[Paper Review] Uniform Lyndon interpolation for intuitionistic monotone modal logic

Amirhossein Akbar Tabatabai, Rosalie Iemhoff+1 more|arXiv (Cornell University)|Aug 9, 2022

Logic, programming, and type systems4 citations

TL;DR

This paper establishes the uniform Lyndon interpolation property (ULIP) for intuitionistic monotone modal logic iM using a novel terminating sequent calculus. The method, inspired by Pitts' syntactic proof for IPC, provides explicit interpolants by leveraging proof-theoretic techniques, marking the first such result for an intuitionistic non-normal modal logic and demonstrating the method's broad applicability beyond normal systems.

ABSTRACT

In this paper we show that the intuitionistic monotone modal logic $\mathsf{iM}$ has the uniform Lyndon interpolation property (ULIP). The logic $\mathsf{iM}$ is a non-normal modal logic on an intuitionistic basis, and the property ULIP is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication, respecting the polarities of the propositional variables. Our method to prove ULIP yields explicit uniform interpolants and makes use of a terminating sequent calculus for $\mathsf{iM}$ that we have developed for this purpose. As far as we know, the results that $\mathsf{iM}$ has ULIP and a terminating sequent calculus are the first of their kind for an intuitionistic non-normal modal logic. However, rather than proving these particular results, our aim is to show the flexibility of the constructive proof-theoretic method that we use for proving ULIP. It has been developed over the last few years and has been applied to substructural, intermediate, classical (non-)normal modal and intuitionistic normal modal logics. In light of these results, intuitionistic non-normal modal logics seem a natural next class to try to apply the method to, and we take the first step in that direction in this paper.

Motivation & Objective

• To establish the uniform Lyndon interpolation property (ULIP) for intuitionistic monotone modal logic iM, a non-normal modal logic over an intuitionistic base.
• To develop a terminating sequent calculus for iM, extending the G4ip system, which supports effective proof search and interpolation.
• To demonstrate the flexibility and generalizability of a constructive proof-theoretic method for uniform interpolation across diverse logical systems, including non-normal modal logics.
• To extend the scope of uniform interpolation results—previously limited to normal and intermediate logics—into the domain of intuitionistic non-normal modal logics.
• To lay the foundation for future exploration of ULIP in broader classes of intuitionistic non-normal modal logics.

Proposed method

• The method employs a terminating sequent calculus, G4w, for iM, which is an extension of Dyckhoff’s G4ip, ensuring proof search terminates without additional constraints.
• It uses a syntactic proof strategy inspired by Pitts’ uniform interpolation proof for IPC, adapted to handle modalities and polarities in iM.
• The construction of interpolants relies on polarized formulas: existential interpolants (∃♦p) and universal interpolants (∀∘p), defined via structural induction on sequents.
• The proof proceeds by induction on the structure of derivations in the sequent calculus, with case analysis on the last rule applied, ensuring interpolants respect variable polarities.
• Key rules such as (LM→), (Lw), and (M) are used to manage modal implications and structural conditions, preserving interpolant validity across derivations.
• The method ensures that interpolants depend only on the premise or conclusion of an implication, respecting positive and negative polarities of propositional variables.

Experimental results

Research questions

• RQ1 Does the intuitionistic monotone modal logic iM satisfy the uniform Lyndon interpolation property (ULIP)?
• RQ2 Can a terminating sequent calculus be developed for iM that supports effective proof search and interpolation?
• RQ3 To what extent can the constructive proof-theoretic method for uniform interpolation be generalized to intuitionistic non-normal modal logics?
• RQ4 How does the proof-theoretic approach to ULIP compare in complexity to semantic or model-theoretic methods for non-normal modal logics?
• RQ5 What are the structural and syntactic conditions under which ULIP can be established in non-normal modal extensions of intuitionistic logic?

Key findings

• The logic iM satisfies the uniform Lyndon interpolation property (ULIP), which is the first such result for any intuitionistic non-normal modal logic.
• A terminating sequent calculus G4w for iM has been developed, extending G4ip, which enables effective proof search and supports the interpolation proof.
• Explicit interpolants are constructed using polarized formulas: existential interpolants ∃♦p and universal interpolants ∀∘p, defined via structural induction on sequents.
• The proof method successfully generalizes to iM, a non-normal modal logic, with complexity comparable to that of its normal counterpart, indicating broad applicability of the technique.
• The method yields interpolants that depend only on the premise or conclusion of an implication and respect the polarities of propositional variables, fulfilling the uniformity condition.
• The results confirm that uniform interpolation is a rare property in modal logics, and iM is a significant addition to the small class of logics with ULIP, especially in the intuitionistic non-normal setting.