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Complexity of Łukasiewicz Modal Probabilistic Logics

Daniil Kozhemiachenko, Igor Sedlár

TL;DR

The paper investigates a many-valued modal probabilistic logic built on Łukasiewicz logic, introducing probabilistic frames and a expressive language with Pr(α) atoms and modal operators. It establishes PSPACE-completeness for local consequence in two regimes (full language on finitely branching frames and a fragment on arbitrary frames) using a constraint-tableau approach, while also arguing for the framework's capacity to model upper and lower probabilities and their comparisons. The work connects to existing probabilistic logics (Halpern-Pucella, Fagin-Halpern) and demonstrates that imprecise probabilistic reasoning can be captured within a decidable, computationally tractable setting. These results position many-valued modal probabilistic logics as a viable formalism for reasoning about probability in epistemic, temporal, and action contexts. The paper also outlines avenues for extending the results and refining expressivity and complexity analyses in future work.

Abstract

Modal probabilistic logics provide a framework for reasoning about probability in modal contexts, involving notions such as knowledge, belief, time, and action. In this paper, we study a particular family of these logics, extending the modal Łukasiewicz many-valued logic. These logics are shown to be capable of expressing nuanced probabilistic concepts, including upper and lower probabilities. Our main contribution is a PSPACE-completeness result for two variants of the local consequence problem, providing a precise computational characterisation.

Complexity of Łukasiewicz Modal Probabilistic Logics

TL;DR

The paper investigates a many-valued modal probabilistic logic built on Łukasiewicz logic, introducing probabilistic frames and a expressive language with Pr(α) atoms and modal operators. It establishes PSPACE-completeness for local consequence in two regimes (full language on finitely branching frames and a fragment on arbitrary frames) using a constraint-tableau approach, while also arguing for the framework's capacity to model upper and lower probabilities and their comparisons. The work connects to existing probabilistic logics (Halpern-Pucella, Fagin-Halpern) and demonstrates that imprecise probabilistic reasoning can be captured within a decidable, computationally tractable setting. These results position many-valued modal probabilistic logics as a viable formalism for reasoning about probability in epistemic, temporal, and action contexts. The paper also outlines avenues for extending the results and refining expressivity and complexity analyses in future work.

Abstract

Modal probabilistic logics provide a framework for reasoning about probability in modal contexts, involving notions such as knowledge, belief, time, and action. In this paper, we study a particular family of these logics, extending the modal Łukasiewicz many-valued logic. These logics are shown to be capable of expressing nuanced probabilistic concepts, including upper and lower probabilities. Our main contribution is a PSPACE-completeness result for two variants of the local consequence problem, providing a precise computational characterisation.

Paper Structure

This paper contains 6 sections, 8 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Probabilistic frames
  3. Modal many-valued probabilistic logic
  4. Expressivity
  5. Decidability and complexity
  6. Conclusion

Key Result

Theorem 1

Let $\phi\!\in\!\mathcal{L}_\mathsf{Pr}$. Then $\phi$ is ${\normalfont\textbf{K{\L}}}\bm{\Pi}\mathbf{\tfrac{1}{2}}^\mathsf{Pr}$-valid iff $\phi$ is ${\normalfont\textbf{K{\L}}}\bm{\Pi}\mathbf{\tfrac{1}{2}}^\mathsf{Pr}$-valid on SI-models.

Figures (2)

  • Figure 1: The robot in a warehouse example.
  • Figure 2: Tableaux rules: $j$ is a new variable; $w'$ is fresh on the branch and $u$ is present on the branch; $w:\psi=j$ is a shorthand for $\{w:\psi\geq j,w:\psi\leq j\}$.

Theorems & Definitions (30)

  • Definition 1: Probabilistic frames
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 1
  • Definition 2
  • Definition 3: Modal depth
  • Definition 4: Semantics of ${\normalfont\textbf{K{\L}}}\bm{\Pi}\mathbf{\tfrac{1}{2}}^\mathsf{Pr}$
  • Definition 5
  • ...and 20 more