GPT-5 Pro Challenges NICD-with-Erasures Majority Optimality with New Counterexample

GPT-5 Pro, the latest iteration of OpenAI's language model series, has made a significant contribution to theoretical computer science by finding a counterexample to the NICD-with-erasures majority optimality conjecture. This conjecture is part of a list of open problems compiled by the Simons Institute for the Theory of Computing (Simons list, p.25). The discovery not only highlights the model's capabilities in solving complex mathematical problems but also opens new avenues for research in this area.

Technical Details and Implications

The NICD-with-erasures majority optimality conjecture posits that under certain conditions, the majority function is optimal for a specific type of decision-making problem. Specifically, it deals with functions ( f: {-1, 1}^n \to {-1, 1} ) and their expected values when some inputs are erased (i.e., set to zero). The conjecture suggests that the majority function should have the highest expected value in these scenarios.

However, GPT-5 Pro found a counterexample at ( p = 0.4 ) and ( n = 5 ), where the function:

[ f(x) = \text{sign}(x_1 - 3x_2 + x_3 - x_4 + 3x_5) ]

yields an expected value of ( E|f(x)| = 0.43024 ), which is higher than the best majority function's expected value of ( 0.42904 ).

Key Points

• Counterexample Discovery: GPT-5 Pro identified a specific function that outperforms the majority function under the given conditions.
• Parameters:
  • ( p = 0.4 ): The probability that an input is erased (set to zero).
  • ( n = 5 ): The number of inputs.
• Function Definition: The counterexample function is defined as ( f(x) = \text{sign}(x_1 - 3x_2 + x_3 - x_4 + 3x_5) ), where the sign function outputs ( 1 ) if the argument is positive and ( -1 ) otherwise.
• Expected Value Comparison:
  • The expected value of the counterexample function: ( E|f(x)| = 0.43024 ).
  • The best majority function's expected value: ( 0.42904 ).

Significance for Practitioners

This discovery has several implications for researchers and practitioners in theoretical computer science and AI:

• Challenging Assumptions: It challenges the long-standing assumption that the majority function is optimal in all cases, which could lead to a re-evaluation of existing algorithms and models.
• Algorithm Design: The counterexample provides a new benchmark for designing more efficient decision-making functions in scenarios with input erasures.
• Model Capabilities: It demonstrates the potential of large language models like GPT-5 Pro to solve complex mathematical problems, suggesting they could be valuable tools in theoretical research.

Implementation and Verification

To verify this result, researchers can implement the function ( f(x) ) and simulate the expected value calculation under the given conditions. The steps would involve:

1. Generating Input Vectors: Create random input vectors ( x \in {-1, 1}^5 ).
2. Applying Erasures: Randomly set some elements of ( x ) to zero with probability ( p = 0.4 ).
3. Evaluating the Function: Compute ( f(x) ) for each modified input vector.
4. Calculating Expected Value: Average the absolute values of ( f(x) ) over a large number of trials to estimate ( E|f(x)| ).

Conclusion

The discovery by GPT-5 Pro of a counterexample to the NICD-with-erasures majority optimality conjecture is a significant milestone in theoretical computer science. It not only challenges existing assumptions but also opens new avenues for research and algorithm design. For practitioners, this finding underscores the importance of continuously testing and validating theoretical models,