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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| ISSN: | 1860-5974, 1860-5974 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 F is a class of pairs (P,L), where P=(Pn:n=1,2,3,…), Pn are probability distributions on the set Wn of all σ-structures with domain {1,…,n} (where σ is a first-order signature) and L is a logic with truth values in the unit interval [0,1]. An inference framework F′ is asymptotically at least as expressive as an inference framework F if for every (P,L)∈F there is (P′,L′)∈F′ such that P is asymptotically total variation equivalent to P′ and for every φ(x¯)∈L there is φ′(x¯)∈L′ such that φ′(x¯) is asymptotically equivalent to φ(x¯) with respect to P. This relation is a preorder. If, in addition, F is at least as expressive as F′ then we say that F and 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. 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. |
| Author | Weitkämper, Felix Koponen, Vera |
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| Snippet | 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... 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... 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... |
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| SubjectTerms | 03c13, 68t27, 68q87 Computer Science computer science - logic in computer science Datavetenskap f.4.1 finite model theory logical convergence laws logical expressivity Machine learning Maskininlärning Matematik Mathematics probabilistic graphical models probability logic scalable inference statistical relational artificial intelligence |
| Title | On the relative asymptotic expressivity of inference frameworks |
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