Shallow Models for Non-Iterative Modal Logics

The methods used to establish PSPACE-bounds for modal logics can roughly be grouped into two classes: syntax driven methods establish that exhaustive proof search can be performed in polynomial space whereas semantic approaches directly construct shallow models. In this paper, we follow the latter approach and establish generic PSPACE-bounds for a large and heterogeneous class of modal logics in a coalgebraic framework. In particular, no complete axiomatisation of the logic under scrutiny is needed. This does not only complement our earlier, syntactic, approach conceptually, but also covers a wide variety of new examples which are difficult to harness by purely syntactic means. Apart from re-proving known complexity bounds for a large variety of structurally different logics, we apply our method to obtain previously unknown PSPACE-bounds for Elgesem’s logic of agency and for graded modal logic over reflexive frames.

💡 Research Summary

The paper addresses the problem of establishing PSPACE upper bounds for a wide class of modal logics without relying on a complete axiomatization. Traditional approaches fall into two categories: syntactic methods, which show that exhaustive proof search can be carried out in polynomial space, and semantic methods, which construct shallow (finite‑depth) models directly. While the syntactic route works well for many normal logics, it becomes cumbersome for non‑normal or otherwise heterogeneous logics where a suitable tableau or Gentzen system is hard to obtain.

The authors adopt the semantic side and work within the coalgebraic modal logic framework, extending it to non‑iterative logics—those whose axioms contain modal operators but no nesting of modalities. To capture a broader range of semantics (including probabilistic, graded, coalition, and conditional logics) they introduce copointed functors ((T,\varepsilon)). A copointed functor consists of a base functor (S_0) (describing the shape of transitions) together with a natural transformation (\varepsilon:T\to \mathrm{Id}) that links each state to its own transition structure, thereby encoding local frame conditions that are typical for non‑normal logics.

The central technical contribution is the definition of two model‑size properties for the one‑step fragment of a logic (the fragment where modal operators appear only once, i.e., without nesting):

1. One‑Step Poly‑size Model Property (OSPMP). For any satisfiable one‑step formula, there exists a one‑step model whose underlying set has size polynomial in the size of the formula. This property is considerably weaker than the classic shallow‑model property but is often much easier to verify.

2. One‑Step Pointwise Poly‑size Model Property (OSPPMP). When OSPMP fails, OSPPMP requires that for each state the “local” one‑step model is polynomially bounded, even if the global model may be exponentially large. This pointwise bound enables a polynomial‑space simulation of exponentially branching shallow models.

The authors prove two main theorems:

• If a non‑iterative modal logic satisfies OSPMP, then its satisfiability problem lies in PSPACE. The construction proceeds by assembling a global shallow model from the polynomial‑size one‑step models, guaranteeing that the overall model can be traversed using only polynomial space.

• If a logic satisfies OSPPMP (and the underlying functor admits a suitable notion of pointwise smallness), then the same PSPACE upper bound holds. The algorithm traverses the exponentially branching shallow model depth‑first, but at each step only a polynomial‑size “local” structure needs to be stored.

A further corollary shows that for any bounded‑rank fragment (formulas whose modal nesting depth is bounded by a constant), OSPMP yields an NP upper bound, because a polynomial‑size model can be guessed nondeterministically and verified in polynomial time. This generalises known NP results for the rank‑restricted fragments of K and T to many non‑normal logics.

The paper applies this framework to several concrete logics, obtaining both known and new complexity results:

• K and T – the classic normal modal logics – are re‑proved to be PSPACE‑complete, illustrating that the semantic method subsumes the traditional tableau approach.
• Conditional logics such as CK, CK+MP, and CK+ID satisfy OSPMP trivially, yielding PSPACE bounds without constructing a resolution‑closed rule set.
• Elgesem’s logic of agency – a non‑normal logic modelling agency actions – is shown to satisfy OSPMP, delivering the first PSPACE upper bound for this system.
• Graded modal logic over reflexive frames (the logic (T_n)) – previously conjectured to be EXPTIME‑hard – is proved PSPACE‑complete via OSPPMP. This also covers the reflexive version of Presburger modal logic.
• Presburger modal logic – which combines modal operators with linear arithmetic constraints – is handled similarly; the reflexive fragment receives a PSPACE bound, extending earlier decidability results.
• Coalition logic and probabilistic modal logic are mentioned as examples where OSPPMP is easier to establish than a full shallow‑model property, confirming that the semantic approach works even when syntactic resolution closure is infeasible.

Methodologically, the paper demonstrates that by reducing the full modal satisfiability problem to the one‑step fragment, one can avoid dealing with nested modalities altogether. The copointed functor machinery cleanly separates the “transition shape” (captured by (S_0)) from the “local frame condition” (captured by (\varepsilon)), allowing the same proof technique to apply uniformly across a diverse set of semantics.

In summary, the authors provide a robust, axiomatization‑free semantic technique for establishing PSPACE upper bounds for a large class of non‑iterative modal logics. The introduction of OSPMP and its pointwise variant offers a practical criterion that is often straightforward to verify, leading to new complexity results for several important logics and unifying existing ones under a single coalgebraic umbrella. This work significantly broadens the toolkit for modal‑logic complexity analysis and opens the door for further extensions to richer, possibly iterative, systems.