On the Strength of Uniqueness Quantification in Primitive Positive Formulas

Victor Lagerkvist; Gustav Nordh

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arXiv · 1906.07031

On the Strength of Uniqueness Quantification in Primitive Positive Formulas

Victor Lagerkvist·Gustav Nordh

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Abstract

Uniqueness quantification ($\exists !$) is a quantifier in first-order logic where one requires that exactly one element exists satisfying a given property. In this paper we investigate the strength of uniqueness quantification when it is used in place of existential quantification in conjunctive formulas over a given set of relations $\Gamma$, so-called primitive positive definitions (pp-definitions). We fully classify the Boolean sets of relations where uniqueness quantification has the same strength as existential quantification in pp-definitions and give several results valid for arbitrary finite domains. We also consider applications of $\exists !$-quantified pp-definitions in computer science, which can be used to study the computational complexity of problems where the number of solutions is important. Using our classification we give a new and simplified proof of the trichotomy theorem for the unique satisfiability problem, and prove a general result for the unique constraint satisfaction problem.

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Victor Lagerkvist, Gustav Nordh. 2019-06-17. On the Strength of Uniqueness Quantification in Primitive Positive Formulas. https://arxiv.org/abs/1906.07031

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