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Logic for Concurrency and Synchronisation pp 3–88Cite as

Geometry of Deduction Via Graphs of Proofs

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Part of the book series: Trends in Logic ((TREN,volume 15))

Abstract

We are here concerned with the study of proofs from a geometric perspective. By first recalling the pioneering work of Statman in his doctoral thesis Structural Complexity of Proofs (1974), we review two recent research programmes which approach the study of structural properties of formal proofs from a geometric perspective: (i) the notion of proof-net, given by Girard in 1987 in the context of linear logic; and (ii) the notion of logical flow graph given by Buss in 1991 and used as a tool for studying the exponential blow up of proof sizes caused by the cut-elimination process, a recent programme (1996–2000) proposed by Carbone in collaboration with Semmes.

Statman’s geometric perspective does not seem to have developed much further than his doctoral thesis, but the fact is that it looks as if the main idea, i.e. extracting structural properties of proofs in natural deduction (ND) using appropriate geometric intuitions, offers itself as a very promising one. With this in mind, and having at our disposal some interesting and rather novel techniques developed for proof-nets and logical flow graphs, we have tried to focus our investigation on a research for an alternative proposal for looking at the geometry of ND systems. The lack of symmetry in ND presents a challenge for such a kind of study. Of course, the obvious alternative is to look at multiple-conclusion calculi. We already have in the literature different approaches involving such calculi. For example, Kneale’s (1958) tables of development (studied in depth by Shoesmith & Smiley (1978)) and Ungar’s (1992) multiple-conclusion ND.

After surveying the main research programmes, we sketch a proposal which is similar to both Kneale’s and Ungar’s in various aspects, mainly in the presentation of a multiple conclusion calculus in ND style. Rather than just presenting yet another ND proof system, we emphasise the use of ‘modern’ graph-theoretic techniques in tackling the ‘old’ problem of adequacy of multiple-conclusion ND. Some of the techniques have been developed for proof-nets (e.g. splitting theorem, soundness criteria, sequentialisation), and have proved themselves rather elegant and useful indeed.

Research partially funded by a grant from PROPESQ/UFPE under the Projeto Enxoval.

Research partially funded by a CNPq Bolsa de Produtividade em Pesquisa (“Pesquisador 1-C”), grant 301492/88-3.

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Authors and Affiliations

  1. Centro de Informática, Universidade Federal de Pernambuco, Brazil

    Anjolina Grisi de Oliveira & Ruy J. G. B. de Queiroz

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  1. Anjolina Grisi de Oliveira
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  1. Universidade Federal de Pernambuco, Recife, Brazil

    Ruy J. G. B. de Queiroz

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de Oliveira, A.G., de Queiroz, R.J.G.B. (2003). Geometry of Deduction Via Graphs of Proofs. In: de Queiroz, R.J.G.B. (eds) Logic for Concurrency and Synchronisation. Trends in Logic, vol 15. Springer, Dordrecht. https://doi.org/10.1007/0-306-48088-3_1

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  • DOI: https://doi.org/10.1007/0-306-48088-3_1

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