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Preprint Number 2154

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2154. Eion Blanchard, Philipp Hieronymi
Decidability bounds for Presburger arithmetic extended by sine

Submission date: 31 March 2022


We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (sin-PA), and systematically study decision problems for sets of sentences in sin-PA. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in sin-PA.

Mathematics Subject Classification: 03B25 (Primary) 03C62 (Secondary)

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Full text arXiv 2204.00099: pdf, ps.

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