A convergence law for continuous logic and continuous structures with finite domains
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916054158016512 |
|---|---|
| author | Koponen, Vera |
| author_facet | Koponen, Vera |
| contents | We consider continuous relational structures with finite domain $[n] := \{1, \ldots, n\}$ and a many valued logic, $CLA$, with values in the unit interval and which uses continuous connectives and continuous aggregation functions. $CLA$ subsumes first-order logic on ``conventional'' finite structures. To each relation symbol $R$ and identity constraint $ic$ on a tuple the length of which matches the arity of $R$ we associate a continuous probability density function $μ_R^{ic} : [0, 1] \to [0, \infty)$.
We also consider a probability distribution on the set $\mathbf{W}_n$ of continuous structures with domain $[n]$ which is such that for every relation symbol $R$, identity constraint $ic$, and tuple $\bar{a}$ satisfying $ic$, the distribution of the value of $R(\bar{a})$ is given by $μ_R^{ic}$, independently of the values for other relation symbols or other tuples.
In this setting we prove that every formula in $CLA$ is asymptotically equivalent to a formula without any aggregation function. This is used to prove a convergence law for $CLA$ which reads as follows for formulas without free variables: If $φ\in CLA$ has no free variable and $I \subseteq [0, 1]$ is an interval, then there is $α\in [0, 1]$ such that, as $n$ tends to infinity, the probability that the value of $φ$ is in $I$ tends to $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A convergence law for continuous logic and continuous structures with finite domains Koponen, Vera Logic in Computer Science Artificial Intelligence Logic 03C13, 03C66, 68T27, 68T30, 68T37 F.4.1; G.3; I.2.4 We consider continuous relational structures with finite domain $[n] := \{1, \ldots, n\}$ and a many valued logic, $CLA$, with values in the unit interval and which uses continuous connectives and continuous aggregation functions. $CLA$ subsumes first-order logic on ``conventional'' finite structures. To each relation symbol $R$ and identity constraint $ic$ on a tuple the length of which matches the arity of $R$ we associate a continuous probability density function $μ_R^{ic} : [0, 1] \to [0, \infty)$. We also consider a probability distribution on the set $\mathbf{W}_n$ of continuous structures with domain $[n]$ which is such that for every relation symbol $R$, identity constraint $ic$, and tuple $\bar{a}$ satisfying $ic$, the distribution of the value of $R(\bar{a})$ is given by $μ_R^{ic}$, independently of the values for other relation symbols or other tuples. In this setting we prove that every formula in $CLA$ is asymptotically equivalent to a formula without any aggregation function. This is used to prove a convergence law for $CLA$ which reads as follows for formulas without free variables: If $φ\in CLA$ has no free variable and $I \subseteq [0, 1]$ is an interval, then there is $α\in [0, 1]$ such that, as $n$ tends to infinity, the probability that the value of $φ$ is in $I$ tends to $α$. |
| title | A convergence law for continuous logic and continuous structures with finite domains |
| topic | Logic in Computer Science Artificial Intelligence Logic 03C13, 03C66, 68T27, 68T30, 68T37 F.4.1; G.3; I.2.4 |
| url | https://arxiv.org/abs/2504.08923 |