Free deduction: An analysis of “Computations” in classical logic
Cut-elimination is a central tool in proof-theory, but also a way of computing with proofs used for constructing new functional languages. As such it depends on the properties of the deduction system in which proofs are written.
For intuitionistic logic, natural deduction allows a cut-elimination procedure which effectively provides a computation mechanism with deep theoretical properties such as confluence and strong normalisation. For classical logic, on the contrary, neither natural deduction nor sequent calculus provide a suitable cut-elimination, and, in fact, the computational meaning of classical proofs is an open problem.
In this paper, a new deduction sytem is introduced: free deduction. Free deduction is an adequate system for classical logic, allowing a global cut-elimination procedure in the style of intuitionistic natural deduction. We prove that, provided a choice of inputs (which corresponds to a fundamental non-determinism of classical logic), the cut-elimination procedure in free deduction provides a computation mechanism for classical logic which satisfies confluence and strong normalisation.
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- BARENDREGT, The lambda calculus, North-Holland, 1985.Google Scholar
- GIRARD, Proof theory and logical complexity, Bibliopolis, 1987.Google Scholar
- HOWARD, The formulae-as-types notion of construction, in “To HB Curry...”, Academic Press, 1980.Google Scholar
- PRAWITZ, Natural deduction, Almqvist&Wiksell, 1965.Google Scholar