Abstract hierarchies and degrees

AbstractThe aim of this paper is to enrich the algebraic-axiomatic approach to recursion theory developed in [1] by an analogue to the classical arithmetical hierarchy and an abstract notion of degree. The results presented here are rather initial and elementary; indeed, the main problem was the very choice of right abstract concepts.

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Alternating Turing machines and the analytical hierarchy

10.29007/t77g ◽

2018 ◽

Author(s):

Daniel Leivant

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Recursion Theory ◽

Turing Machines ◽

Arithmetical Hierarchy ◽

Definition Of ◽

Alternating Turing Machines ◽

Insight Into ◽

Computably Enumerable

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

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It’s in My Bones

The Blind Storyteller ◽

10.1093/oso/9780190061920.003.0009 ◽

2020 ◽

pp. 131-149

Author(s):

Iris Berent

Keyword(s):

Embodied Cognition ◽

The Body ◽

Computational Theory Of Mind ◽

Abstract Concepts ◽

Computational Theory ◽

Bodily Experiences ◽

Abstract Notion ◽

The Mind ◽

Material Bodies

When I point to an object, you and I can agree on what it is (a red, round cup). How does our brain (matter) represent such notions? And how do we (distinct material bodies) apparently converge so we can talk about the same things? Cognitive scientists and philosophers have long assumed that people share abstract concepts (e.g., a cup); to explain how such abstract concepts can give rise to thinking, they further proposed the computational theory of mind. But theories of “embodied cognition” assert that cognition is all “in people’s bones.” What we know as a cup is not an abstract notion but rather the bodily experiences of our sensory and motor interactions with a cup—its shiny color, how it feels in our hands, the smoothness of its surface, its weight, and shape. I suggest that “Embodiment” is alluring because it promises to resolve the mysteries of Dualism (how can material bodies encode the immaterial notion of a cup?) and the origins of ideas (how do we all converge on an understanding that allows us to talk about the same things?). The solution is strikingly simple—just remove the “mind” from the equation. If there is no (immaterial) knowledge, then we no longer need to worry about how knowledge arises from the body and how knowledge can be learned. As discussed in the previous chapter, people erroneously believe that “if it’s in my body” then “it’s inborn.” Dualism and essentialism thus explain some of the lure of embodied cognition.

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Ramsey's theorem and recursion theory

Journal of Symbolic Logic ◽

10.2307/2272972 ◽

1972 ◽

Vol 37 (2) ◽

pp. 268-280 ◽

Cited By ~ 74

Author(s):

Carl G. Jockusch

Keyword(s):

Recursion Theory ◽

The Other ◽

Natural Numbers ◽

Arithmetical Hierarchy ◽

Original Proof ◽

Ramsey's Theorem ◽

Ramsey’S Theorem ◽

Infinite Sets ◽

Recursive Partition ◽

Positive Results

Let N be the set of natural numbers. If A ⊆ N, let [A]n denote the class of all n-element subsets of A. If P is a partition of [N]n into finitely many classes C1, …, Cp, let H(P) denote the class of those infinite sets A ⊆ N such that [A]n ⊆ Ci for some i. Ramsey's theorem [8, Theorem A] asserts that H(P) is nonempty for any such partition P. Our purpose here is to study what can be said about H(P) when P is recursive, i.e. each Ci, is recursive under a suitable coding of [N]n. We show that if P is such a recursive partition of [N]n, then H(P) contains a set which is Πn0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [N]n into two classes such that H(P) contains no Σn0 set. These results answer a question raised by Specker [12].A basic partition is a partition of [N]2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.

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Jens E. Fenstad. General recursion theory. An axiomatic approach. Perspectives in mathematical logic. Springer-Verlag, Berlin, Heidelberg, and New York, 1980, XI + 225 pp.

Journal of Symbolic Logic ◽

10.2307/2273600 ◽

1982 ◽

Vol 47 (3) ◽

pp. 696-698

Author(s):

Douglas Cenzer

Keyword(s):

New York ◽

Mathematical Logic ◽

Recursion Theory ◽

Axiomatic Approach

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Forcing and reducibilities

Journal of Symbolic Logic ◽

10.2307/2273547 ◽

1983 ◽

Vol 48 (2) ◽

pp. 288-310 ◽

Cited By ~ 6

Author(s):

Piergiorgio Odifreddi

Keyword(s):

Set Theory ◽

Recursion Theory ◽

Jump Operator ◽

Previous Knowledge ◽

Arithmetical Hierarchy ◽

Closed Sets ◽

Turn On

We see far away, Newton said, if we stand on giants' shoulders. We take him seriously here and moreover (as appropriate to recursion-theorists) we will jump from one giant to another, since this paper is mostly an exegesis of two fundamental works: Feferman's Some applications of the notions of forcing and generic sets [4] and Sacks' Forcing with perfect closed sets [19]. We hope the reader is not afraid of heights: our exercises are risky ones, since the two giants are in turn on the shoulders of others! Feferman [4] rests on the basic works of Cohen [2], who introduced forcing with finite conditions in the context of set theory; Sacks [19] relies on Spector [24], who realized—in recursion theory—the necessity of more powerful approximations than the finite ones.To minimize the risk we will try to keep technicalities to a minimum, choosing to give priority to the methodology of forcing. We do not suppose any previous knowledge of forcing in the reader, but we do require some acquaintance with recursion theory. After all, our interest lies in the applications of the forcing method to the study of various recursion-theoretic notions of degrees. The farther we go, the deeper we plunge into recursion theory.In Part I only very basic notions and results are used, like the definitions of the arithmetical hierarchy and of the jump operator and their relationships.

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