Quantifier elimination for counting extensions of Presburger arithmetic

Abstract

AbstractWe give a new quantifier elimination procedure for Presburger arithmetic extended with a unary counting quantifier $$\exists ^{= x} y\, \mathrm {\Phi }$$ ∃ = x y Φ that binds to the variable $$x$$ x the number of different $$y$$ y satisfying $$\mathrm {\Phi }$$ Φ . While our procedure runs in non-elementary time in general, we show that it yields nearly optimal elementary complexity results for expressive counting extensions of Presburger arithmetic, such as the threshold counting quantifier $$\exists ^{\ge c} y\, \mathrm {\Phi }$$ ∃ ≥ c y Φ that requires that the number of different y satisfying $$\mathrm {\Phi }$$ Φ be at least $$c\in \mathbb {N}$$ c ∈ N , where c can succinctly be defined by a Presburger formula. Our results are cast in terms of what we call the monadically-guarded fragment of Presburger arithmetic with unary counting quantifiers, for which we develop a 2ExpSpace decision procedure.