Lifting to Parity Decision Trees Via Stifling

TL;DR

This paper demonstrates that deterministic decision tree complexity can be lifted to parity decision tree size complexity using small gadgets satisfying the stifling property, advancing understanding of complexity lifting and proof systems.

Contribution

It introduces the concept of stifling gadgets for lifting to PDT size and shows that constant-size gadgets suffice, improving upon previous randomized and larger gadget approaches.

Findings

Constant-size gadgets enable PDT size lifting.

Systematic reduction of lower bounds from resolution width to size.

First known method to lift deterministic decision tree complexity to PDT size.

Abstract

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple gadgets of constant size, like Indexing on 3 input bits, Inner Product on 4 input bits, Majority on 3 input bits and random functions, satisfy this property. It can be shown that existing randomized communication lifting theorems ([G\"{o}\"{o}s, Pitassi, Watson. SICOMP'20], [Chattopadhyay et al. SICOMP'21]) imply PDT-size lifting. However there are two shortcomings of this approach: first they lift randomized decision tree complexity of $f$, which could be exponentially smaller than its deterministic counterpart when either $f$ is a partial function or even a total search problem. Second, the size of the gadgets in such lifting theorems are as large as logarithmic in the size of the input to $f$. Reducing the gadget size to a constant is an important open problem at the frontier of current research. Our result shows that even a random constant-size gadget does enable lifting to PDT size. Further, it also yields the first systematic way of turning lower bounds on the width of tree-like resolution proofs of the unsatisfiability of constant-width CNF formulas to lower bounds on the size of tree-like proofs in the resolution with parity system, i.e., $\textit{Res}$($\oplus$), of the unsatisfiability of closely related constant-width CNF formulas.