On the Complexity of the Numerically Definite Syllogistic and Related Fragments

TL;DR

This work investigates the satisfiability and finite satisfiability of logics extending the syllogistic with counting quantifiers, distinguishing two main tiers: $\mathcal{N}^1$–$\mathcal{C}^1$ and $\mathcal{N}^2$–$\mathcal{C}^2$. It establishes that all logics between $\mathcal{N}^1$ and $\mathcal{C}^1$ are strongly NP-complete for satisfiability, while all logics between $\mathcal{N}^2$ and $\mathcal{C}^2$ are NEXPTIME-complete for both satisfiability and finite satisfiability; the results rely on reductions from $3$-coloring and tiling problems, and on finite-model arguments for the fragments involved. The paper also analyzes probabilistic satisfiability (PSAT) to illustrate incompleteness in several proposed syllogistic proof-systems, proving that no complete deductive system exists for the numerically definite syllogistic and its relatives under these frameworks. Overall, the results delineate sharp complexity boundaries for these counting-augmented syllogistics and highlight fundamental limits of syllogistic proof theory in this domain.

Abstract

In this paper, we determine the complexity of the satisfiability problem for various logics obtained by adding numerical quantifiers, and other constructions, to the traditional syllogistic. In addition, we demonstrate the incompleteness of some recently proposed proof-systems for these logics.

Figures (1)

• Figure 1: Models of $\Phi$: (a) standard semantics; (b) probabilistic semantics. In (b), one of the sets $R_j = R' \cap T_j$ is empty.

Theorems & Definitions (26)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1
• Lemma 3
• proof
• Proposition 1
• proof
• Lemma 4