Calculability of the primitive recursive functionals of finite type over the natural numbers

1975 ◽  
pp. 152-163
Author(s):  
Yoshito Hanatani
Keyword(s):  
Finite Type ◽  
Natural Numbers ◽  
Primitive Recursive ◽  
Primitive Recursive Functionals

A system of abstract constructive ordinals

1972 ◽  
Vol 37 (2) ◽  
pp. 355-374 ◽  
Author(s):  
W. A. Howard
Keyword(s):  
Finite Type ◽  
Free Variable ◽  
Transfinite Induction ◽  
Functional Interpretation ◽  
Variable Theory ◽  
Inductive Definitions ◽  
Heyting Arithmetic ◽  
Abstract Notion ◽  
Primitive Recursive ◽  
Primitive Recursive Functionals

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.

Ordinal analysis of terms of finite type

1980 ◽  
Vol 45 (3) ◽  
pp. 493-504 ◽  
Author(s):  
W. A. Howard
Keyword(s):  
Finite Type ◽  
Axiom Of Choice ◽  
Transfinite Induction ◽  
Critical Number ◽  
Ordinal Analysis ◽  
Primitive Recursion ◽  
Successor Relation ◽  
Ordinal Notations ◽  
Primitive Recursive ◽  
Primitive Recursive Functionals

Sanchis [Sa] and Diller [Di] have introduced an interesting relation between terms for primitive recursive functionals of finite type in order to obtain a computability proof. The method is as follows. First the relation is shown to be well-founded. Then computability of each term is obtained by transfinite induction over the relation.Their relation is given by various clauses which define the (immediate) successors of a term. Hence, starting with a term H and repeatedly applying the successor relation one generates the tree of H (say). The problem is to prove that the trees are well-founded. In their proofs Sanchis and Diller use the axiom of bar induction. Diller also gives a proof using transfinite induction: each tree is shown to have a length b < φω(0) (the first ω-critical number) and the induction is over ordinals less than or equal to b.For proof-theoretic purposes it would be desirable to employ a method of proof which is more elementary than the axiom of bar induction and also to obtain a smaller ordinal bound. The purpose of the following is to provide such a method. We show that for a primitive recursive term H of finite type the tree generated by the relation of Sanchis and Diller has length less than ε0. Our method is similar to that used by Tait [Ta] in his theory of infinite terms. The immediately natural metamathematics is elementary intuitionistic analysis with the axiom of choice and transfinite induction over the relevant ordinal notations. We give several versions of the basic list of successor clauses, and for one version we show how to carry out the treatment in Skolem arithmetic (with primitive recursion of lowest type).

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

Finite axiomatizability for equational theories of computable groupoids

1989 ◽  
Vol 54 (3) ◽  
pp. 1018-1022 ◽  
Author(s):  
Peter Perkins
Keyword(s):  
Binary Operation ◽  
Finite Algebra ◽  
Equational Theory ◽  
Natural Numbers ◽  
Finite Axiomatizability ◽  
Finite Algebras ◽  
Equational Theories ◽  
Primitive Recursive ◽  
Positive Integers ◽  
Recursive Set

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.

The countably based functionals

1983 ◽  
Vol 48 (2) ◽  
pp. 458-474 ◽  
Author(s):  
John P. Hartley
Keyword(s):  
Finite Type ◽  
Recursion Theory ◽  
Type Structure ◽  
Topological Properties ◽  
Natural Numbers ◽  
Finite Amount ◽  
Amount Of Information ◽  
Natural Concept ◽  
Higher Types

In [5], Kleene extended previous notions of computations to objects of higher finite type in the maximal type-structure of functionals given by:Tp(0) = N = the natural numbers,Tp(n + 1) = NTp(n) = the set of total maps from Tp(n) to N.He gave nine schemata, S1–S9, for describing algorithms for computations from a finite list of functionals, and indices to denote these algorithms. It is generally agreed that S1-S9 give a natural concept of computations in higher types.The type-structure of countable functions, Ct(n) for n ϵ N, was first developed by Kleene [6] and Kreisel [7]. It arises from the notions of ‘constructivity’, and has been extensively studied as a domain for higher type recursion theory. Each countable functional is globally described by a countable amount of information coded in its associate, and it is determined locally by a finite amount of information about its argument. The countable functionals are well summarised in Normann [9], and treated in detail in Normann [8].The purpose of this paper is to discuss a generalisation of the countable functionals, which we shall call the countably based functions, Cb(n) for n ϵ N. It is suggested by the notions of ‘predicativity’, in which we view N as a completed totality, and build higher types on it in a constructive manner. So we shall allow quantification over N and include application of 2E in our schemata. Each functional is determined locally by a countable amount of information about its argument, and so is globally described by a continuum of information coded in its associate, which will now be a type-2 object. This generalisation, obtained via associates, was proposed by Wainer, and seems to reflect topological properties of the countable functionals.

scholarly journals KMS STATES ON NICA–TOEPLITZ ALGEBRAS OF PRODUCT SYSTEMS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250123 ◽  
Author(s):  
JEONG HEE HONG ◽  
NADIA S. LARSEN ◽  
WOJCIECH SZYMAŃSKI
Keyword(s):  
Finite Type ◽  
Lattice Ordered Group ◽  
Toeplitz Algebra ◽  
Product System ◽  
Natural Numbers ◽  
Ordered Group ◽  
Kms States ◽  
The Core ◽  
Product Systems ◽  
Affine Semigroup

We investigate KMS states of Fowler's Nica–Toeplitz algebra [Formula: see text] associated to a compactly aligned product system X over a semigroup P of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If (G, P) is a lattice ordered group and X is a product system of finite type over P satisfying certain coherence properties, we construct KMSβ states of [Formula: see text] associated to a scalar dynamics from traces on the coefficient algebra of the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.

Parametrized bar recursion: a unifying framework for realizability interpretations of classical dependent choice

2019 ◽  
Vol 29 (4) ◽  
pp. 519-554 ◽  
Author(s):  
Thomas Powell
Keyword(s):  
Large Family ◽  
Classical Analysis ◽  
Wide Range ◽  
Simple Corollary ◽  
The Many ◽  
Primitive Recursive ◽  
Dependent Choice ◽  
Primitive Recursive Functionals ◽  
Negative Translation

Abstract During the last 20 years or so, a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with some form of bar recursion, which realizes the negative translation of either countable or countable dependent choice. In this work, we present the many variants of bar recursion used in this context as instantiations of a parametrized form of backward recursion, and give a uniform proof that under certain conditions this recursor realizes a corresponding family of parametrized dependent choice principles. From this proof, the soundness of most of the existing bar recursive realizability interpretations of choice, including those based on the Berardi–Bezem–Coquand functional, modified realizability and the more recent products of selection functions of Escardó and Oliva, follows as a simple corollary. We achieve not only a uniform framework in which familiar realizability interpretations of choice can be compared, but show that these represent just simple instances of a large family of potential interpretations of dependent choice principles.

Primitive recursive ordinal functions with added constants

1977 ◽  
Vol 42 (1) ◽  
pp. 77-82
Author(s):  
Stanley H. Stahl
Keyword(s):  
Constant Function ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Primitive Recursive

One of the basic differences between the primitive recursive functions on the natural numbers and the primitive recursive ordinal functions (PR) is the nearly complete absence of constant functions in PR. Since ω is closed under all of the functions in PR, if α is any infinite ordinal, then λξ·α is not in PR. It is easily seen, however, that if one adds to the initial functions of PR the constant function λξ·ω, then all of the ordinals up to ω#, the next largest PR-closed ordinal, have their constant functions in this class. Since, however, such collections of functions are always countable, it is also the case that if one adds to the initial functions of PR the function λξ. α for uncountable α, then there are ordinals β < α whose constant functions are not in this collection. Because of this, the following objects are of interest:Definition. For all α,(i)PR(α) is the collection of functions obtained by adding to the initial primitive recursive ordinal functions, the function λξ· α;(ii) PRsp(α), the primitive recursive spectrum of α, is the set {β < α ∣ λξβ ∈ PR(α);(iii) Λ (α)= μρ(ρ∉ PRsp(α)).

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