Computing minimal generators in mixed contexts: A direct approach

This paper proposes a direct algorithm for computing the minimal generators of a mixed context, avoiding prior decomposition. It uses the simplification logic for mixed attributes to define and identify these generators natively, offering a more streamlined and efficient path.

Keywords

Mixed minimal generators, Direct algorithm, Simplification logic, Mixed contexts

While our previous work characterized mixed minimal generators based on the generators of the separate positive and negative contexts, a direct computational method is still needed. This paper proposes a direct algorithm for computing the minimal generators of a mixed context without requiring prior decomposition. We leverage the Simplification Logic for Mixed Attributes to define and identify “mixed minimal generators” natively. Our approach avoids the overhead of computing two separate sets of generators and then combining them, offering a more streamlined and potentially more efficient path. We prove the correctness of our direct approach and provide an algorithm for its implementation.

Introduction

Our work in established a theoretical link between the minimal generators of a mixed context and those of its positive and negative components. This is a characterization, not an algorithm. The implied procedure—compute positive generators, compute negative generators, then combine—is indirect and potentially inefficient.

This paper proposes a direct method. We want to answer the question: given a mixed closed set C, how can we find its minimal generators by operating directly within the mixed attribute space M \cup \bar{M}?

Our contributions are:

1. A formal, self-contained definition of minimality for generators in the mixed setting.
2. A direct algorithm for computing mixed minimal generators, leveraging the mixed simplification logic.

Methodology and expected theoretical results

Theoretical foundation

The key will be to integrate the constraints from the mixed logic (e.g., m \wedge \bar{m} = \bot) into the definition of a generator. A set G \subseteq M \cup \bar{M} is a mixed minimal generator of C if G^{mixed-closure} = C and no proper subset has this property. * Main theoretical result: We will develop an algorithm based on the logic that, for a given mixed closed set C, can efficiently find all its minimal generators by iteratively testing and reducing subsets of C. This will be more direct than the logic-based algorithm for binary generators.

Work plan

• Months 1-5: Develop the full theory and algorithm for direct computation of mixed minimal generators.
• Months 6-9: Implement the algorithm and compare its performance to the indirect, decomposition-based approach implied by.
• Months 10-12: Write the manuscript.

Potential target journals

1. International Journal on Computational Intelligence Systems (Q3): A natural venue to publish a direct follow-up and improvement on the work from.
2. Information Sciences (Q1): A higher-impact alternative.

MVA strategy

• Paper 1 (The MVA): The full paper as described. The novelty is the direct computational method, which is a clear and self-contained contribution.