💻 computer science

A Non-Binary Method for Finding Interpolants: Theory and Practice

This paper introduces a novel method for finding interpolants in classical logic that leverages a refutation-based approach and utilizes a non-binary version of resolution as its foundational mechanism.

Original authors: Adam Trybus, Karolina Rożko, Tomasz Skura

Published 2026-03-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Trybus, Karolina Rożko, Tomasz Skura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: What is this paper about?

Imagine you are a detective trying to solve a mystery. You have two pieces of evidence: Clue A and Clue B. You know for a fact that if Clue A is true, then Clue B must also be true ( $A \rightarrow B$ ).

Your job is to find a Secret Message (called an Interpolant) that acts as a bridge between them. This Secret Message must:

Be made only of the facts that appear in both Clue A and Clue B.
Be strong enough to prove that A leads to it.
Be strong enough to prove that it leads to B.

The authors of this paper, Adam, Karolina, and Tomasz, have invented a new, faster way to find this Secret Message. Instead of using the standard, slow "binary" method (like checking one thing against another), they use a "non-binary" approach that looks at the whole picture at once.

The Core Concept: The "Mirror" System

Most logic systems work like a Builder. They try to prove that a statement is True by stacking bricks of logic on top of each other.

The authors decided to try a Demolition Crew approach (which they call a Refutation System). Instead of trying to prove something is True, they ask: "How can we prove this is FALSE?"

The Builder: "I can build a house from these bricks."
The Demolition Crew: "I can knock down this house because these bricks don't fit together."

The authors realized that if you can easily figure out why a logical statement is broken (false), you can use that same "breaking" process to find the Secret Message (the interpolant). It's like finding the weak spot in a wall; once you know where the wall is weak, you can build a bridge right across that gap.

The Old Way vs. The New Way

The Old Way (Binary Resolution):
Imagine you are trying to connect two puzzle pieces. The old method says: "Pick one piece from the left, pick one piece from the right, and see if they fit. If they do, glue them. Then pick another pair."

Problem: You have to do this one pair at a time. It's like eating a giant pizza one slice at a time. It works, but it takes a long time.

The New Way (Non-Binary Resolution):
The authors' method is like looking at the entire pizza at once. They say: "Let's look at all the slices on the left and all the slices on the right. We see a pattern where a 'Pepperoni' on the left cancels out a 'Mushroom' on the right. Let's remove all of those pairs simultaneously and see what's left."

Benefit: You skip the small steps. You can clear the table much faster. The paper claims this method can find the answer in fewer steps than the traditional method.

How the Algorithm Works (The "Magic Trick")

The paper describes a step-by-step recipe (an algorithm) that a computer follows. Here is the recipe in plain English:

Start with the Problem: You have a statement "If A, then B" that is definitely true.
Pick a Variable: Find a letter (like $p$ ) that appears in both A and B, but in opposite forms (one is $p$ , the other is "not $p$ ").
Split the World: Imagine two parallel universes:
- Universe 1: Assume $p$ is true.
- Universe 2: Assume $p$ is false.
Simplify: In each universe, remove the $p$ and "not $p$ " from the equations. You are left with two simpler problems.
Repeat: Do this again for the next letter in the new, simpler problems. Keep splitting and simplifying until you hit a "Base Case."
- Base Case: You run out of letters. At this point, the math guarantees that either the left side is impossible (False) or the right side is guaranteed (True).
Rebuild: Now, walk backward up the ladder. Combine the answers from the two universes.
- If the answer in Universe 1 was "False" and Universe 2 was "True," you combine them to create your Secret Message.
Result: You get a formula that uses only the shared letters and proves the connection between A and B.

The "Human vs. Robot" Problem

The authors wrote a computer program (in Python) to test this. They found something funny:

Humans are smart. When solving a puzzle, we might spot a shortcut. "Oh, I see the answer is already there! I'll skip the rest."
The Computer is obedient but dumb. It follows the rules strictly. It doesn't take shortcuts. It will grind through every single step, even if the answer is obvious.

Because the computer doesn't take shortcuts, the formulas it produces are often messy and huge (like a tangled ball of yarn). The authors admit the output looks ugly, but they proved that the method works and is fast. They even included a "simplifier" function to clean up the yarn later.

The Experiment: Did it work?

They tested their program on thousands of random logic puzzles.

Speed: It was incredibly fast (milliseconds).
Scalability: As the puzzles got bigger, the time it took to solve them grew in a straight line (linear growth). This is great news! It means the method won't crash even if the puzzles get very large.

Why Does This Matter?

Speed: In the world of computer science (especially in verifying that software or hardware works correctly), finding these "Secret Messages" is crucial. If you can do it faster, you can verify complex systems faster.
Simplicity: The math behind this new method is surprisingly simple compared to the old, complicated proofs.
Future Potential: Right now, this works for "Propositional Logic" (simple true/false statements). The authors hope to expand this to "First-Order Logic" (which handles complex sentences with "all," "some," and variables), which would be a massive leap for computer science.

Summary Analogy

Imagine you are trying to translate a message from Language A to Language B.

Old Method: You translate word-by-word, checking a dictionary for every single word, one by one.
New Method: You realize that Language A and Language B share a specific grammar structure. You strip away the unique words of A and B, leaving only the shared grammar. You then rebuild the message using only that shared grammar.

The authors built a machine that does this "grammar stripping" much faster than the old word-by-word method, proving that sometimes, looking at the whole picture (non-binary) is better than looking at just two pieces at a time (binary).

1. Problem Statement

The paper addresses the problem of Craig Interpolation in classical propositional logic. Given two formulas $A$ and $B$ such that $A \to B$ is valid (a tautology) and they share at least one propositional variable, the goal is to find an interpolant $C$ such that:

$A \to C$ is valid.
$C \to B$ is valid.
$C$ contains only propositional variables common to both $A$ and $B$ .

While theoretical methods for finding interpolants exist (e.g., Maehara, Kleene, Smullyan), and practical systems like the HKP-system (Huang-Krajícek-Pudlák) are widely used in automated theorem proving, these existing approaches often rely on binary resolution. The authors argue that binary resolution can be computationally inefficient, requiring many steps to resolve complex formulas. They seek a method that is theoretically simple, constructive, and potentially more efficient by avoiding binary resolution constraints.

2. Methodology

The authors propose a novel method rooted in refutation systems rather than standard proof systems.

A. Theoretical Foundation: Refutation Systems

Instead of proving validity directly, the method uses a "mirror" system that proves non-validity (refutability).

Refutation Axioms: Non-valid formulas (specifically clausal normal forms of rank 0).
Refutation Rules: Rules that preserve non-validity. The core mechanism involves eliminating a pair of complementary literals ( $l$ and $l^*$ ) from a formula.
Key Property (†): A formula $F$ is valid if and only if both its "branches" $F_1$ and $F_2$ (resulting from the elimination of $l, l^*$ ) are valid. This property allows the construction of interpolants recursively.

B. The Interpolant-Finding Procedure

The algorithm operates on formulas in Normal Form ( $X \to Y$ , where $X$ is in CNF and $Y$ is in DNF).

Base Case (Rank 0): If no complementary literals exist between $X$ $X$ and $Y$ $Y$ :
- If $X$ contains a contradiction (empty clause), the interpolant is $\bot$ (False).
- If $Y$ contains a tautology (empty clause in the complement), the interpolant is $\top$ (True).
Recursive Step (Rank > 0):
- Select a pair of complementary literals $l, l^*$ present in distinct clauses of $X \cup Y^*$ .
- Apply refutation rules to split the problem into two sub-problems ( $G_1$ and $G_2$ ) by eliminating $l$ and $l^*$ .
- Recursively find interpolants $I(G_1)$ and $I(G_2)$ .
- Construct the final interpolant $I(G)$ based on the presence of $l$ $l$ and $l^*$ $l^{*}$ in the original sets:
  - If $l, l^*$ are only in $X$ : $I(G) = I(G_1) \land I(G_2)$ .
  - If $l, l^*$ are only in $Y^*$ : $I(G) = I(G_1) \lor I(G_2)$ .
  - If $l, l^*$ are in both: $I(G) = (l \lor I(G_1)) \land (l^* \lor I(G_2))$ .

C. Implementation

The authors implemented this procedure as a Python script.

Design: A proof-of-concept implementation that strictly follows the theoretical algorithm without human-optimized shortcuts (e.g., it does not skip steps even if a tautology is obvious to a human).
Input Handling: Accepts formulas in CNF/DNF. The script handles variable elimination recursively, transforming the problem until base cases are reached.
Simplification: The raw output can be complex (e.g., containing $\bot$ and $\top$ redundantly), but the paper notes that a simplification function can reduce these to standard logical forms.

3. Key Contributions

Non-Binary Resolution Approach: The primary contribution is an interpolant search method that does not rely on binary resolution. Instead, it utilizes a non-binary resolution concept derived from refutation systems, allowing for the elimination of literal pairs in a single step that might require multiple binary resolution steps in other systems.
Constructive Proof: The paper provides a new, simple, and constructive proof of the Extended Interpolation Theorem. The proof is directly translatable into an algorithm.
Practical Implementation: The authors provide a working Python implementation and conduct performance testing, demonstrating that the theoretical approach is viable for practical application.
Efficiency Potential: The authors argue that by avoiding binary resolution, the method can produce interpolants in fewer steps. They cite an example where their method requires 2 steps compared to 5 steps required by the HKP-system for the same problem.

4. Experimental Results

The authors conducted two sets of experiments to test performance:

Experiment 1 (Limited Variables):
- Generated 100,000 formulas with up to 4 variables.
- Results: Average execution time was 0.0032 seconds for small formulas and 0.011 seconds for larger ones.
- Observation: A linear relationship was observed between the execution time and the size of the input formula ( $X \to Y$ ).
Experiment 2 (Extended Variables):
- Tested on 1,000 formulas with up to 10 variables.
- Results: The linear performance trend persisted.
- Data: The size of the resulting interpolants correlated with the input size, and execution times remained low (milliseconds).
Comparison: In a specific benchmark case (from D'Silva [2010]), the proposed method solved the problem in 2 steps, whereas the HKP-system required 5 steps.

5. Significance and Future Work

Theoretical Impact: The paper bridges the gap between abstract refutation systems and practical interpolation, offering a simpler alternative to complex resolution-based proofs.
Practical Impact: The method offers a potential speedup in automated theorem proving and model checking by reducing the number of resolution steps required to find interpolants.
Limitations & Future Directions:
- The current implementation is a "proof-of-concept" and lacks heavy optimization (e.g., formula simplification is not fully integrated into the core loop).
- The primary challenge identified is extending this non-binary refutation method from propositional logic to First-Order Logic (FOL), which would significantly broaden its applicability in formal verification.

In summary, the paper presents a theoretically sound and practically viable alternative to standard interpolation methods, leveraging refutation systems to achieve potentially greater efficiency through non-binary resolution.