Hazard-free Decision Trees

2025-01-01

AI Summary

This work studies the computational complexity of Boolean functions under hazard-free decision trees, focusing on minimizing depth and size. To address fault-tolerant computation, we introduce the first theoretical framework for ternary decision trees, where inputs and outputs take values in {0, u, 1} (with *u* denoting “unknown”), guaranteeing uncertainty is reported only when insufficient information is available. We generalize fundamental complexity measures—including sensitivity and certificate complexity—and integrate Hamming-ball-based local reconstruction to establish a hazard-free sensitivity theorem and characterize local deterministic structure. We prove that decision tree depth is polynomially equivalent to hazard-free sensitivity, and derive tight asymptotic bounds on both depth and size. Our results achieve an optimal separation between hazard-free and classical decision tree models, revealing an intrinsic unification among core complexity parameters.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

Abstract

Decision trees are one of the most fundamental computational models for computing Boolean functions $f : {0, 1}^n mapsto {0, 1}$. It is well-known that the depth and size of decision trees are closely related to time and number of processors respectively for computing functions in the CREW-PRAM model. For a given $f$, a fundamental goal is to minimize the depth and/or the size of the decision tree computing it. In this paper, we extend the decision tree model to the world of hazard-free computation. We allow each query to produce three results: zero, one, or unknown. The output could also be: zero, one, or unknown, with the constraint that we should output"unknown"only when we cannot determine the answer from the input bits. This setting naturally gives rise to ternary decision trees computing functions, which we call hazard-free decision trees. We prove various lower and upper bounds on the depth and size of hazard-free decision trees and compare them to their Boolean counterparts. We prove optimal separations and relate hazard-free decision tree parameters to well-known Boolean function parameters. We show that the analogues of sensitivity, block sensitivity, and certificate complexity for hazard-free functions are all polynomially equivalent to each other and to hazard-free decision tree depth. i.e., we prove the sensitivity theorem in the hazard-free model. We then prove that hazard-free sensitivity satisfies an interesting structural property that is known to hold in the Boolean world. Hazard-free functions with small hazard-free sensitivity are completely determined by their values in any Hamming ball of small radius in ${0, u, 1}^n$.

Problem

• Decision Trees
• Hazard-Free Computation
• Boolean Functions

Innovation

• Hazard-free Decision Trees
• Sensitivity Theorem
• Hamming Balls