Approximate Degree Composition for Recursive Functions

2024-07-11

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This work investigates the preservation of approximation degree under composition—particularly recursive self-composition—of Boolean functions. We address the long-standing open problem of whether approximation degree composes strictly for recursively composed structures where either the inner or outer function is the $d$-fold self-composition of a base function $h$ (with $h$ neither AND nor OR) and $d = Omega(log log n)$. We establish, for the first time, that strict composition holds: the approximation degree of the composed function equals the product of the approximation degrees at each level. Technically, we construct a lower bound via a circuit with intermediate majority gates and combine polynomial approximation with algebraic elimination to efficiently decouple the dependence on majority gates. Our result introduces the new paradigm of “recursion-depth-driven composition of approximation degree,” significantly expanding the known class of composable functions and resolving a central open question in this line of research.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: egin{itemize} item The outer function $f:{0,1}^n o {0,1}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= Omega(loglog n)$. item The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= Omega(loglog n)$, where $n$ is the arity of the outer function. end{itemize} In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be emph{efficiently eliminated} if the inner or outer function is a recursive function.

Problem

• Boolean functions
• Complexity approximation
• Recursive calls

Innovation

• Recursive Function Approximations
• Majority Voting Assistance
• Efficient Composition Conditions