Polynomial-time abelian groups

Cenzer, Douglas; Remmel, Jeffrey B.

doi:10.1016/0168-0072(92)90076-c

Cited by 45 publications

(6 citation statements)

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“…According to [12], a structure in a finite language is punctual if its domain is N, and its operations and relations are primitive recursive. This definition is similar to the notion of a primitive recursive structure suggested by Maltsev [1] and then used by Cenzer and Remmel [16]. The only subtle but important difference is that the domain has to be the whole of N and not merely a primitive recursive subset of N. This subtle strengthening of the definition led to an unexpectedly rich theory of punctual structures; we cite the survey [17] for a detailed exposition of the new emerging theory.…”

Section: Introductionmentioning

confidence: 88%

Punctual Categoricity Relative to a Computable Oracle

Kalimullin

Melnikov

2021

Lobachevskii J Math

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We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle $$f$$ means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with $$f$$. We will prove that the punctual categoricity relative to a computable oracle can hold only for finitely generated or locally finite structures. We will show that the punctual categoricity of finitely generated structures is exhaused by the computable oracles with primitive recursive graph. We also present an example of locally finite structure where the punctual categoricity is provided by a primitive recursively bounded computable oracle.

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Section: Introductionmentioning

confidence: 88%

Punctual Categoricity Relative to a Computable Oracle

Kalimullin

Melnikov

2021

Lobachevskii J Math

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“…Proposition 5. 5. There is a computable injection structure A, consisting of infinitely many Z-orbits, such that, for each e, there are infinitely many orbits which are Turing equivalent to W e .…”

Section: Injection Structuresmentioning

confidence: 99%

“…Let A = (A, f ) be the injection structure from Proposition 5. 5. Fixing any finite number of orbits, there are still two orbits of different degree so that the isomorphism between these two orbits cannot be extended to a computable automorphism.…”

Section: Injection Structuresmentioning

confidence: 99%

Computability and categoricity of weakly ultrahomogeneous structures

Adams

Cenzer

2017

COM

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This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is ∆ 0 2 categorical. A structure A is said to be weakly ultrahomogeneous if there is a finite (exceptional ) set of elements a1, . . . , an such that A becomes ultrahomogeneous when constants representing these elements are added to the language. Characterizations are obtained for weakly ultrahomogeneous linear orderings, equivalence structures, injection structures and trees, and these are compared with characterizations of the computably categorical and ∆ 0 2 categorical structures. Index sets are used to determine the complexity of the notions of ultrahomegenous and weakly ultrahomogeneous for various families of structures.

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“…Although a primitive recursive algorithm does not have to be computationally feasible, it serves as a useful abstraction which unites most common complexity classes of interest. In fact, as discussed in [2,22], very often eliminating unbounded search is the crucial step in turning a general Turing computable algebraic procedure into, say, a polynomial time or a polylogspace one; see, for example, [5,6,7,16]. A nontrivial illustration of this phenomenon is the recent solution [3] to a problem of Khouissainov and Nerode on the characterization of automatic structures ( [24], Question 4.9).…”

mentioning

confidence: 99%

“…Another useful role of primitive recursion is in proving that no feasible procedure is possible at all. Indeed, it is often easiest to argue that a primitive recursive procedure fails to exist, let alone a polynomial or exponential time procedure; see, for example, [5,7,22]. In such proofs one can typically diagonalize even against all total (Turing) computable procedures, that is, against those procedures which eventually halt [25,26].…”

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confidence: 99%

Punctual Categoricity and Universality

Downey¹,

Greenberg²,

Melnikov³

et al. 2020

J. symb. log.

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We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

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Polynomial-time abelian groups

Abstract: Annals of Pure and Applied Logic 56 (1992) 313-363. doi:10.1016/0168-0072(92)90076-CReceived by publisher: 1991-06-22Harvest Date: 2016-01-04 12:21:35DOI: 10.1016/0168-0072(92)90076-CPage Range: 313-36

Cited by 45 publications

References 2 publications

Punctual Categoricity Relative to a Computable Oracle

Punctual Categoricity Relative to a Computable Oracle

Computability and categoricity of weakly ultrahomogeneous structures

Punctual Categoricity and Universality

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