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.Comment: In Proceedings LSFA 2011, arXiv:1203.542

On Graphical Calculi for Modal Logics

We present a graphical approach to classical and intuitionistic modal logics, which provides uniform formalisms for expressing, analysing and comparing their semantics. This approach uses the flexibility of graphical calculi to express directly and intuitively the semantics for modal logics. We illustrate the benefits of these ideas by applying them to some familiar cases of classical and intuitionistic multi-modal logics.Cálculos Gráficos para lógicas modais Apresentamos uma abordagem gráfica para as lógicas modais clássica e intuicionista, capaz de fornecer formalismos uniformes para expressar, analisar e comparar suas respectivas semânticas. Tal abordagem utiliza a flexibilidade dos cálculos gráficos para expressar, direta e intuitivamente, a semântica das lógicas modais. Ilustramos os benefícios dessas ideias aplicando-as a alguns casos conhecidos de lógicas multimodais clássica e intuicionista.---Artigo em inglês

An Agg Application Supporting Visual Reasoning1

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A graphical approach to relational reasoning

Relational reasoning is concerned with relations over an unspecified domain of discourse. Two limitations to which it is customarily subject are: only dyadic relations are taken into account; all formulas are equations, having the same expressive power as first-order sentences in three variables. The relational formalism inherits from the Peirce-Schröder tradition, through contributions of Tarski and many others. Algebraic manipulation of relational expressions (equations in particular) is much less natural than developing inferences in first-order logic; it may in fact appear to be overly machine-oriented for direct hand-based exploitation. The situation radically changes when one resorts to a convenient representation of relations based on labeled graphs. The paper provides details of this representation, which abstracts w.r.t. inessential features of expressions. Formal techniques illustrating three uses of the graph representation of relations are discussed: one technique deals with translating first-order specifications into the calculus of relations; another one, with inferring equalities within this calculus with the aid of convenient diagram-rewriting rules; a third one with checking, in the specialized framework of set theory, the definability of particular set operations. Examples of use of these techniques are produced; moreover, a promising approach to mechanization of graphical relational reasoning is outlined

On Graphical Calculi for Modal Logics

We present a graphical approach to classical and intuitionistic modal logics, which provides uniform formalisms for expressing, analysing and comparing their semantics. This approach uses the flexibility of graphical calculi to express directly and intuitively the semantics for modal logics. We illustrate the benefits of these ideas by applying them to some familiar cases of classical and intuitionistic multi-modal logics

Proofs with Graphs

We present a graphical calculus, which allows mathematical formulae to be represented and reasoned about using a visual representation. We define how a formula may be represented by a graph, and present a number of laws for transforming graphs, and describe the effects these transformations have on the corresponding formulae. We then use these transformation laws to perform proofs. We illustrate the graphical calculus by applying it to the relational and sequential calculi. The graphical calculus makes formulae easier to understand, and so often makes the next step in a proof more obvious. Furthermore, it is more expressive, and so allows a number of proofs that cannot otherwise be undertaken in a point-free way. 1 Introduction Traditionally, mathematical formulae are written down on a single line. For example, in the relational calculus [12], given four relations P , Q, R and S, we can write P ;Q " R;S to represent the relation that relates two elements x and y iff there exist u and..