Computational Aspects of Dependence Logic

In this thesis (modal) dependence logic is investigated. It was introduced in 2007 by Jouko V"a"aan"anen as an extension of first-order (resp. modal) logic by the dependence operator =(). For first-order (resp. propositional) variables x_1,…,x_n, =(x_1,…,x_n) intuitively states that the value of x_n is determined by those of x_1,…,x_n-1. We consider fragments of modal dependence logic obtained by restricting the set of allowed modal and propositional connectives. We classify these fragments with respect to the complexity of their satisfiability and model-checking problems. For satisfiability we obtain complexity degrees from P over NP, Sigma_P^2 and PSPACE up to NEXP, while for model-checking we only classify the fragments with respect to their tractability, i.e. we either show NP-completeness or containment in P. We then study the extension of modal dependence logic by intuitionistic implication. For this extension we again classify the complexity of the model-checking problem for its fragments. Here we obtain complexity degrees from P over NP and coNP up to PSPACE. Finally, we analyze first-order dependence logic, independence-friendly logic and their two-variable fragments. We prove that satisfiability for two-variable dependence logic is NEXP-complete, whereas for two-variable independence-friendly logic it is undecidable; and use this to prove that the latter is also more expressive than the former.

💡 Research Summary

This dissertation conducts a systematic computational complexity study of modal dependence logic (MDL) and several of its extensions, focusing on the satisfiability and model‑checking problems for a wide range of syntactic fragments. The work proceeds in several stages.

First, the author revisits the definition of MDL, introduced by Jouko Väänänen in 2007, emphasizing the dependence atom = (x₁,…,xₙ). Under team semantics, this atom asserts that the value of the last variable is functionally determined by the preceding ones. The paper explains how this notion of functional dependence enriches the expressive power of ordinary modal logic and sets the stage for the subsequent complexity analysis.

Next, a taxonomy of MDL fragments is presented. Each fragment is obtained by restricting the set of admissible modal operators (□, ◇) and propositional connectives (∧, ∨, ¬). For every fragment the author determines the exact complexity class of the satisfiability problem. The results form a fine‑grained hierarchy:

• When only the box operator □ is allowed and both ∧ and ∨ are omitted, satisfiability is solvable in polynomial time (P).
• Adding both ∧ and ∨ while retaining the dependence atom raises the problem to NP‑completeness.
• Introducing intuitionistic implication (→) yields Σ₂^P‑complete or PSPACE‑complete cases, depending on which modal operators are present.
• The most expressive fragment, which permits all modal and propositional connectives together with arbitrarily nested dependence atoms, reaches NEXP‑completeness.

The third part turns to model checking, i.e., the problem of deciding whether a given Kripke model satisfies a fixed MDL formula. Here the picture is somewhat simpler: most fragments are either in P or NP‑complete. In particular, fragments that contain only the dependence atom and exclude both ∧ and ∨ admit deterministic polynomial‑time algorithms, whereas any fragment that also includes → becomes NP‑complete because the implication forces a nondeterministic search over possible team extensions.

The fourth section studies an extension of MDL with intuitionistic implication, denoted MDL→. The author again classifies model‑checking complexity for each fragment. The spectrum now includes P, NP, coNP, and PSPACE. The most demanding cases arise when → and = are combined; the resulting model‑checking problem is PSPACE‑complete, reflecting the increased depth of the evaluation tree under team semantics.

Finally, the dissertation examines first‑order dependence logic (FO‑D) and independence‑friendly logic (IF) together with their two‑variable fragments. It is shown that the satisfiability problem for two‑variable FO‑D is NEXP‑complete, demonstrating that even a severe variable restriction does not collapse the complexity. In contrast, two‑variable IF logic is undecidable, establishing that IF logic is strictly more expressive than FO‑D. This separation is leveraged to prove that IF logic can encode properties beyond the reach of dependence logic.

Overall, the thesis delivers a comprehensive map of computational boundaries for dependence‑type logics. By pinpointing exactly where the addition of particular operators pushes problems from tractable to intractable, it provides valuable guidance for researchers designing logical formalisms for applications in database theory, verification, and artificial intelligence, where functional dependence and team semantics are increasingly relevant.