First-order spectra with one binary predicate

Durand, Arnaud; Ranaivoson, Solomampionona

doi:10.1016/0304-3975(95)00112-3

Cited by 9 publications

(6 citation statements)

References 14 publications

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“…For all i, β ∈ N, The relationships between spectra of i-ary functions and spectra of i + 1-ary relations can be made more precise. In [35], it is shown that a spectrum of a first-order formula involving any number of unary function symbols is also the spectrum of a formula using only one binary relation. This can be generalized to any arity.…”

Section: Beyond One Unary Function and Transfer Theoremsmentioning

confidence: 99%

Fifty years of the spectrum problem: survey and new results

Durand

Jones

Makowsky³

et al. 2012

Bull. symb. log

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Abstract. In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz' question was answered in various ways,

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Section: Beyond One Unary Function and Transfer Theoremsmentioning

confidence: 99%

Fifty years of the spectrum problem: survey and new results

Durand

Jones

Makowsky³

et al. 2012

Bull. symb. log

Self Cite

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“…Note that Corollary 1.2 strengthens the result by Fagin [8] which states that Asser's conjecture can be reduced to sentences (with arbitrary number of variables) over graphs. We also note the difference between Theorem 1.1 and the result by Durand and Ranaivoson [5] mentioned above. In [5], multiple unary functions are encoded using only one binary relation (with the graphs being restricted to those with bounded outdegree), whereas in Theorem 1.1, multiple binary relations are encoded with one binary relation (albeit with linear blowup in the size of the model).…”

Section: Introductionmentioning

confidence: 64%

“…They also showed that there is a sentence ϕ using two unary functions such that the language {1 n | n ∈ Spec(ϕ)} is NP-complete. That two unary functions are necessary to obtain an NP-complete language is shown immediately by Durand, Fagin and Loescher [5,3], where they show that the spectrum of a first-order sentence using only one unary function symbol is a semilinear set.…”

Section: Introductionmentioning

confidence: 94%

“…Durand and Ranaivoson [5] considered the class of spectra of sentences using only unary function symbols and proved that it is included in the class of spectra of sentences using only one binary relation. In particular, they established that the spectra of sentences using only unary function symbols are exactly the spectra of sentences using one binary relation when the models for the latter are restricted to directed graphs of bounded outdegree.…”

Section: Introductionmentioning

confidence: 99%

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A note on first-order spectra with binary relations

Kopczyński¹,

Tan²

2018

Logical Methods in Computer Science ; Volume 14

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The spectrum of a first-order sentence is the set of the cardinalities of its finite models. In this paper, we consider the spectra of sentences over binary relations that use at least three variables. We show that for every such sentence Φ, there is a sentence Φ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of Φ. Moreover, the models of Φ are all bipartite graphs. As a corollary, we obtain that to settle Asser's conjecture, i.e., whether the class of spectra is closed under complement, it is sufficient to consider only sentences using only three variables whose models are restricted to undirected bipartite graphs.

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“…Durand et al [DFL98] investigate the spectra of sentences in vocabularies with unary relations and one unary function and show that the class of such spectra is exactly the ultimately periodic (or semilinear) sets of natural numbers. On the other hand, if we allow two unary functions or one binary relation in the vocabulary, there are spectra that are NEXPTIME-complete [Fag74,DR96]. However, since the class of these spectra is not closed under polynomial-time reductions, it does not follow that it includes all of NEXPTIME or indeed even all of NE.…”

Section: Introductionmentioning

confidence: 99%

Bounded degree and planar spectra

Dawar¹,

Kopczyński²

2017

Logical Methods in Computer Science ; Volume 13

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The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra.

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First-order spectra with one binary predicate

Cited by 9 publications

References 14 publications

Fifty years of the spectrum problem: survey and new results

Fifty years of the spectrum problem: survey and new results

A note on first-order spectra with binary relations

Bounded degree and planar spectra

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