[Paper Review] The Expressive Power of Modal Dependence Logic

Lauri Hella, Kerkko Luosto+2 more|arXiv (Cornell University)|Jun 24, 2014

Logic, Reasoning, and Knowledge13 references22 citations

TL;DR

This paper establishes that the expressive power of extended modal dependence logic (EMDL) and modal logic with intuitionistic disjunction (ML(∨)) are equivalent, characterizing definable team properties as those that are downward closed and closed under team k-bisimulation for some finite k. It further proves that any translation from EMDL to ML(∨) incurs an exponential blow-up in formula size, resolving an open problem in the field.

ABSTRACT

We study the expressive power of various modal logics with team semantics. We show that exactly the properties of teams that are downward closed and closed under team k-bisimulation, for some finite k, are definable in modal logic extended with intuitionistic disjunction. Furthermore, we show that the expressive power of modal logic with intuitionistic disjunction and extended modal dependence logic coincide. Finally we establish that any translation from extended modal dependence logic into modal logic with intuitionistic disjunction increases the size of some formulas exponentially.

Motivation & Objective

• To characterize the expressive power of modal logics with team semantics, particularly ML(∨) and EMDL.
• To resolve the open question of whether EMDL is strictly more expressive than ML(∨), or equivalent in expressive power.
• To introduce and formalize team bisimulation as a generalization of k-bisimulation for team semantics.
• To establish semantical invariants—upper and lower dimensions—for formulas in EMDL and ML(∨) to analyze translation complexity.
• To prove that translations from EMDL to ML(∨) require exponential size increase in the worst case.

Proposed method

• Introduce team bisimulation as a canonical extension of k-bisimulation to team semantics, capturing invariance under team-based structure preservation.
• Define and analyze the upper and lower dimensions of formulas as semantical invariants to measure the complexity of team properties.
• Use induction on formula structure to bound the dimension of ML(∨) formulas in terms of the number of intuitionistic disjunctions (∨) they contain.
• Prove that the dimension of a dependence atom = (p₁,…,pₙ,q) is 2^(2ⁿ), which establishes a lower bound on the number of ∨ symbols required in any equivalent ML(∨) formula.
• Leverage the dimension bounds to show that any formula in EMDL equivalent to a dependence atom requires at least 2ⁿ intuitionistic disjunctions in ML(∨), implying exponential blow-up.
• Establish the equivalence of EMDL and ML(∨) by showing that both logics define exactly the same class of downward closed, team k-bisimulation-closed properties.

Experimental results

Research questions

• RQ1 Is the expressive power of extended modal dependence logic (EMDL) strictly greater than that of modal logic with intuitionistic disjunction (ML(∨))?
• RQ2 Can the expressive power of ML(∨) be fully characterized using team semantics and bisimulation-like invariants?
• RQ3 What is the minimal size increase when translating EMDL formulas into equivalent ML(∨) formulas?
• RQ4 Do semantical invariants such as upper and lower dimension provide a precise measure of formula complexity in team semantics?
• RQ5 Can team bisimulation serve as a canonical generalization of k-bisimulation for characterizing definable team properties in modal logics?

Key findings

• The expressive power of EMDL and ML(∨) are equivalent: both define exactly the team properties that are downward closed and closed under team k-bisimulation for some finite k.
• The dimension of a dependence atom = (p₁,…,pₙ,q) is 2^(2ⁿ), which is the minimal dimension required to express functional dependence over n propositional variables.
• Any formula in ML(∨) expressing the dependence atom = (p₁,…,pₙ,q) must contain at least 2ⁿ intuitionistic disjunction symbols, implying a worst-case exponential blow-up.
• The dimension of any ML(∨) formula φ is bounded by 2^(occ_∨(φ)), where occ_∨(φ) is the number of ∨ symbols in φ, establishing a tight upper bound on formula complexity.
• The characterization of definable properties via downward closure and team k-bisimulation closure provides a complete and canonical semantics for EMDL and ML(∨).
• The results imply that EMDL can be seen as a canonical extension of modal logic for expressing dependence, with expressive power fully captured by team-based invariance.