This paper is about an extension of monadic second-order logic over the full binary tree, which has a quantifier saying ``almost surely a branch { \pi} \in { 0, 1}^w satisfies a formula { \phi}({ \pi})''. This logic was introduced by Michalewski and Mio; we call it MSO+nabla following notation of Shelah and Lehmann. The logic MSO+nabla subsumes many qualitative probabilistic formalisms, including qualitative probabilistic CTL, probabilistic LTL, or parity tree automata with probabilistic acceptance conditions. We show that it is undecidable to check if a given sentence of MSO+nabla is true in the full binary tree.
Categories: cs.LO