On Constructing Most General Solutions for Parametric Constraints (Extended Preprint)

2026-07-09

AI Summary

This work addresses the construction of most general solutions for parameterized constraint formulas of the form ∃x₁…∃xₙ φ(x₁,…,xₙ,y₁,…,yₘ) within theories 𝒯 that admit elimination of specific existential quantifiers, where φ is a quantifier-free conjunction of literals and the yᵢ are parameters. By introducing conditional function symbols that capture “if-then-else” constructs, the authors generalize existing results on the existence of most general unifiers in discriminator clusters. Integrating parameterized constraint solving with algebraic semantic characterizations, they establish a unified framework for constructing most general solutions. This approach substantially broadens the scope of applicability compared to prior methods, and its effectiveness and generality are demonstrated through illustrative examples.

Abstract

Let ${\cal T}$ be a theory allowing a form of elimination of existential quantifiers (possibly for formulae in a certain class). We analyze possibilities of constructing (most general) solutions w.r.t.\ ${\cal T}$ for formulae of the form $\exists x_1 \dots \exists x_n φ(x_1, \dots, x_n, y_1, \dots, y_m)$, where $φ$ is a quantifier-free conjunction of literals in the signature of ${\cal T}$, and the free variables $y_1, \dots, y_m$ are regarded as parameters. We show that in the presence of function symbols which describe ``{\sf if}-{\sf then}-{\sf else}'' constructions in certain models of ${\cal T}$, we can describe the most general solution of such formulae, thus generalizing results about the existence of most general unifiers in discriminator varieties. We illustrate the ideas on examples.

Problem

• parametric constraints
• most general solutions
• existential quantifiers
• quantifier elimination
• discriminator varieties

Innovation

• most general solution
• existential quantifier elimination
• parametric constraints
• if-then-else functions
• discriminator varieties