Realizability interpretation of PA by iterated limiting PCA

For any partial combinatory algebra (PCA for short) $\mathcal{A}$, the class of $\mathcal{A}$-representable partial functions from $\mathbb{N}$ to $\mathcal{A}$ quotiented by the filter of cofinite sets of $\mathbb{N}$ is a PCA such that the representable partial functions are exactly the limiting partial functions of $\mathcal{A}$-representable partial functions (Akama 2004). The n-times iteration of this construction results in a PCA that represents any n-iterated limiting partial recursive function, and the inductive limit of the PCAs over all n is a PCA that represents any arithmetical partial function. Kleene's realizability interpretation over the former PCA interprets the logical principles of double negation elimination for Σ0n-formulae, and over the latter PCA, it interprets Peano's arithmetic (PA for short). A hierarchy of logical systems between Heyting's arithmetic (HA for short) and PA is used to discuss the prenex normal form theorem, relativised independence-of-premise schemes, and the statement ‘PA is an unbounded extension of HA’.

On Generalized Quantum Turing Machine and Its Applications

Ohya and Volovich discussed a quantum algorithm for the SAT problem with a chaos amplification process (OMV SAT algorithm) and showed that the number of steps it performed was polynomial in input size. In this paper, we define a generalized quantum Turing machine (GQTM) and related computational complexity. Then we show that there exists a GQTM which recognizes the SAT problem in polynomial time. Moreover, we discuss the problem of finding the quantum algorithm for a partial recursive function.

Type reconstruction in Fω

We investigate Girard's calculus Fω as a ‘Curry style’ type assignment system for pure lambda terms. First we show an example of a strongly normalizable term that is untypable in Fω. Then we prove that every partial recursive function is nonuniformly represented in Fω (even if quantification is restricted to constructor variables of level 1). It follows that the type reconstruction problem is undecidable and cannot be recursively separated from normalization.

Theoretical Pearls:Representing ‘undefined’ in lambda calculus

Let ψ be a partial recursive function (of one argument) with λ-defining termF∈Λ°. This meansThere are several proposals for whatF⌜n⌝ should be in case ψ(n) is undefined: (1) a term without a normal form (Church); (2) an unsolvable term (Barendregt); (3) an easy term (Visser); (4) a term of order 0 (Statman).These four possibilities will be covered by one ‘master’ result of Statman which is based on the ‘Anti Diagonal Normalization Theorem’ of Visser (1980). That ingenious theorem about precomplete numerations of Ershov is a powerful tool with applications in recursion theory, metamathematics of arithmetic and lambda calculus.

Π01-classes and Rado's selection principle

There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded -classes have arisen. For our purposes we may define a -class to be a set Path(T) of all infinite paths through a recursive tree T. Here a recursive tree T is just a recursive subset of ω<ω, the set of all finite sequences of the natural numbers ω = {0,1,2,…}, which is closed under initial segments. If the tree T is finitely branching, then we say the -class Path(T) is bounded. If T is highly recursive, i.e., if there exists a partial recursive function f: T→ω such that for each node ηЄ T, f(η) equals the number of immediate successors of η, then we say that the -class Path(T) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded -class and the set of proper marriages for a highly recursive society was always an r.b. -class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. -class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. Π1-class C there is a society Sc such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of Sc. Jockusch conjectured that the set of marriages of a recursive society can represent any bounded -class and the set of marriages of a highly recursive society can represent any r.b. -class. These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. -class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. -class.

Some properties of invariant sets

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures (ω, ·) and (ω, E) respectively. Here, · and E are defined by n · m ≃ φn (m) and nEm if and only if n Є Wm, where {φn} and {Wn} are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers.We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of (ω,E). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of (ω, E) cannot be effectively enumerated.We will next discuss representations of r.e. sets invariant with respect to automorphisms of (ω, ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of (ω, E) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of (ω, ·), then for every a in the domain of f, there is a term t(a) built up from a and · only such that f(a) ≃ t(a). This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of (ω, ·) cannot be effectively enumerated.

Recursive isomorphism types of recursive Boolean algebras

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

On complexity properties of recursively enumerable sets

An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity.As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative.In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

A note on frame extensions

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R.In the following, the notation and terminology is that of [1] and [2].Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ.Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.