# Hereditariness in the finite and prefix classes of first order logic

- 11:00 7th November 2019 ( week 4, Michaelmas Term 2019 )LTB Wolfston building

In a recent paper [2], a strengthening of the above failure was shown: in the finite, for every k, there is a hereditary FO sentence that is not equivalent to any \Sigma^0_2 sentence that contains k existential quantifiers. We generalize this result by showing in the finite, that for every n, there is a hereditary FO sentence \varphi_n that is not equivalent to any \Sigma^0_n sentence. In short, (FO-)hereditariness over all finite structures is not capturable by any prefix class of FO. We show further that \neg \varphi_n is expressible in Datalog(\neq, \neg), thus answering in the negative an open question posed by Rosen and Weinstein [1], that asks whether FO \cap Datalog(\neq, \neg) is contained (semantically) inside some prefix class of FO. The methods used to show our result extend the ideas from [2] in a significant way, and involve the construction of structures that exhibit self-similarity.

This is joint work with Anuj Dawar.

[1] Eric Rosen and Scott Weinstein. Preservation theorems in finite model theory. In International Workshop on Logic and Computational Complexity, pages 480–502. Springer, 1994.

[2] Abhisekh Sankaran. Revisiting the generalized Los-Tarski theorem. In Logic and Its Applications - 8th Indian Conference, ICLA 2019, Delhi, India, March 1-5, 2019, Proceedings, pages 76–88, 2019.