Completeness and Incompleteness for Expanding Gödel-Löb Logics

Abstract: Expanding products of modal logics are bimodal logics obtained from the combination of a horizontal component' logic and avertical component' logic, lying between the fusion and the Cartesian product of the two logics. Gabelaia et al. showed that expanding products are often decidable when the first component is Noetherian, although their methods are semantical and do not yield complete axiomatisations. They do, however, propose a candidate, dubbed the expanding commutator of the two logics and known to be complete in many `non-Noetherian' cases. In this paper, we consider various expanding products of modal logics whose vertical component is $\sf GL$. We show that the standard axiomatisation is complete when the horizontal component is either ${\sf K4}$ or ${\sf GL}$, but incomplete when it is ${\sf Grz}$ or any logic between ${\sf K4.3}$ and ${\sf Grz.3}$, thus yielding a partial solution to a question posed by Gabelaia et al. more than two decades ago.

• The paper establishes completeness for expanding products with vertical GL when the horizontal logic is K4 or GL, employing canonical models and simulation formulas.
• The paper demonstrates incompleteness for cases with horizontal logic Grz or axioms between K4.3 and Grz.3 by constructing explicit counterexamples in forward-confluent models.
• The paper introduces innovative methods such as quasimodel extraction via Yankov-Fine simulation formulas, thereby advancing decidability and modal logic analysis.

Completeness and Incompleteness for Expanding Gödel-Löb Logics

Introduction

The paper "Completeness and Incompleteness for Expanding Gödel-Löb Logics" (2606.20229) develops an extensive theory of bimodal logics arising from expanding products, focusing on combinations where the vertical component is Gödel-Löb logic ($\mathsf{GL}$). Expanding products reside between the fusion and Cartesian product of modal logics and are motivated by applications in temporal description logics and dynamic topological logics, where domain growth or dynamic behaviour requires modalities to interact in nontrivial ways. The principal concern is to analyze completeness and incompleteness phenomena for these expanding products, particularly relating to the standard axiomatisations provided by the so-called expanding commutator.

Formal Framework and Semantics

The theory is situated within bimodal languages generated by propositional variables with two modal operators ($\Diamond_h$, $\Diamond_v$), corresponding to horizontal and vertical interactions. The paper details several semantic variants:

• Expanding Domain Semantics: Horizontal frames ($\mathsf{L}_h$) are paired with vertical frames ($\mathsf{L}_v$) at each node, with the constraint that expansions along the horizontal relation embed previous vertical frames as subframes in successors.
• Embedding Domain Semantics: An extension allowing injective embeddings instead of inclusions, facilitating technical constructions especially with transitive frames.
• Forward-Confluent Semantics: A further generalization, relying on two binary relations with left-commutativity and Church-Rosser properties, which capture the interplay between modalities in canonical models.

These semantic regimes are compared and shown to be essentially equivalent under tree-like horizontal structures, a feature exploited throughout the completeness proofs.

Expanding Commutators: Hilbert-Style Calculi

A Hilbert-style calculus, the expanding commutator $[\mathsf{L}_h, \mathsf{L}_v]^{\mathsf{e}}$, combines axioms from both component logics and enforces commutativity principles ($$\Diamond_v \Diamond_h p \to \Diamond_h \Diamond_v p$$ and $\Diamond_v \Box_h p \to \Box_h \Diamond_v p$). Crucially, the calculus is sound for the forward-confluent frames and for expanding domain and embedding domain models when the horizontal logic extends $\mathsf{K4}$ or $\mathsf{GL}$.

Main Results: Completeness and Incompleteness

The central findings are as follows:

• Completeness: For expanding products with vertical $\Diamond_h$0, the standard axiomatisation is complete when the horizontal component is $\Diamond_h$1 or $\Diamond_h$2. This is demonstrated via canonical model techniques adapted to the forward-confluent setting and refined via quasimodel extraction and simulations. The completeness proof for $\Diamond_h$3 requires the construction of finite quasimodels using simulation formulas (Yankov-Fine), leveraging properties of Noetherianity and the finite model property. For $\Diamond_h$4, completeness follows from well-quasi-ordering (Kruskal’s theorem) and appropriate extraction arguments.
• Incompleteness: If the horizontal logic is $\Diamond_h$5, or includes the weak connectedness axiom ($\Diamond_h$6), or lies strictly between $\Diamond_h$7 and $\Diamond_h$8, then $\Diamond_h$9 is incomplete for its expanding domain semantics. Strong explicit counterexamples are constructed in the forward-confluent models, showing that certain formulas valid in expanding domain models are not derivable in the commutator, thus providing partial resolution to Gabelaia et al.’s decades-old questions about Noetherian cases [pml].

Technical Contributions

• Canonical Forward-Confluent Models: The construction and properties of canonical models for expanding commutators are established, including forward-confluence, transitivity, and the truth lemma.
• Quasimodel Extraction via Simulations: The paper rigorously develops extraction procedures based on moments—finite, tree-like structures labeled by types—and simulations to convert canonical models into expanding domain models.
• Yankov-Fine Simulation Formulas: Definitions of formulas characterizing the simulability of moments in the canonical model are provided, crucial for ensuring finite quasimodels in $\Diamond_v$0-based expanding domains.
• Model-Theoretic Decidability: The finite model property and decidability (but not primitive recursive decidability) are established for key expanding commutators.

Implications and Open Problems

Practically, these results refine the understanding of tractability and axiomatisation for complex multimodal logics with applications in knowledge representation, temporal reasoning, and spatial logics. The analysis delineates a border where completeness transitions to incompleteness, pinpointing the precise impact of Noetherian and linear modalities. The theoretical apparatus developed enriches modal logic's toolkit, especially for handling domain expansion dynamics and interactions of modal operators.

Open questions posed include:

• Determining for which decidable expanding products (as shown by Gabelaia et al.) the commutator is indeed complete.
• The existence of natural or finite axiomatisations in cases where completeness fails.
• The status of intuitionistic companions and modal translations for various expanding commutator logics, and their finite axiomatisability and complexity bounds.

Conclusion

The paper provides a rigorous account of completeness and incompleteness for expanding Gödel-Löb logics, mapping out the intricate landscape of modal bimodal logics with expanding domains. The results reached resolve long-standing questions about axiomatisability in expanding products, establish strong completeness in core cases, and delineate sharp boundaries for incompleteness, substantially advancing the theory and its applications within modal and temporal logic.