# Lambda representation of operations between different term algebras

## 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 algebras*A*_{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}×...x*A*_{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]

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## References

- [BöB85]Corrado Böhm and Allessandro Berarducci,
*Automatic synthesis of typed λ programs on term algebras*, Theoretical Computer Science 39 (1985) 135–154CrossRefGoogle Scholar - [Lei89]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 - [Mad91]Madry M,
*On the λ definable functions between numbers, words and trees*Fundamenta Informaticae, 1991Google Scholar - [Sch75]Schwichtenberg H.,
*Definierbare Funktionen im λ-Kalkül mit Typen*, Arch Math. Logik Grundlagenforsch 17 (1975–76) pp 113–114.Google Scholar - [Sta79]Statman R.,
*Intuitionistic propositional logic is polynomial-space complete*, Theoretical Computer Science 9, 67–72 (1979)CrossRefGoogle Scholar - [Zai87]Zaionc M.,
*Word operations definable in the typed λ calculus*, Theoretical Computer Science 52 (1987) pp. 1–14CrossRefGoogle Scholar - [Zai90]Zaionc M.,
*A Characteristic of λ definable Tree Operations*, Information and Computation 89 No.1, (1990) 35–46CrossRefGoogle Scholar - [Zai91]Zaionc M.,
*λ definability on free algebras*, Annals of Pure and Applied Logic 51 (1991) pp 279–300.CrossRefGoogle Scholar