Department of

# Mathematics

Seminar Calendar
for Logic Seminar events the year of Monday, May 17, 2021.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 27, 2021

1:00 pm in Zoom (email Philipp Hieronymi),Tuesday, April 27, 2021

#### Decidability bounds for Presburger arithmetic extended by sine

###### Eion Blanchard (Illinois)

Abstract: Since 1929, when Presburger formalized his eponymous arithmetic $(\mathbb{Z}, <, +)$—integers with order and addition, but not multiplication—various extensions of this theory have been studied. Presburger arithmetic (PA) is decidable, as are some of its extensions, such as that by base-2 exponentiation. In joint work with Philipp Hieronymi, we extend PA by the sine function. Let sine-Presburger arithmetic ($\sin$-PA) be the theory of $(\mathbb{R}, <, +, \mathbb{Z}, \sin)$ with quantification only over integers. We show that the $\sin$-PA fragment with 4 alternating quantifier blocks is undecidable by encoding the halting problem. On the other hand, we present a decision procedure for the existential fragment of $\sin$-PA, under assumption of Schanuel’s Conjecture from transcendental number theory.