Preservation and decomposition theorems for bounded degree structures

Harwath, Frederik; Heimberg, Lucas; Schweikardt, Nicole

doi:10.1145/2603088.2603130

Cited by 7 publications

(12 citation statements)

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“…Notably, the classes of structures that satisfy our property are incomparable to those studied in [7,21,38].…”

Section: The Above Results Motivates Us To Ask the Following Question:mentioning

confidence: 93%

“…For a vocabulary τ , let τ disj-un,2 be the vocabulary ob-tained by expanding τ with 2 fresh unary predicates P 1 and P 2 . Given structures A 1 and A 2 whose cartesian product we intend to take, we first construct the 2-disjoint sum [38] of A 1 and A 2 , denoted A 1 ⊕ A 2 , which is the τ disj-un,2 -structure obtained upto isomorphism, by expanding the disjoint union A 1 ⊔ A 2 with P 1 and P 2 interpreted respectively as the universes of the isomorphic copies of A 1 and A 2 that are used in constructing A 1 ⊔ A 2 . The cartesian product A 1 ⊗ A 2 is then the structure Ξ(A 1 ⊕ A 2 ) where Ξ is the (2, τ disj-un,2 , τ, FO)-translation scheme given by Ξ = (ξ, (ξ R ) R∈τ ) where ξ(x, y) = (P 1 (x) ∧ P 2 (y)) and for R ∈ τ of arity r, we have ξ R (x 1 , y 1 , . .…”

Section: Closure Of L-ebsp(· ·) Under Operations Implemented Using Tmentioning

confidence: 99%

“…To "recover" classical preservation theorems in the finite model theory setting, recent research [6,7,15,16,21,38] in the last ten years, has focussed attention on studying these theorems over "well-behaved" classes of finite structures. In particular, Atserias, Dawar and Grohe showed in [7] that under suitable closure assumptions, classes of structures that are acyclic or of bounded degree admit the Łoś-Tarski theorem for sentences.…”

mentioning

confidence: 99%

“…(Note that this theorem being true over all finite structures does not imply that it would be true over subclasses of finite structures; restricting attention to a subclass weakens both the hypothesis and the consequent of the statement of the theorem). Subsequently, Harwath, Heimberg and Schweikardt [38] studied the bounds for an effective version of the Łoś-Tarski theorem and the homomorphism preservation theorem over bounded degree structures. In [21], Duris showed that the Łoś-Tarski theorem holds for structures that are acyclic in a more general sense.…”

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confidence: 99%

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A generalization of the Łoś–Tarski preservation theorem

Sankaran

Adsul

Chakraborty

2016

Annals of Pure and Applied Logic

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Chairman Date:Place: DeclarationI declare that this written submission represents my ideas in my own words and where others' ideas or words have been included, I have adequately cited and referenced the original sources.I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsified any idea/data/fact/source in my submission. I understand that any violation of the above will be cause for disciplinary action by the Institute and can also evoke penal action from the sources which have thus not been properly cited or from whom proper permission has not been taken when needed.(Signature of student) and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Łoś-Tarski theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Łoś-Tarski theorem for sentences at level k, abbreviated GLT(k). To the best of our knowledge, GLT(k) is the first to relate counts of quantifiers appearing in the sentences of the Σ In summary, the properties introduced in this thesis are interesting in both the classical and finite model theory contexts, and yield in both these contexts, a new parameterized generalization of the Łoś-Tarski preservation theorem.viii

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“…Notably, the classes of structures that satisfy our property are incomparable to those studied in [7,21,38].…”

Section: The Above Results Motivates Us To Ask the Following Question:mentioning

confidence: 93%

Section: Closure Of L-ebsp(· ·) Under Operations Implemented Using Tmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A generalization of the Łoś–Tarski preservation theorem

Sankaran

Adsul

Chakraborty

2016

Annals of Pure and Applied Logic

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“…This runtime is thus nonelementary in the size of ϕ, and cannot be improved in general, owing to a non-elementary lower bound for the size of the decomposition over all finite structures (and hence also arbitrary structures) [2]. The time complexity can however be improved by considering special classes of finite structures, such as those of bounded degree, where it takes at most 3-fold exponential time to compute the decomposition if the degree is at least 3, and 2-fold exponential time if the degree is at most 2 [14].…”

Section: Introductionmentioning

confidence: 99%

Feferman-Vaught Decompositions for Prefix Classes of First Order Logic

Sankaran¹

2021

Preprint

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The Feferman-Vaught theorem provides a way of evaluating a first order sentence ϕ on a disjoint union of structures by producing a decomposition of ϕ into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be nonelementarily larger than ϕ. We show that for first order sentences in prenex normal form with a fixed number of quantifier alternations, such a decomposition, further with the same number of quantifier alternations, can be obtained in time elementary in the size of ϕ. We obtain this result as a consequence of a more general decomposition theorem that we prove for a family of infinitary logics we define. We extend these results by considering binary operations other than disjoint union, in particular sumlike operations such as ordered sum and NLC-sum, that are definable using quantifierfree interpretations.

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A Generalization of the Łoś-Tarski Preservation Theorem over Classes of Finite Structures

Sankaran

Adsul

Chakraborty

2014

Mathematical Foundations of Computer Science 2014

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We investigate a generalization of the Loś-Tarski preservation theorem via the semantic notion of preservation under substructures modulo k-sized cores. It was shown earlier that over arbitrary structures, this semantic notion for first-order logic corresponds to definability by ∃ k ∀ * sentences. In this paper, we identify two properties of classes of finite structures that ensure the above correspondence. The first is based on well-quasi-ordering under the embedding relation. The second is a logic-based combinatorial property that strictly generalizes the first. We show that starting with classes satisfying any of these properties, the classes obtained by applying operations like disjoint union, cartesian and tensor products, or by forming words and trees over the classes, inherit the same property. As a fallout, we obtain interesting classes of structures over which an effective version of the Loś-Tarski theorem holds.

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Preservation and decomposition theorems for bounded degree structures

Cited by 7 publications

References 18 publications

A generalization of the Łoś–Tarski preservation theorem

A generalization of the Łoś–Tarski preservation theorem

Feferman-Vaught Decompositions for Prefix Classes of First Order Logic

A Generalization of the Łoś-Tarski Preservation Theorem over Classes of Finite Structures

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