A unified view of inter-contextual similarity: Bonds, intermediate quantifiers, and metrics on concept lattices

This paper develops a unified theory of FCA bonds, moving beyond the simple universal/existential dichotomy. We introduce ‘intermediate quantifiers’ for more flexible bond construction and formalize the notion of a bond as a similarity metric that quantifies the structural coherence between two concept lattices.

Keywords

Inter-contextual similarity, Bonds, Intermediate quantifiers, Generalized quantifiers, Metrics on concept lattices, Recommender systems

Bonds in FCA provide a powerful mechanism for relating two distinct formal contexts, with applications in recommender systems. The existing framework, based on rigorous (universal) and benevolent (existential) quantifiers, represents two extremes of a spectrum. This paper develops a unified theory of bonds by introducing a framework for “intermediate quantifiers,” allowing for the definition of bonds with varying degrees of permissiveness. Furthermore, we formalize the notion of a bond as a metric of similarity between the concept lattices of the two contexts. We define an operator that, given a bond, produces a value in a result lattice quantifying the structural coherence between the two underlying conceptual structures. This work provides a richer theoretical understanding of inter-contextual relationships and opens the door to more nuanced and flexible recommender systems.

Introduction

Bonds serve as a “conceptual glue” between two contexts, \mathbb{K}_1 and \mathbb{K}_2, by establishing a relationship between the objects of one and the attributes of the other. The construction of bonds induced by external information has been defined using the two logical extremes: the universal quantifier (leading to a pessimistic bond) and the existential quantifier (leading to an optimistic bond). This binary choice is too restrictive for many real-world applications where a degree of flexibility between these two poles is desirable.

Furthermore, while bonds intuitively capture a form of “similarity” between contexts, this notion has never been formalized. How similar are two concept lattices if they admit a dense, rigorous bond versus a sparse, benevolent one?

This paper addresses these two open theoretical questions. Our contributions are:

1. A general framework for constructing bonds using intermediate quantifiers (e.g., “for at least n% of…”, “for most…”), moving beyond the simple universal/existential dichotomy.
2. A formal definition of a bond-induced similarity metric, which maps a bond \mathbb{B} between two contexts to a value representing the degree of structural alignment between their respective concept lattices, \mathfrak{B}(\mathbb{K}_1) and \mathfrak{B}(\mathbb{K}_2).
3. An analysis of how the properties of bonds (e.g., those generated by different quantifiers) correlate with different interpretations of inter-lattice similarity.

Methodology and expected theoretical results

Part 1: Intermediate quantifiers for bonds

We will leverage the theory of generalized quantifiers to define a parametric class of bond construction operators. Let \mathbb{P}=(M_1, M_2, I_P) be the external information context. The current bond definition can be seen as: g_1 I_{bond} g_2 \iff Q m_1 \in g_1^\uparrow, Q' m_2 \in g_2^\downarrow: m_1 I_P m_2. Where Q, Q' are \forall or \exists. We will replace Q, Q' with generalized quantifiers. * Main theoretical result 1: We will prove that bonds constructed with certain classes of intermediate quantifiers (e.g., monotone quantifiers) still form a complete lattice. We will characterize the structure of this “lattice of bonds”.

Part 2: Bonds as a similarity metric

The core idea is to measure how well a bond preserves conceptual structure. * Definition of the similarity operator \mu: We will define an operator \mu: \text{Bonds}(\mathbb{K}_1, \mathbb{K}_2) \to \mathbb{D}, where \mathbb{D} is a suitable result lattice (e.g., [0,1]). A possible definition for \mu(\mathbb{B}) could be an aggregation of how well the concepts of the bond context \mathbb{B} align with the concepts of \mathbb{K}_1 and \mathbb{K}_2. For instance, it could measure the average “conceptual integrity” of the mappings induced by the bond. * Main theoretical result 2: We will prove that \mu satisfies desirable metric-like properties (e.g., \mu is maximal for the identity bond, monotonic with respect to the bond ordering). We will analyze how \mu behaves for bonds generated by different quantifiers, formally showing that rigorous bonds lead to higher similarity scores.

Work plan

• Months 1-4: Develop the theory of intermediate quantifiers for bonds. Prove the lattice structure of the resulting bond sets.
• Months 5-8: Formally define the similarity operator \mu and prove its key properties. This will involve collaboration with all named researchers.
• Months 9-10: Analyze the relationship between the quantifier used to build a bond and its resulting similarity score \mu.
• Months 11-12: Prepare a manuscript for a top journal, focusing on the unified theoretical framework presented.

Potential target journals

1. Information Sciences (Q1): An ideal venue, as you have already published high-impact work on bonds there. This would be a natural and significant follow-up.
2. Fuzzy Sets and Systems (Q1): A strong candidate, especially for the fuzzy extension of these ideas.
3. Journal of Logic and Computation (Q2): Suitable for a version of the paper that focuses heavily on the generalized quantifier aspect.

Minimum viable article (MVA) strategy

The two main ideas (intermediate quantifiers and similarity metrics) can be separated into two strong papers.

• Paper 1 (The MVA - intermediate quantifiers):
  • Scope: This paper focuses exclusively on generalizing bond construction. It introduces the framework for using intermediate quantifiers, proves the resulting algebraic structures (e.g., the lattice of bonds), and discusses the practical implications for creating more flexible recommender systems.
  • Goal: To expand the foundational theory of bond construction.
  • Target venue: A strong theoretical conference like IPMU or a journal like International Journal of Approximate Reasoning.
• Paper 2 (Bonds as a metric):
  • Scope: Citing Paper 1 for the construction methods, this paper introduces the novel idea of formalizing inter-lattice similarity via bonds. It would define the operator \mu, prove its mathematical properties, and analyze how it provides a quantitative measure of conceptual alignment.
  • Goal: To introduce a new theoretical tool for comparing and analyzing conceptual structures.
  • Target venue: A high-impact journal with a theoretical focus like Information Sciences or Fuzzy Sets and Systems.