Lifting theorems for parity decision trees

15^th November 2022, 15:00 Meeting room 101 Ashton Building
Nikhil Mande
CWI, Amsterdam

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 o g as long as the gadget g satisfies a property that we call stifling. It can be shown that existing randomized communication lifting theorems (Goos, 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, of the unsatisfiability of closely related constant-width CNF formulas.

Based on joint work with Arkadev Chattopadhyay (TIFR, Mumbai), Swagato Sanyal (IIT, Kharagpur) and Suhail Sherif (Vector Institute, Toronto)