First order complexity of finite random structures
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916378364084224 |
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| author | Demin, Danila Zhukovskii, Maksim |
| author_facet | Demin, Danila Zhukovskii, Maksim |
| contents | For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $φ$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $φ$ validating 0--1 law is not recursively enumerable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First order complexity of finite random structures Demin, Danila Zhukovskii, Maksim Logic in Computer Science Discrete Mathematics Combinatorics Logic Probability For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $φ$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $φ$ validating 0--1 law is not recursively enumerable. |
| title | First order complexity of finite random structures |
| topic | Logic in Computer Science Discrete Mathematics Combinatorics Logic Probability |
| url | https://arxiv.org/abs/2402.02567 |