A Short Note on Two-Variable Logic with a Linear Order Successor and a Preorder Successor

The finite satisfiability problem of two-variable logic extended by a linear order successor and a preorder successor is shown to be undecidable. The decidability of the finite satisfiability problem of two-variable logic extended by orders and preorders as well as their corresponding successor relations has been currently investigated by several papers [2, 5, 3]. This short note extends [3] which is work under submission. In the setting of [3] only two cases remained open. One of them, namely the extension of two-variable logic by one successor relation of a linear order and one successor relation of a preorder, is settled here. We refer to [2] and [5] for more background and motivation. In the rest of this note we first introduce the necessary notation and then prove that the finite satisfiability problem of the extension of two-variable logic by one successor relation of a linear order and one successor relation of a preorder is undecidable.