Hazard-free Decision Trees

TL;DR

This work extends decision tree complexity to a hazard-free, three-valued logic where inputs may be uncertain, introducing the hazard-free extension $\widetilde{f}$ and studying depth/size tradeoffs. It establishes that hazard-free MUX functions can have exponential depth growth (depth $=2^n+n$) while identifying easy non-degenerate hazard-free functions with small depth, and provides general procedures to derive hazard-free trees from Boolean trees with explicit size bounds. The authors develop hazard-free analogues of sensitivity, block sensitivity, and certificate complexity, prove a hazard-free sensitivity theorem, and show these parameters are polynomially related to hazard-free depth, alongside learning consequences under uncertainty. The work also analyzes size lower/upper bounds via prime implicants/implicates and discusses extensions, structural insights, and open problems, laying groundwork for a robust theory of hazard-free computation with practical implications for learning and circuit design under uncertainty.

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 $\{0, u, 1\}^n$.

Figures (2)

• Figure 1: Decision trees for $\mathsf{MUX}_1$
• Figure 2: Hazard-free decision tree for $f_n$

Theorems & Definitions (68)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Corollary 1.10