On the relative asymptotic expressivity of inference frameworks
We consider logics with truth values in the unit interval $[0,1]$. Such logics are used to define queries and to define probability distributions. In this context the notion of almost sure equivalence of formulas is generalized to the notion of asymptotic equivalence. We prove two new results about...
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| Vydané v: | Logical methods in computer science Ročník 20, Issue 4; číslo 4; s. 13:1 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Logical Methods in Computer Science e.V
01.01.2024
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| Predmet: | |
| ISSN: | 1860-5974, 1860-5974 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We consider logics with truth values in the unit interval $[0,1]$. Such
logics are used to define queries and to define probability distributions. In
this context the notion of almost sure equivalence of formulas is generalized
to the notion of asymptotic equivalence. We prove two new results about the
asymptotic equivalence of formulas where each result has a convergence law as a
corollary. These results as well as several older results can be formulated as
results about the relative asymptotic expressivity of inference frameworks. An
inference framework $\mathbf{F}$ is a class of pairs $(\mathbb{P}, L)$, where
$\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$, $\mathbb{P}_n$ are
probability distributions on the set $\mathbf{W}_n$ of all $\sigma$-structures
with domain $\{1, \ldots, n\}$ (where $\sigma$ is a first-order signature) and
$L$ is a logic with truth values in the unit interval $[0, 1]$. An inference
framework $\mathbf{F}'$ is asymptotically at least as expressive as an
inference framework $\mathbf{F}$ if for every $(\mathbb{P}, L) \in \mathbf{F}$
there is $(\mathbb{P}', L') \in \mathbf{F}'$ such that $\mathbb{P}$ is
asymptotically total variation equivalent to $\mathbb{P}'$ and for every
$\varphi(\bar{x}) \in L$ there is $\varphi'(\bar{x}) \in L'$ such that
$\varphi'(\bar{x})$ is asymptotically equivalent to $\varphi(\bar{x})$ with
respect to $\mathbb{P}$. This relation is a preorder. If, in addition,
$\mathbf{F}$ is at least as expressive as $\mathbf{F}'$ then we say that
$\mathbf{F}$ and $\mathbf{F}'$ are asymptotically equally expressive. Our third
contribution is to systematize the new results of this paper and several
previous results in order to get a preorder on a number of inference systems
that are of relevance in the context of machine learning and artificial
intelligence. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-20(4:13)2024 |