Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by V"a"an"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,…,p_n, =(p_1,…,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,…,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend V"a"an"anen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by V"a"an"anen and Sevenster.

💡 Research Summary

Modal Dependence Logic (MDL) was introduced by Väänänen as an extension of basic modal logic with a new atomic operator “=()”. For propositional variables (p_1,\dots ,p_n) the atom (=(p_1,\dots ,p_{n-1};p_n)) asserts that the truth value of (p_n) is functionally determined by the values of (p_1,\dots ,p_{n-1}). MDL is interpreted under team semantics, i.e., formulas are evaluated over sets of worlds rather than a single world. This global perspective makes the satisfiability problem considerably harder than for ordinary modal logic; Sevenster (2009) proved that full MDL‑SAT is NEXPTIME‑complete.

The present paper undertakes a systematic classification of the complexity of MDL‑SAT for a variety of syntactic fragments obtained by restricting the set of propositional connectives that may be used together with dependence atoms. The authors consider six main fragments, each corresponding to a different combination of the following operators: literals, dependence atoms, conjunction (∧), disjunction (∨), classical disjunction (⊕), negation (¬), and the modal operators □ (necessity) and ◇ (possibility). For each fragment they establish both lower and upper bounds, thereby obtaining exact completeness results.

1. Poor‑Man’s Dependence Logic (PMDL) – formulas built from literals, dependence atoms, ∧, □, ◇, but without any form of disjunction. Despite the apparent restriction, the authors show that PMDL‑SAT remains NEXPTIME‑complete. The hardness proof reduces the exponential‑tiling problem to PMDL by encoding the tiling grid into a Kripke structure and using dependence atoms to enforce the functional constraints that mimic the tiling rules. The upper bound follows from Sevenster’s original NEXPTIME algorithm, which can be applied unchanged because the fragment is a syntactic subset of full MDL.

2. Monotone Fragment (MF) – formulas may contain literals, dependence atoms, only disjunction (∨) (no conjunction), and the modal operators, but no negation. Here the complexity drops to PSPACE‑complete. The reduction uses quantified Boolean formulas (QBF) as the source problem: existential quantifiers are simulated by ◇, universal quantifiers by □, and the disjunction ∨ captures the choice of truth assignments for the quantified variables. Because QBF is PSPACE‑complete, the reduction yields PSPACE‑hardness, while the standard tableau‑based PSPACE algorithm for modal logic provides the matching upper bound.

3. Extension with Classical Disjunction – the language is enriched by allowing a classical disjunction operator (⊕) in addition to the dependence‑disjunction (∨). The authors prove that this addition does not increase the complexity: satisfiability remains NEXPTIME‑complete. The proof shows that classical disjunction can be eliminated via a polynomial‑time translation into the existing operators, so the hardness and upper‑bound arguments from the full MDL case still apply.

4. Negation‑Free, Dependence‑Disjunction‑Free Fragment (NFD) – the only Boolean connectives permitted are ∧ and classical ∨, together with □ and ◇; both negation and the dependence‑disjunction are disallowed. For this fragment the satisfiability problem lands in the second level of the polynomial hierarchy, specifically Σ₂^P‑complete. The lower bound is obtained by a reduction from the canonical Σ₂^P‑hard problem ∃∀‑SAT, where the outer existential quantifiers are simulated by ◇, the inner universal quantifiers by □, and the conjunction/disjunction structure mirrors the quantifier alternation. The upper bound follows from the observation that checking a candidate model can be expressed as an ∃∀‑formula, which is decidable in Σ₂^P.

5. Other Combinations – the paper also treats intermediate cases (e.g., allowing both ∧ and ∨ but still forbidding negation) and shows that they either collapse to one of the four complexities already identified or are subsumed by them. In each case the authors give a concise reduction or a straightforward simulation argument to place the fragment in the appropriate class.

The technical contributions are twofold. First, the authors develop a suite of reductions that respect the peculiarities of team semantics: dependence atoms can encode functional relationships across worlds, allowing them to simulate exponential‑size structures even when the underlying syntax seems weak. Second, they adapt classic complexity‑theoretic tools (tiling, QBF, Σ₂^P‑hard problems) to the modal setting, carefully handling the interaction between modal operators and dependence atoms.

Beyond the theoretical classification, the paper discusses practical implications. Since the presence of dependence‑disjunction or negation is the main driver of NEXPTIME‑hardness, any implementation of MDL‑based verification tools that aims for tractable performance should consider restricting these operators. Conversely, when expressive power is paramount (e.g., modelling functional dependencies in multi‑agent systems), the high complexity is unavoidable, and one must rely on heuristics or incomplete methods.

In summary, this work completes the complexity landscape for MDL satisfiability under all syntactic restrictions originally considered by Väänänen and Sevenster. It shows a clear hierarchy: from Σ₂^P (most restricted) through PSPACE (monotone fragment) up to NEXPTIME (any fragment containing dependence‑disjunction or negation). The results deepen our understanding of how team‑based dependencies interact with modal operators and provide a solid foundation for future research on algorithmic aspects of dependence logics.