[Paper Review] Semantic interpolation

Dov Gabbay, Karl Schlechta|ArXiv.org|Jun 22, 2009

Natural Language Processing Techniques4 citations

TL;DR

This paper introduces a novel semantic framework for interpolation in non-monotonic logics using product spaces and generalized Hamming relations, demonstrating that interpolants exist when abstract size laws and Hamming distance structures are satisfied. The key contribution is a unified algebraic approach that separates logical from algebraic interpolation, enabling interpolation in logics where it previously failed due to definability constraints.

ABSTRACT

We treat interpolation for various logics.

Motivation & Objective

• To develop a general semantic framework for interpolation in non-monotonic logics beyond classical systems.
• To establish conditions under which interpolants exist using abstract size laws and generalized Hamming relations.
• To unify algebraic and logical interpolation by separating definability constraints from structural properties.
• To extend interpolation to many-valued and non-monotonic settings through product space representations.
• To investigate the connection between revision operators (e.g., Parikh-style) and semantic interpolation via distance-based relations.

Proposed method

• Representing logical states as sequences over variable domains, enabling abstraction from classical truth values to many-valued systems.
• Defining relevance and irrelevance of variables via sets R(Σ) and I(Σ), where R(Σ) contains essential variables and I(Σ) those whose values do not affect membership in Σ.
• Introducing product size and Hamming distance structures to formalize 'closeness' between interpretations, enabling interpolation via minimal-distance relations.
• Using the relation σ ∣ τ to denote minimal-distance pairs, which generalizes classical and non-monotonic consequence relations.
• Proving that Hamming-based relations decompose over product spaces, ensuring consistency under restriction and projection.
• Establishing that revision operators based on Hamming distances preserve interpolation properties under structural constraints.

Experimental results

Research questions

• RQ1 Under what conditions does an interpolant exist in non-monotonic logics when consequence relations differ?
• RQ2 How can abstract size laws and generalized Hamming relations ensure the existence of interpolants?
• RQ3 Can interpolation be achieved in logics where the set of models is not definable in the language?
• RQ4 What is the role of product space decomposition in enabling interpolation for non-monotonic reasoning?
• RQ5 To what extent can Hamming distance-based revision operators be used to generate interpolants?

Key findings

• Interpolation is possible in non-monotonic logics when the consequence relation is defined via Hamming distance and product space structure.
• The relation σ ∣ τ, defined as minimal-distance pairs, satisfies decomposition properties: (Σ₁′×Σ₁″)∣(Σ₂′×Σ₂″) = (Σ₁′∣Σ₂′)×(Σ₁″∣Σ₂″).
• If Σ₁∣Σ₂ = (Σ₁′∣Σ₂′)×(Σ₁″∣Σ₂″), then the interpolant is preserved under restriction to subspaces, ensuring consistency in reduced models.
• Hamming distance-based revision operators are decomposable and generate interpolants, generalizing Parikh’s approach.
• The paper shows that algebraic interpolation can exist even when logical interpolation fails due to undefinable model sets, highlighting the need for richer languages.
• A representation result for distance-based revision is shown to be impossible in general due to arbitrary choices in distance assignment, leaving a uniform representation as an open problem.