[Paper Review] Monadic Second Order Logic with Measure and Category Quantifiers

Matteo Mio, Michał Skrzypczak+1 more|arXiv (Cornell University)|Feb 15, 2017

Advanced Algebra and Logic31 references3 citations

TL;DR

This paper investigates extensions of Monadic Second-Order logic (MSO) over infinite words and trees with second-order quantifiers for 'almost all' subsets, interpreted via Baire category (forall^*) and Lebesgue measure (forall^{=1}). It proves that MSO with the category quantifier is decidable (S1S + forall^* = S1S), while MSO with the measure quantifier is undecidable (S1S + forall^{=1} is undecidable), and establishes connections to probabilistic automata and definability in S2S.

ABSTRACT

We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized "for almost all" quantifiers interpreted using the notions of Baire category and Lebesgue measure.

Motivation & Objective

• To study the expressive and logical properties of Monadic Second-Order logic (MSO) extended with second-order quantifiers for 'almost all' subsets, interpreted via Baire category and Lebesgue measure.
• To determine the decidability of the resulting logics over the structures (N,<) and the full binary tree.
• To clarify the relationship between MSO with measure/category quantifiers and automata-theoretic interpretations, particularly probabilistic tree automata.
• To resolve open problems regarding definability and complexity in extensions of S2S with these quantifiers.

Proposed method

• The paper uses automata-theoretic techniques and game-theoretic reasoning, particularly Banach–Mazur games, to analyze the behavior of quantifiers over Cantor space 2^ω.
• It applies results from probabilistic automata theory to prove the undecidability of S1S + forall^{=1}, leveraging connections to Bojańczyk’s undecidable logic S1S + U.
• For the category quantifier, it provides an effective quantifier elimination procedure for S1S + forall^*, showing equivalence to plain S1S.
• It introduces a new logic S2S + forall^{=1}_π, where π ranges over paths, and proves its equivalence to S2S + U^{=1} via automata-based interpretations.
• It analyzes the definability of languages via probabilistic tree automata and compares them to S2S + forall^{=1}_π definable languages.
• It identifies limitations of probabilistic automata by showing that S2S + forall^{=1}_π can define non-regular languages that are not captured by such automata.

Experimental results

Research questions

• RQ1 Is the theory of S2S + forall^* decidable, despite a flawed proof in prior work?
• RQ2 Does S2S + forall^{=1}_π define only sets in the Δ¹₂ class of the projective hierarchy?
• RQ3 Can every S2S + forall^{=1}_π definable language be captured by a probabilistic tree automaton?
• RQ4 Is the theory of S2S + forall^{=1} decidable?
• RQ5 Can an algorithm be designed to compute the measure of an S2S-definable set?

Key findings

• The logic S1S + forall^* is decidable and equivalent to plain S1S, as the forall^* quantifier can be effectively eliminated.
• The logic S1S + forall^{=1} is undecidable, proven via an interpretation of Bojańczyk’s undecidable logic S1S + U.
• The theory S2S + forall^{=1} is undecidable, as it interprets S1S + forall^{=1} via a translation from S1S to S2S.
• The logic S2S + forall^{=1}_π is equivalent to S2S + U^{=1}, establishing a connection between path-based measure quantifiers and unbounding quantifiers.
• There exist S2S + forall^{=1}_π definable languages that are not definable by probabilistic tree automata, showing a strict increase in expressive power.
• The class of S2S + forall^{=1}_π definable sets is not necessarily contained in the Δ¹₂ class, though all known examples are.