Sheila R. M. Veloso

On graph reasoning

In this paper, we study the (positive) graph relational calculus. The basis for this calculus was introduced by Curtis and Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easy to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

Reasoning with Graphs

In this paper we study the (positive) graph relational calculus. The basis for this calculus was introduced by S. Curtis and G. Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easier to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

A calculus for graphs with complement

We present a system for deriving inclusions between graphs from a set of inclusions between graphs taken as hypotheses. The novel features are the extended notion of graph with an explicitly representation of complement, the more involved definition of the system, and its completeness proof due to the embedding of complements. This is an improvement on former work, where complement was introduced by definition. Our calculus provides a basis on which one can construct a wide range of graph calculi for several algebras of relations.

On Positive Relational Calculi

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed.

On a Graph Calculus for Algebras of Relations

We present a sound and complete logical system for deriving inclusions between graphs from inclusions between graphs, taken as hypotheses. Graphs provide a natural tool for expressing relations and reasoning about them. Here we extend this system to a sound and complete one to cope with proofs from hypotheses. This leads to a system dealing with complementation. Other approaches using pictures for relations use as bases the theory of allegories or rewriting systems. Our formalism is more widely applicable and provides a common denominator of these approaches.

On Graph Refutation for Relational Inclusions

We introduce a graphical refutation calculus for relational inclusions: it reduces establishing a relational inclusion to establishing that a graph constructed from it has empty extension. This sound and complete calculus is conceptually simpler and easier to use than the usual ones.

Positive Fork Graph Calculus

We introduce and illustrate a graph calculus for proving and deciding the positive identities and inclusions of fork algebras, i.e., those without occurrences of complementation. We show that this graph calculus is sound, complete and decidable. Moreover, the playful nature of this calculus renders it much more intuitive than its equational counterpart.

On Ultrafilter Logic and Special Functions

Logics for “generally” were introduced for handling assertions with vague notions,such as “generally”, “most”, “several”, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable elimination of the generalized quantifier. These devices permit using methods for classical first-order logic to reason about consequence in ultrafilter logic.

On Graph Calculi for Multi-modal Logics

We introduce a method, based on graphical representation, for formulating sound and complete calculi for multi-modal logics. This approach provides uniform tools for expressing and manipulating modal formulas. We illustrate the method by constructing, in a natural manner, correct graph calculi for some multi-modal logics, which may include the global and difference modalities, and may have some special properties.

A Tool for Analysing Logics

We introduce and examine a tool for analysing logics. This algebraic tool, coming from some ideas introduced by J. Piaget, provides condensed information about a logic (with emphasis on the behavior of a unary symbol), as such, it can be employed for analysing and, to some extent, comparing logics.