A Calculus for Graphs with Complement

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.…”

“…1 Discussions with Petrucio Viana and Renata de Freitas are gratefully acknowledged. 2 Complementation may be introduced by definition, if one can reason from hypotheses [9], or it can be handled via arcs labeled by boxes [12].…”

A Graph Calculus for Predicate Logic

“…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)).…”

“…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].…”