Abstract
We present an algorithm for simplifying Fitch-style natural-deduction proofs in classical first-order logic. We formalize Fitch-style natural deduction as a denotational proof language,\({\user1{\mathcal{N}\mathcal{D}\mathcal{L}}}\), with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transformations and show them to be terminating and to respect the formal semantics of the language. We also show that the transformations never increase the size or complexity of a deduction – in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of superfluous “detours,” and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the transformations are fully implemented in SML-NJ, and the complete code listing is available on the Web.
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Arkoudas, K. Simplifying Proofs in Fitch-Style Natural Deduction Systems. J Autom Reasoning 34, 239–294 (2005). https://doi.org/10.1007/s10817-005-9000-3
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DOI: https://doi.org/10.1007/s10817-005-9000-3