A sequent calculus for a semi-associative law (1803.10080v3)

Abstract: We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, right rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. We then describe two main applications of the coherence theorem, including: 1. A new proof of the lattice property for the Tamari order, and 2. A new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice $Y_n$.

• The paper establishes a coherence theorem ensuring each valid entailment in the Tamari order has a unique focused derivation.
• It refines Lambek's calculus by restricting the left rule to the leftmost context, accurately capturing the semi-associative law.
• The work proves the lattice properties of the Tamari order and validates combinatorial formulas linking binary trees with planar maps.

Sequent Calculus for a Semi-Associative Law

The paper under review presents a detailed exploration of a sequent calculus tailored to capture the classical Tamari order, which is a partially ordered set of fully-bracketed words (or equivalently, binary trees) that emerges from a semi-associative law. This work extends upon a preliminary presentation at FSCD 2017 by providing a proof for the lattice properties of the Tamari order through a coherence theorem. The paper successfully integrates discussions on historical and related contemporary research to ensure a comprehensive understanding of the subject.

Overview of the Sequent Calculus

The sequent calculus proposed in this paper is a refinement of Lambek's original calculus, adjusted to encapsulate the Tamari order. The main alteration involves a restriction of Lambek's product rules, which confines the applicability of the left rule to the leftmost context only. This restriction is crucial for accurately modeling semi-associativity.

The calculus consists of:

• Left Rule (L): Addresses leftmost context elements.
• Right Rule (R): Maintains Gentzen's format but with restricted validity.
• Structural Rules: Comprising identity and restricted cut rules.

The core result is a coherence theorem demonstrating that any valid entailment in the Tamari order admits a unique focused derivation. This focused derivation is significant as it simplifies the complexity typically associated with cut-elimination procedures in proof theory.

Key Contributions and Claims

The paper presents multiple contributions:

• Coherence Theorem: The claim that each valid derivation in the Tamari order has one and only one focused representation, providing an efficient tool for verifying entailments.
• Lattice Property Proof: The sequent calculus effectively proves the lattice nature of Tamari orders, a foundational property showing that every pair of elements has a least upper bound (join) and greatest lower bound (meet).
• Tutte--Chapoton Formula Verification: Provides an alternative proof for Chapoton's formula about the number of intervals in the Tamari lattice, thus linking lambda calculus with combinatorial structures like planar maps.

Practical and Theoretical Implications

The implications of this research are multi-faceted:

• Theoretical Insights: The sequent calculus introduces a paradigmatic shift in understanding semi-associativity in algebraic structures. The coherence theorem not only simplifies the logical structure but also offers insights into the deeper categorical connections with associativity and non-associativity.
• Practical Applications: The canonical nature of focused derivations serves as a potent tool for computational implementations in proof verification systems and could inspire architectural designs in programming languages where associativity plays a crucial role.
• Combinatorial and Graph-Theoretical Connections: By establishing relationships with lambda calculus and planar maps, the paper extends the utility of the Tamari lattice in computational fields such as complexity analysis and symbolic computation.

Speculations on Future Developments

The ramifications of this research provide a basis for several future pursuits:

1. Extension to Skew-Monoidal Categories: The paper hints at possible connections between the skew-monoidal categories and the semi-associative structures explored here, suggesting a potential pathway for uniting logics under a broader categorical umbrella.
2. Further Algebraic Structures: Exploration into mixed-associativity frameworks combining the disclosed sequent calculus could offer new inroads into multi-modal logics pertinent to linguistic and semantic models.
3. Deeper Computational Implementations: With the understanding of Tamari lattices, developing algorithms for efficient computation in related fields such as data sorting and tree enumeration could be envisioned.

In conclusion, the paper provides a comprehensive and academically focused examination of a sequent calculus that advances our understanding of semi-associativity and its applications within mathematical logic and computer science. The verification of strong numerical claims like the Tutte--Chapoton formula testifies to the robustness of the theoretical framework, while the paper's methodological innovations signal potential advancements in both theory and practice within the field.