Logics Capturing Local Properties
Leonid Libkin
- 01 Feb 2000
- pp 217-229
TL;DR: This paper gives logical characterizations of local properties behind Hanf's and Gaifman's theorems, and deals with an infinitary logic with counting terms and quantifiers, that is known to capture Hanflocality on structures of bounded valence.
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Abstract: Well-known theorems of Hanf's and Gaifman's establishing locality of first-order definable properties have been used in many applications. These theorems were recently generalized to other logics, which led to new applications in descriptive complexity and database theory. However, a logical characterization of local properties that correspond to Hanf's and Gaifman's theorems, is still lacking. Such a characterization only exists for structures of bounded valence.
In this paper, we give logical characterizations of local properties behind Hanf's and Gaifman's theorems. We first deal with an infinitary logic with counting terms and quantifiers, that is known to capture Hanflocality on structures of bounded valence. We show that testing isomorphism of neighborhoods can be added to it without violating Hanflocality, while increasing its expressive power. We then show that adding local second-order quantification to it captures precisely all Hanf-local properties. To capture Gaifman-locality, one must also add a (potentially infinite) case statement. We further show that the hierarchy based on the number of variants in the case statement is strict.
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Citations
Counting Bounded Tree Depth Homomorphisms
Martin Grohe
- 08 Jul 2020
TL;DR: In this paper, it was shown that G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at mostk.
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•Posted Content
Counting Bounded Tree Depth Homomorphisms
TL;DR: It is proved that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at mostk.
18
On locality and uniform reduction
H. Niemisto
- 26 Jun 2005
TL;DR: It is shown that all regular Hanf-local logics not tolerantlyHanf- local have weak version of uniform reduction for pairs and relationship between different forms of locality, regularity and uniform reduction is studied in the class of finite directed trees.
14
Logics Capturing Local Properties
Leonid Libkin
- 01 Feb 2000
TL;DR: This paper gives logical characterizations of local properties behind Hanf's and Gaifman's theorems, and deals with an infinitary logic with counting terms and quantifiers, that is known to capture Hanflocality on structures of bounded valence.
A logical approach to locality in pictures languages
Etienne Grandjean,Frédéric Olive +1 more
TL;DR: This paper generalizes to any dimension the characterization by Giammarresi et al. (1996) of the class of recognizable picture languages in existential monadic second-order logic, a robust complexity class that contains, for d = 1 , all the natural NP-complete problems.
11
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