Abstract
Let [0,1] be the real unit interval. A Schauder hat is a Λ-shaped function h:[0,1] → [0,1] whose four pieces are given by linear polynomials with integral coefficients. Rose and Rosser gave an effective method to represent every Schauder hat by a sentence in the infinite-valued calculus of Lukasiewicz. We give an effective method to reduce every sentence ψ with one variable, to an equivalent sentence φ which is a disjunction of Schauder hat sentences. Since the equivalence between ψ and φ holds in all n-valued calculi, our normal form reduction may be used for a uniform (i.e., n-free) treatment of deduction in these calculi. For the case under consideration, our methods already yield a self-contained and constructive proof of McNaughton's theorem stating that in the infinite-valued calculus every piecewise linear function with integral coefficients is representable by some sentence.
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© 1992 Springer-Verlag Berlin Heidelberg
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Mundici, D. (1992). Normal forms in infinite-valued logic: The case of one variable. In: Börger, E., Jäger, G., Kleine Büning, H., Richter, M.M. (eds) Computer Science Logic. CSL 1991. Lecture Notes in Computer Science, vol 626. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023773
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DOI: https://doi.org/10.1007/BFb0023773
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