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Lambda representation of operations between different term algebras

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Computer Science Logic (CSL 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 933))

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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 Aτ 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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References

  1. Corrado Böhm and Allessandro Berarducci, Automatic synthesis of typed λ programs on term algebras, Theoretical Computer Science 39 (1985) 135–154

    Article  Google Scholar 

  2. Daniel Leivant Subrecursion and lambda representation over free algebras, in S. Buss and P Scott (eds.), Feasible Mathematics (Proceedings of June 1988 Workshop at Cornell)

    Google Scholar 

  3. Madry M, On the λ definable functions between numbers, words and trees Fundamenta Informaticae, 1991

    Google Scholar 

  4. Schwichtenberg H., Definierbare Funktionen im λ-Kalkül mit Typen, Arch Math. Logik Grundlagenforsch 17 (1975–76) pp 113–114.

    Google Scholar 

  5. Statman R., Intuitionistic propositional logic is polynomial-space complete, Theoretical Computer Science 9, 67–72 (1979)

    Article  Google Scholar 

  6. Zaionc M., Word operations definable in the typed λ calculus, Theoretical Computer Science 52 (1987) pp. 1–14

    Article  Google Scholar 

  7. Zaionc M., A Characteristic of λ definable Tree Operations, Information and Computation 89 No.1, (1990) 35–46

    Article  Google Scholar 

  8. Zaionc M., λ definability on free algebras, Annals of Pure and Applied Logic 51 (1991) pp 279–300.

    Article  Google Scholar 

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Leszek Pacholski Jerzy Tiuryn

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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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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60017-6

  • Online ISBN: 978-3-540-49404-1

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