Logic isn’t just about what’s there: simplifying rules with positive and negative data
In data, knowing ‘what is absent’ is as important as ‘what is present’. We developed a new ‘mixed simplification logic’ to clean and remove redundancy from rules that use both positive and negative attributes.
When we analyze data, we usually focus on what is there: “this patient has a fever,” or “this product was purchased.” But in many cases, what isn’t there is just as important. A medical diagnosis, for example, relies heavily on knowing a patient “has no cough” or “has no rash.”
This “negative” information is incredibly valuable. The problem is that the traditional logic used in data mining (like Formal Concept Analysis) wasn’t built to handle it properly.
In our 2022 paper in Mathematics, we built the missing piece: a mixed simplification logic to handle both positive and negative attributes.
🧐 The problem: a logic that only sees half the picture
For years, we’ve used “Simplification Logic” (SL) to manage and clean up sets of logical rules (implications) found in data. It’s the “engine” that allows us to remove redundant rules (like if A, then B when we already know if A, then B and C) and find a minimal, clean basis.
But this engine only worked with positive information. When researchers tried to introduce negative attributes, they didn’t have the right set of formal rules to manage them. The result? You’d end up with a messy, complex, and highly redundant set of rules that was impossible to use for automated reasoning.
💡 Our solution: a new, complete logic for “mixed” data
We went back to the foundations of the logic itself. We asked, “what new algebraic rules do we need to prove that we can safely simplify implications that contain both positive and negative facts?”
Our paper formally introduces a mixed simplification logic. This is a new, sound set of equivalence-preserving rules that, for the first time, allows a computer to understand and manipulate “mixed implications” correctly.
🚀 The results: an algorithm that automatically cleans up rules
A logic is a great theoretical step, but its real power comes when you build a tool with it.
We used our new logic as the “brain” for a new automated method. This method takes any set of mixed implications (positive and negative) and automatically applies our simplification rules to remove all redundancy.
The output is a smaller, cleaner, and more manageable set of rules that contains the exact same amount of knowledge as the original, bloated set.
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🔬 Why does this matter?
This paper provides the foundational engine for all future work in automated reasoning with mixed attributes.
Before this, it was hard to build smart systems that could reason with both “has fever” and “no cough” because the underlying logic was messy and incomplete. Now, we have the formal “grammar” to do it. This is the essential first step for building more powerful and expressive AI tools in fields like medical diagnosis, e-learning, and any domain where what isn’t there matters as much as what is.
📖 The full paper
For the complete set of logical rules, the formal proofs, and the algorithm for automated redundancy removal, you can read the original open-access article in Mathematics.
Simplifying implications with positive and negative attributes: A logic-based approach. Authors: Francisco Pérez-Gámez, Domingo López-Rodríguez, Pablo Cordero, Ángel Mora, Manuel Ojeda-Aciego. Journal: Mathematics (vol. 10, issue 4, 607)