PSPACE Bounds for Rank-1 Modal Logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

💡 Research Summary

The paper tackles the long‑standing problem of establishing uniform complexity bounds for large families of modal logics. It introduces a general theory that shows every rank‑1 modal logic—that is, any logic axiomatizable by formulas whose modal depth is uniformly one—enjoys a shallow model property and, under mild syntactic conditions on the axiomatisation, can be decided within PSPACE.

The authors work in the coalgebraic semantics framework. A modal signature Λ (a set of unary modal operators) is interpreted over a Set‑functor T, called the signature functor. Coalgebras for T serve as abstract transition systems; each modal operator L∈Λ is given a predicate lifting ⟦L⟧ : Q → Q∘T (where Q is the contravariant powerset functor). This abstraction subsumes ordinary Kripke frames, neighbourhood frames, game frames, weighted multigraphs, probabilistic transition systems, etc. The satisfaction relation x ⊨ φ is defined inductively on the coalgebraic structure, and the central decision problem is local satisfiability: does there exist a T‑coalgebra and a state satisfying a given formula?

The technical core is a rule‑based tableau construction. From a given axiomatization the authors derive a (possibly infinite) set of inference rules. Two meta‑properties of the rule set are required:

1. Resolution closedness – any resolvent of two instantiated rule conclusions can be derived directly by a third rule. This guarantees that the rule set is deductively complete for one‑step reasoning.
2. Contraction closedness – any rule conclusion containing duplicate literals can be replaced by a rule whose conclusion has all literals distinct.

When both properties hold, the set of applicable rules at any tableau step can be bounded polynomially in the size of the input formula. Consequently, the tableau explores only a polynomial‑space search tree: each step removes one modal layer, builds a shallow (depth‑1) model fragment, and proceeds recursively on the sub‑formulas. Because the depth of the constructed model never exceeds the modal depth of the original formula, the algorithm needs to keep only the current and previous layers in memory, yielding a PSPACE algorithm.

The tableau also enjoys a weak subformula property: every formula appearing in the proof is a Boolean combination of sub‑formulas of the original goal. This makes the generated proofs compact and amenable to proof‑search optimisations.

The authors instantiate the generic framework for a variety of well‑known logics:

• K and KD – standard normal modal logics with the powerset functor P (or its non‑empty variant P*). The rule set consists of the usual K‑axiom and the seriality axiom for KD; both are resolution‑ and contraction‑closed.
• Coalition logic – interpreted over a game‑frame functor T that records strategy sets for each agent; the coalition operator