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Asymptotic truth-value laws in many-valued logics

Guillermo Badia, Xavier Caicedo, Carles Noguera

TL;DR

The paper addresses which truth-values arise asymptotically for sentences in many-valued predicate logics on finite structures, extending Fagin's zero-one law to finite lattice-valued algebras and to certain infinitely valued logics, notably Łukasiewicz logic. It develops a general translation to classical logic and constructs countable random A-valued models via extension axioms, enabling transfer of zero-one phenomena and analysis of almost-sure values, including PSPACE-completeness results. The authors characterize the sets of almost-sure values for various finite algebras (MV- and Gödel-type chains, De Morgan algebras) and prove zero-one laws for infinite-valued Łukasiewicz logic and related De Morgan-satisfying logics, with further results describing how these values can vary across logics. The work unifies finite and infinite settings, clarifies independence from atomic distributions, and raises open questions on irrational almost-sure values and algebraic characterizations of the almost-sure set.

Abstract

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. We obtain generalizations of Fagin's classical zero-one law for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp's generalization of Fagin's result. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete, and for some logics we may describe completely the set of truth-values that can be taken by sentences almost surely.

Asymptotic truth-value laws in many-valued logics

TL;DR

The paper addresses which truth-values arise asymptotically for sentences in many-valued predicate logics on finite structures, extending Fagin's zero-one law to finite lattice-valued algebras and to certain infinitely valued logics, notably Łukasiewicz logic. It develops a general translation to classical logic and constructs countable random A-valued models via extension axioms, enabling transfer of zero-one phenomena and analysis of almost-sure values, including PSPACE-completeness results. The authors characterize the sets of almost-sure values for various finite algebras (MV- and Gödel-type chains, De Morgan algebras) and prove zero-one laws for infinite-valued Łukasiewicz logic and related De Morgan-satisfying logics, with further results describing how these values can vary across logics. The work unifies finite and infinite settings, clarifies independence from atomic distributions, and raises open questions on irrational almost-sure values and algebraic characterizations of the almost-sure set.

Abstract

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. We obtain generalizations of Fagin's classical zero-one law for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp's generalization of Fagin's result. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete, and for some logics we may describe completely the set of truth-values that can be taken by sentences almost surely.
Paper Structure (10 sections, 18 theorems, 24 equations)

This paper contains 10 sections, 18 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Zero-one laws in two-valued predicate logic
  3. The case of finitely valued predicate logics
  4. The zero-one law for finitely valued predicate logics
  5. Finitely valued random models and extension axioms
  6. Complexity of almost sure values
  7. The set of almost sure values in finitely valued predicate logics
  8. A zero-one law for infinitely valued Ł ukasiewicz predicate logic
  9. Further results for infinitely valued predicate logics
  10. Final remarks

Key Result

Theorem 1

For any sentence $\varphi \in \mathcal{L}^{\boldsymbol A}(\tau )$ and $a\in A$, $\lim_{n}\mu _{n}(\varphi =a)=1$ or $\lim_{n}\mu _{n}(\varphi =a)=0$. Equivalently, for any sentence $\varphi \in \mathcal{L}^{\boldsymbol A}(\tau )$, there is a (unique) $a\in A$ such that $\lim_{n}\mu _{n}(\varphi =a)=

Theorems & Definitions (54)

  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof
  • Example 6
  • Corollary 7
  • Remark 8
  • ...and 44 more