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Bridging Weighted First Order Model Counting and Graph Polynomials

Authors Qipeng Kuang , Ondřej Kuželka , Yuanhong Wang , Yuyi Wang

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LIPIcs.CSL.2026.7.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • Weighted First-Order Model Counting
  • Axiom
  • Enumerative Combinatorics
  • Tutte Polynomial

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Abstract

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. It can be solved in time polynomial in the domain size for sentences from the two-variable fragment with counting quantifiers, known as C^2. This polynomial-time complexity is known to be retained when extending C^2 by one of the following axioms: linear order axiom, tree axiom, forest axiom, directed acyclic graph axiom or connectedness axiom. An interesting question remains as to which other axioms can be added to the first-order sentences in this way. We provide a new perspective on this problem by associating WFOMC with graph polynomials. Using WFOMC, we define Weak Connectedness Polynomial and Strong Connectedness Polynomials for first-order logic sentences. It turns out that these polynomials have the following interesting properties. First, they can be computed in polynomial time in the domain size for sentences from C^2. Second, we can use them to solve WFOMC with all of the existing axioms known to be tractable as well as with new ones such as bipartiteness, strong connectedness, having k connected components, etc. Third, the well-known Tutte polynomial can be recovered as a special case of the Weak Connectedness Polynomial, and the Strict and Non-Strict Directed Chromatic Polynomials can be recovered from the Strong Connectedness Polynomials.

Cite As Get BibTex

Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang. Bridging Weighted First Order Model Counting and Graph Polynomials. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.7

Author Details

Qipeng Kuang
  • The University of Hong Kong, China
Ondřej Kuželka
  • Czech Technical University in Prague, Czech Republic
Yuanhong Wang
  • Jilin University, Changchun City, China
Yuyi Wang
  • CRRC Zhuzhou Institute, China

Funding

  • Kuželka, Ondřej: Czech Science Foundation project 24-11820S ("Automatic Combinatorialist").
  • Wang, Yuanhong: National Natural Science Foundation of China (No. 62506141).
  • Wang, Yuyi: partially supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2024JJ5128).

Supplementary Materials

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