Abstract
We consider the lambda definability problem over an arbitrary free algebra. There is a natural notion of primitive recursive function in such algebras and at the same time a natural notion of a lambda definable function. We shown that the question: ”For a given free algebra and a primitive recursive function within this algebra decide whether this function is lambda definable” is undecidable if the algebra is infinite. The main part of the paper is dedicated to the algebra of numbers in which lambda definability is described by the Schwichtenberg theorem. The result for an arbitrary infinite free algebra has been obtained by a simple interpretation of numerical functions as recursive functions in this algebra. This result is a counterpart of the distinguished result of Loader in which lambda terms are evaluated in finite domains for which undecidability of lambda definability is proved.
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Małolepszy, J., Moczurad, M., Zaionc, M. (1997). Schwichtenberg-style lambda definability is undecidable. In: de Groote, P., Roger Hindley, J. (eds) Typed Lambda Calculi and Applications. TLCA 1997. Lecture Notes in Computer Science, vol 1210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62688-3_41
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DOI: https://doi.org/10.1007/3-540-62688-3_41
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