Finite type structures within combinatory algebras

Finite Type Structures within Combinatory Algebras

Inside a combinatory algebra, there are 'internal' versions of the nite type structure over !, which form models of various systems of nite type arithmetic. This paper compares internal representations of the intensional and extensional functionals. If these classes coincide, the algebra is called ft-extensional. Some criteria for ft-extensionality are given and a number of well-known ca's are ...

Collapsing Partial Combinatory Algebras

Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or re-cursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identiication of elements in a pca. We establish several facts concerning nal collapses (maximal identiication of elements). `En passan...

Resource Combinatory Algebras

ion of variables A useful combinator: • linear identity: I ≡ S(1)K[ ] satisfies Ix̄ = { x0 if |x̄| = 1 0 otherwise Abstraction on monomials (i) λ∗x.t ≡ Kt if degx(t) = 0 (ii) λ∗x.x ≡ Iion on monomials (i) λ∗x.t ≡ Kt if degx(t) = 0 (ii) λ∗x.x ≡ I (iii) λ∗x.t0[t1, . . . , tk] ≡ Sn̄[λ∗x.t0][λ∗x.t1, . . . , λx.tk] with n̄ = (degx(t0), . . . , degx(tk)) and ∃i. degx(ti) 6= 0. (iv) ... and on polynomials...

Extending partial combinatory algebras

We give a negative answer to the question whether every partial combinatory algebra can be completed. The explicit counterexample will be an intricately constructed term model, the construction and the proof that it works heavily depending on syntactic techniques. In particular, it is a nice example of reasoning with elementary diagrams and descendants. We also include a domain-theoretic proof ...

Extending partial combinatory algebras