Abstract
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.
H. Michalewski—Author supported by Poland’s National Science Centre grant no. 2012/07/D/ST6/02443
M. Mio—Author supported by grant “Projet Émergent PMSO” of the École Normale Supérieure de Lyon and Poland’s National Science Centre grant no. 2014-13/B/ST6/03595.
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Notes
- 1.
See [15] for an overview of Friedman’s research.
- 2.
Further open problems regarding \(\mathrm {MSO}+\forall ^{=1}_\pi \) are formulated in Sect. 8.
- 3.
As observed in [1, Remark 7.3], a proof of Theorem 2 can be derived from the decidability of a similar problem for finite probabilistic automata obtained by Gimbert and Oualhadj in [7, Theorem 4]. In [7, Proposition 2] the authors notice that the problem remains undecidable even if all probabilities appearing in the automaton belongs to \(\{0,\frac{1}{4},\frac{2}{4},\frac{3}{4},1\}\).
- 4.
In fact, following the work of [16], the logic pCTL* is also definable in a weaker logic such as Thomas’ chain logic extended with the quantifier \(\forall ^{=1}_\pi \).
- 5.
Result announced by Scott during a seminar entitled “Mixing Modality and Probability” given in Edinburgh, June 2010.
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Michalewski, H., Mio, M. (2016). Measure Quantifier in Monadic Second Order Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_19
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