Quantifier elimination for counting extensions of Presburger arithmetic
We give a new quantifier elimination procedure for Presburger arithmetic extended with a unary counting quantifier ∃=xyΦ that binds to the variable x the number of different y satisfying Φ. While our procedure runs in non-elementary time in general, we show that it yields nearly optimal elementary c...
| Príomhchruthaitheoirí: | , , |
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| Formáid: | Conference item |
| Teanga: | English |
| Foilsithe / Cruthaithe: |
Springer
2022
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| Achoimre: | We give a new quantifier elimination procedure for Presburger arithmetic extended with a unary counting quantifier ∃=xyΦ that binds to the variable x the number of different y satisfying 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 ∃≥cyΦ that requires that the number of different y satisfying Φ be at least 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. |
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