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A general approach to asymptotic elimination of aggregation functions and generalized quantifiers

Vera Koponen, Felix Weitkämper

TL;DR

This work develops a general framework, $PLA^*$, for logics with truth values in $[0,1]$ that use aggregation functions in place of quantifiers. It shows generalized Mostowski quantifiers can be modeled by aggregation functions and introduces two local continuity notions, ct-continuity and up-continuity, to enable asymptotic elimination of aggregation functions. The main theorem proves that, under suitable assumptions, any $PLA^*$ formula is asymptotically equivalent to an $L_0$-basic formula, effectively reducing complex formulas to a 0/1-valued base fragment on large finite structures. This provides a modular, broadly applicable approach to quantifier elimination in probabilistic/logical settings and clarifies when and how aggregation-driven formulas converge to simpler descriptions. The results unify and extend previous work (KW1, KW2) and underpin potential future applications to other logics with aggregation constructs.

Abstract

We consider a logic with truth values in the unit interval and which uses aggregation functions instead of quantifiers, and we describe a general approach to asymptotic elimination of aggregation functions and, indirectly, of asymptotic elimination of Mostowski style generalized quantifiers, since such can be expressed by using aggregation functions. The notion of ``local continuity'' of an aggregation function, which we make precise in two (related) ways, plays a central role in this approach.

A general approach to asymptotic elimination of aggregation functions and generalized quantifiers

TL;DR

This work develops a general framework, , for logics with truth values in that use aggregation functions in place of quantifiers. It shows generalized Mostowski quantifiers can be modeled by aggregation functions and introduces two local continuity notions, ct-continuity and up-continuity, to enable asymptotic elimination of aggregation functions. The main theorem proves that, under suitable assumptions, any formula is asymptotically equivalent to an -basic formula, effectively reducing complex formulas to a 0/1-valued base fragment on large finite structures. This provides a modular, broadly applicable approach to quantifier elimination in probabilistic/logical settings and clarifies when and how aggregation-driven formulas converge to simpler descriptions. The results unify and extend previous work (KW1, KW2) and underpin potential future applications to other logics with aggregation constructs.

Abstract

We consider a logic with truth values in the unit interval and which uses aggregation functions instead of quantifiers, and we describe a general approach to asymptotic elimination of aggregation functions and, indirectly, of asymptotic elimination of Mostowski style generalized quantifiers, since such can be expressed by using aggregation functions. The notion of ``local continuity'' of an aggregation function, which we make precise in two (related) ways, plays a central role in this approach.
Paper Structure (9 sections, 10 theorems, 44 equations)

This paper contains 9 sections, 10 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Logic with aggregation functions
  3. Expressing generalized quantifiers with aggregation functions
  4. Notions of continuity for aggregation functions
  5. Convergence test continuity
  6. Uniform point continuity
  7. Asymptotic equivalence to basic formulas
  8. Proof of Theorem \ref{['main result']}
  9. Conclusion

Key Result

Proposition 3.3

Let $\varphi(\bar{x}) \in FOGQ(\sigma)$. Then there is $\psi(\bar{x}) \in PLA^*(\sigma)$ such that for every finite $\sigma$-structure $\mathcal{A}$ and every $\bar{a} \in A^{|\bar{a}|}$, $\mathcal{A} \models \varphi(\bar{a})$ if and only if $\mathcal{A}(\psi(\bar{a})) = 1$, and $\mathcal{A} \not\mo

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Remark 2.10
  • Definition 3.1
  • ...and 28 more