Abstract
We give a new quantifier elimination procedure for Presburger arithmetic extended with a unary counting quantifier \(\exists ^{= x} y\, \mathrm {\Phi }\) that binds to the variable \(x\) the number of different \(y\) satisfying \(\mathrm {\Phi }\). 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 \(\exists ^{\ge c} y\, \mathrm {\Phi }\) that requires that the number of different y satisfying \(\mathrm {\Phi }\) be at least \(c\in \mathbb {N}\), 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.
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Chistikov, D., Haase, C., Mansutti, A. (2022). Quantifier elimination for counting extensions of Presburger arithmetic. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_12
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DOI: https://doi.org/10.1007/978-3-030-99253-8_12
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