Complexity of Model Checking for Modal Dependence Logic

Modal dependence logic (MDL) was introduced recently by V"a"an"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,…,p_n the atomic formula =(p_1,…,p_(n-1),p_n) intuitively states that the value of p_n is determined solely by those of p_1,…,p_(n-1). We show that model checking for MDL formulae over Kripke structures is NP-complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives. It turns out that several fragments, e.g., the one without modalities or the one without propositional connectives, remain NP-complete. We also consider the restriction of MDL where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the model checking problem for this bounded MDL is still NP-complete. We additionally extend MDL by allowing classical disjunction - introduced by Sevenster - besides dependence disjunction and show that classical disjunction is always at least as computationally bad as bounded arity dependence atoms and in some cases even worse, e.g., the fragment with nothing but the two disjunctions is NP-complete. Furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as bounded MDL.

💡 Research Summary

The paper conducts a thorough complexity analysis of the model‑checking problem for Modal Dependence Logic (MDL), a recent extension of basic modal logic that incorporates dependence atoms of the form = (p₁,…,pₙ₋₁,pₙ). A dependence atom asserts that the truth value of the last proposition is functionally determined by the values of the preceding ones, evaluated over a team (a set of worlds) in a Kripke structure. The authors first establish that MDL‑MC (the decision problem “given a Kripke structure, a team, and an MDL formula, does the formula hold in the team?”) belongs to NP. They present a nondeterministic top‑down algorithm that evaluates any MDL formula in polynomial time by recursively handling each connective: atomic propositions and their negations are checked pointwise; dependence atoms are verified by scanning all pairs of worlds in the team; the two disjunctions (dependence‑disjunction ∨ and classical disjunction >) are treated by guessing a partition or a choice, respectively; conjunction, box (□) and diamond (◇) are handled in the usual modal fashion; and the algorithm respects the restriction that negation may only appear in front of atomic formulas. This shows membership in NP for the unrestricted logic as well as for all its fragments.

To prove NP‑hardness, the authors give a polynomial‑time reduction from 3‑SAT. For a 3‑CNF formula ϕ = C₁∧…∧Cₘ over variables x₁,…,xₙ they construct a Kripke structure with one world per clause and no accessibility relation. Each world is labelled with propositional atoms rⱼ and pⱼ that encode whether variable xⱼ appears positively, negatively, or not at all in the corresponding clause. They then define the MDL formula ψ = ∧_{j=1}^{n}( rⱼ ∧ = (pⱼ) ). The team consists of all worlds. It is shown that ϕ is satisfiable iff ψ holds in the team. The reduction uses only the connectives ∧, ∨, and the dependence atom =, establishing NP‑completeness for any fragment that contains these three operators. Consequently, even when modal operators are completely omitted, the model‑checking problem remains NP‑hard.

The paper proceeds to a systematic classification of all fragments obtained by arbitrarily restricting the set of allowed operators from the full set {□, ◇, ∧, ∨, ¬, =, >, ⊥}. The results are summarised in Table 1. Key observations include:

• Removing both □ and ◇ does not lower the complexity; fragments with only ∧, ∨, = are already NP‑complete.
• Allowing only ◇ (no □) still yields NP‑hardness, whereas allowing only □ (no ◇) drops the problem to P, because □ can be evaluated deterministically on the whole team.
• The presence of classical disjunction > is at least as hard as dependence atoms; even the fragment consisting solely of ∨ and > (without any dependence atoms) is NP‑complete.
• Negation, being restricted to atomic formulas, does not affect the overall complexity classification.

Next, the authors study a bounded‑arity version MDLₖ, where each dependence atom may involve at most k propositional variables (k is a fixed constant for the whole logic). They prove that for any fixed k ≥ 0, the unrestricted MDLₖ‑MC remains NP‑complete, but certain fragments become tractable. In particular, when only ◇ is allowed (and k ≥ 1), the problem falls into P; otherwise, the presence of ∧ or ∨ together with ◇ restores NP‑hardness. These findings are collected in Table 2.

A further contribution is the incorporation of the classical disjunction operator > (introduced by Sevenster). The authors show that > is “computationally bad”: any fragment that contains > is at least as hard as the corresponding fragment without >, and in many cases strictly harder. The fragment with only the two disjunctions (∨ and >) is already NP‑complete, demonstrating that > alone can generate the full hardness of the logic. Conversely, fragments that contain only ∧ and □ (or only □) stay in P even when > is present, because the modal structure eliminates the nondeterministic splitting that > would otherwise introduce.

Finally, the paper nearly completes the complexity landscape for all possible operator restrictions. The only unresolved case is the fragment built from atomic propositions, dependence atoms, the ordinary disjunction ∨, and atomic negation ¬; for this fragment the authors could not yet establish NP‑hardness nor a polynomial‑time algorithm. All other combinations are classified as either NP‑complete or solvable in polynomial time, with precise conditions spelled out.

In summary, the study demonstrates that model checking for MDL is intrinsically hard: NP‑completeness persists under a wide variety of syntactic restrictions, and only very limited fragments (essentially those lacking any form of nondeterministic team splitting or modal branching) admit efficient algorithms. The work also clarifies the impact of bounded arity and classical disjunction on the complexity, providing a detailed map that can guide practitioners in choosing suitable fragments of MDL for applications where tractable verification is required.