Measure Quantifier in Monadic Second Order Logic

We study the extension of Monadic Second Order logic with the “for almost all” quantifier \(\forall ^{=1}\) whose meaning is, informally, that \(\forall ^{=1}X.\phi (X)\) holds if \(\phi (X)\) holds almost surely for a randomly chosen X. We prove that the theory of \(\mathrm {MSO}+\forall ^{=1}\) is undecidable both when interpreted on \((\omega ,<)\) and the full binary tree. We then identify a fragment of \(\mathrm {MSO}+\forall ^{=1}\), denoted by \(\mathrm {MSO}+\forall ^{=1}_\pi \), and reduce some interesting problems in computer science and mathematical logic to the decision problem of \(\mathrm {MSO}+ \forall ^{=1}_\pi \). The question of whether \(\mathrm {MSO}+\forall ^{=1}_\pi \) is decidable is left open.