graph', 'slice' and 'arc'), together with some ideas from Peirce's diagrams for relations [11,4].…”