Hereditarily Sequential Functionals
نویسنده
چکیده
In order to deene models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the deenability problem for PCF, posed in Sco72], and the full abstraction problem for PCF, raised in Plo77]. The presented computation model, forming the class of hereditarily sequential func-tionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. We characterize the computable elements in this model in two diierent ways: (a) by recursiveness requirements for the game, and (b) as deenability with the schemata (S1){ (S8), (S11), which is related to deenability in PCF. It turns out that both deenitions give the same class of computable functionals. So a robust notion of (sequential) computability on higher types is presented.
منابع مشابه
On the product of a compact space with an hereditarily absolutely countably compact space
We show that the product of a compact, sequential T2 space with an hereditarily absolutely countably compact T3 space is hereditarily absolutely countably compact, and further that the product of a compact T2 space of countable tightness with an hereditarily absolutely countably compact ω-bounded T3 space is hereditarily absolutely countably compact.
متن کاملMonotone Majorizable Functionals
Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus deene MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z:Z(1 ? sg) is not MMF. It is proved that extensionality is not ...
متن کاملOn the ubiquity of certain total type structures
It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene-Kreisel continuous functionals, its effective substructure C, and the type structure HEO of the hereditaril...
متن کاملExact real number computations relative to hereditarily total functionals
We show that the continuous existential quantifier ∃ω is not definable in Escardó’s Real-PCF from all functionals equivalent to a given total one in a uniform way. We further prove that relative to any total functional of type (I → I) → I which gives the maximum-value for any total input, we may, given a computable, total functional Φ of type (R → R) → R find a Real-PCF definable total Ψ equiva...
متن کاملTHE SEQUENTIAL FUNCTIONALS OF TYPE ( ι → ι ) n → ι FORM
We prove that the sequential functionals of some fixed types at type level 2, taking finite sequences of unary functions as arguments, form a directed complete partial ordering. This gives a full characterisation of the types for which the partially ordered set of sequential functionals has this property. As a tool, we prove a normal form theorem for the finite sequential functionals of the typ...
متن کامل