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Essential Structure of Proofs as a Measure of Complexity

Abstract

The essential structure of proofs is proposed as the basis for a measure of complexity of formulas in FOL. The motivating idea was the recognition that distinct theorems can have the same derivation modulo some non essential details. Hence the difficulty in proving them is identical and so their complexity should be the same. We propose a notion of complexity of formulas capturing this property. With this purpose, we introduce the notions of schema calculus, schema derivation and description complexity of a schema formula. Based on these concepts we prove general robustness results that relate the complexity of introducing a logical constructor with the complexity of the component schema formulas as well as the complexity of a schema formula across different schema calculi.

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Acknowledgements

The authors dedicate this work to the memory of Amílcar Sernadas who had the idea for this paper.

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Correspondence to Cristina Sernadas.

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The work was supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020 granted to CMAFcIO (Centro de Matemática, Aplicações Fundamentais e Investigação Operacional) of Universidade de Lisboa and Project: UIDB/EEA/50008/2020 granted to Instituto de Telecomunicações.

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Ramos, J., Rasga, J. & Sernadas, C. Essential Structure of Proofs as a Measure of Complexity. Log. Univers. 14, 209–242 (2020). https://doi.org/10.1007/s11787-020-00251-4

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Keywords

  • Description complexity
  • Schema calculus
  • Schema derivation
  • Uniform and non-uniform robustness

Mathematics Subject Classification

  • 03F20
  • 03F03
  • 03B10
  • 03B22