A Short Tutorial on Order-Invariant First-Order Logic

Schweikardt, Nicole

doi:10.1007/978-3-642-38536-0_10

Cited by 12 publications

(11 citation statements)

References 22 publications

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“…The remainder of this subsection is devoted to the proof of Theorem 3.7. We follow the overall method of [1] for the case of disjoint neighbourhoods (see [16] for an overview) and make use of the connection between arb-inv-FO+MOD p and MOD p -circuits [3], along with a circuit lower bound by Smolensky [17].…”

Section: Shift Localitymentioning

confidence: 99%

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

Harwath

Schweikardt

2017

Logical Methods in Computer Science

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Section: Shift Localitymentioning

confidence: 99%

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

Harwath

Schweikardt

2017

Logical Methods in Computer Science

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“…As mentioned by [18], order-invariance on Fin σ is decidable if the signature σ contains only unary relation symbols. An ordered σ-structure in which the unary relations partition the universe can be regarded as a word.…”

Section: Undecidability Of Order-invariance On Structures Of Tree-depmentioning

confidence: 99%

Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Eickmeyer

Elberfeld

Harwath

2017

ACM Trans. Comput. Logic

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We study the expressive power and succinctness of order-invariant sentences of first-order (FO) and monadic second-order (MSO) logic on structures of bounded tree-depth. Orderinvariance is undecidable in general and, thus, one strives for logics with a decidable syntax that have the same expressive power as order-invariant sentences. We show that on structures of bounded tree-depth, order-invariant FO has the same expressive power as FO. Our proof technique allows for a fine-grained analysis of the succinctness of this translation. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. We obtain similar results for MSO. It is known that the expressive power of MSO and FO coincide on structures of bounded tree-depth. We provide a translation from MSO to FO and we show that this translation is essentially optimal regarding the formula size. As a further result, we show that order-invariant MSO has the same expressive power as FO with modulo-counting quantifiers on bounded tree-depth structures.

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“…The question of its efficiency is in the spirit of the eminent problem on the power of encoding-independent computations; see, e.g., [5]. If we extend the vocabulary with the order relation <, comparison of the two settings is interesting in the context of order-invariant definitions; see, e.g., [10,14]. Note also that the setting where arbitrary numerical relations are allowed brings us in the field of circuit complexity; see [8,10].…”

Section: Introductionmentioning

confidence: 99%

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Verbitsky

Zhukovskii

2019

ACM Trans. Comput. Logic

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Let v(F ) denote the number of vertices in a fixed connected pattern graph F . We show an infinite family of patterns F such that the existence of a subgraph isomorphic to F is expressible by a first-order sentence of quantifier depth 2 3 v(F ) + 1, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any F with using less than 2 3 v(F ) − 2 first-order variables.

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A Short Tutorial on Order-Invariant First-Order Logic

Abstract: Abstract. This paper gives a short introduction to order-invariant first-order logic and arb-invariant first-order logic. We present separating examples demonstrating the expressive power, as well as tools for proving certain expressive weaknesses of these logics.

Cited by 12 publications

References 22 publications

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

On the locality of arb-invariant first-order formulas with modulo counting quantifiers

Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

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