Abstract
There is a natural isomorphism identifying second order types of the simple typed λ calculus with free homogeneous term algebras. Let τ A and τ B be types representing algebras A and B respectively. Any closed term of the type τ A → τ B represents a computable function between algebras A and B. The problem investigated in the paper is to find and characterize the set of all λ definable functions between structures A and B. The problem is presented in a more general setting. If algebrasA 1,..., A n ,B are represented respectively by second order types \(\tau ^{A_l } ,...,\tau ^{A_n } \), τ B then \(\tau ^{A_l } \)→ (...(\(\tau ^{A_n } \)→ τ B...) is a type of functions from the product A 1×...xA n into algebra B. Any closed term of this type is a representation of algorithm which transforms the tuple of terms of types \(\tau ^{A_l } ,...,\tau ^{A_n } \) respectively into a term of type τ B, which represents an object in algebra B (see [BöB85]). The problem investigated in the paper is to find an effective computational characteristic of the λ definable functions between arbitrary free algebras and the expressiveness of such transformations. As an example we will consider λ definability between well known free structures such as: numbers, words and trees. The result obtained in the paper is an extension of the results concerning λ definability in various free structures described in [Sch75] [Sta79] [Lei89] [Zai87] [Zai90] and [Zai91]
This research was supported by KBN Grant 0384/P4/93
This paper was partially prepared while author was visiting Computer Science Department at State University of New York at Buffalo, USA
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© 1995 Springer-Verlag Berlin Heidelberg
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Zaionc, M. (1995). Lambda representation of operations between different term algebras. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022249
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DOI: https://doi.org/10.1007/BFb0022249
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