MSO+nabla is undecidable
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This paper is about an extension of monadic second-order logic over infinite trees, which adds a quantifier that says "the set of branches π which satisfy a formula ϕ(π) has probability one". This logic was introduced by Michalewski and Mio; we call it MSO+nabla following 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 consider the decision problem: decide if a sentence of MSO+nabla is true in the infinite binary tree? For sentences from the weak variant of this logic (set quantifiers range only over finite sets) the problem was known to be decidable, but the question for the full logic remained open. In this paper we show that the problem for the full logic MSO+nabla is undecidable.
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