[Paper Review] The Expressive Power of Epistemic $μ$-Calculus

Cătălin Dima, Bastien Maubert+1 more|arXiv (Cornell University)|Jul 19, 2014

Logic, Reasoning, and Knowledge3 citations

TL;DR

This paper establishes that the epistemic μ-calculus is expressively equivalent to jumping parity tree automata (JTA), and uses this equivalence to show that, under bounded-memory semantics, the epistemic μ-calculus is no more expressive than the standard μ-calculus and has Exptime-complete satisfiability. Crucially, it proves that ATL with imperfect information (ATL_i) is not subsumed by the epistemic μ-calculus under synchronous perfect-recall semantics, demonstrating a fundamental expressiveness gap.

ABSTRACT

While the $μ$-calculus notoriously subsumes Alternating-time Temporal Logic (ATL), we show that the epistemic $μ$-calculus does not subsume ATL with imperfect information (ATL$_i$) for the synchronous perfect-recall semantics. To prove this we first establish that jumping parity tree automata (JTA), a recently introduced extension of alternating parity tree automata, are expressively equivalent to the epistemic $μ$-calculus, and this for any knowledge semantics. Using this result we also show that, for bounded-memory semantics, the epistemic $μ$-calculus is not more expressive than the standard $μ$-calculus, and that its satisfiability problem is EXPTIME-complete.

Motivation & Objective

• To clarify the expressive power of the epistemic μ-calculus in comparison to strategic logics like ATL with imperfect information.
• To establish a formal equivalence between the epistemic μ-calculus and jumping parity tree automata (JTA), extending the classic μ-calculus–automata correspondence.
• To investigate the expressiveness and complexity of the epistemic μ-calculus under bounded-memory (recognizable) indistinguishability relations.
• To determine whether ATL with imperfect information can be expressed within the epistemic μ-calculus, particularly under synchronous perfect-recall semantics.
• To explore the relationship between epistemic μ-calculus, jumping automata, and monadic second-order logic enriched with the equal-level predicate.

Proposed method

• Prove that the epistemic μ-calculus is expressively equivalent to jumping parity tree automata (JTA) for any knowledge semantics, extending the classic μ-calculus–automata correspondence.
• Leverage the fact that JTA with recognizable relations translate in linear time into two-way tree automata to derive complexity results.
• Construct a game-theoretic argument using parity games and game bisimulations to compare winning strategies in automata and logical formulas.
• Use a counterexample formula ⟨⟨a⟩⟩Fp to demonstrate that a JTA accepting all models of this formula must also accept a model where Alice only has a non-uniform strategy, violating the uniformity requirement of ATL_i.
• Apply game bisimulation and color preservation arguments to show that a winning strategy in a related game for ATL_i can be lifted to a winning strategy in a JTA game, leading to a contradiction if the automaton were to accept all ATL_i models.
• Use the equivalence between JTA and epistemic μ-calculus to transfer results on expressiveness and complexity from automata to logic.

Experimental results

Research questions

• RQ1 Is the epistemic μ-calculus as expressive as ATL with imperfect information (ATL_i) under synchronous perfect-recall semantics?
• RQ2 Can jumping parity tree automata (JTA) express the same properties as the epistemic μ-calculus, and is this equivalence preserved across all knowledge semantics?
• RQ3 Does the epistemic μ-calculus have the same expressive power as the standard μ-calculus when restricted to bounded-memory semantics?
• RQ4 Is the satisfiability problem for the epistemic μ-calculus decidable under bounded-memory semantics, and what is its complexity?
• RQ5 Can the epistemic μ-calculus express properties of strategic abilities under imperfect information, such as uniform strategies in ATL_i?

Key findings

• The epistemic μ-calculus is expressively equivalent to jumping parity tree automata (JTA) for any knowledge semantics, extending the classical μ-calculus–automata correspondence.
• Under bounded-memory semantics (i.e., recognizable indistinguishability relations), the epistemic μ-calculus is not more expressive than the standard μ-calculus.
• The satisfiability problem for the epistemic μ-calculus is Exptime-complete under bounded-memory semantics.
• ATL with imperfect information (ATL_i) is not subsumed by the epistemic μ-calculus under synchronous perfect-recall semantics.
• A counterexample formula ⟨⟨a⟩⟩Fp demonstrates that any JTA accepting all models of this formula must also accept a model where the agent only has a non-uniform strategy, violating the uniformity constraint of ATL_i.
• The results suggest that MSO enriched with the equal-level predicate is strictly more expressive than the epistemic μ-calculus, as it can express any ATL_i formula but not all JTA languages.