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Pseudodeterministic Communication Complexity

Mika Göös, Nathaniel Harms, Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov, Weiqiang Yuan

TL;DR

This work constructs a partial two-party boolean function with randomized communication complexity $O(\log n)$ whose every total completion has randomized complexity $n^{\Omega(1)}$, establishing an exponential separation between randomized and pseudodeterministic communication protocols. The authors extend Gavinsky’s lower bound from parity decision trees to the full communication model via a white-box lifting approach using the Inner Product gadget, and develop a robust protocol-transformation and density-restoring framework. Central to the argument are the Stage Lemma and Closeness Lemma, which track a deficit/entropy potential across iterative stages and ensure that sprinkling fixed-1 gadgets remains undetectable by shallow protocols. The construction and analysis yield explicit insights into the structure of pseudodeterministic search in communication, with open directions including tighter bounds for GapHD, TFNP-like, and BPP-verifiable variants, and connections to monochromatic-rectangle phenomena. The results demonstrate that FP_s is strictly smaller than FBPP in the partial-function setting, highlighting fundamental limits of pseudodeterminism in efficient communication.

Abstract

We exhibit an $n$-bit partial function with randomized communication complexity $O(\log n)$ but such that any completion of this function into a total one requires randomized communication complexity $n^{Ω(1)}$. In particular, this shows an exponential separation between randomized and \emph{pseudodeterministic} communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

Pseudodeterministic Communication Complexity

TL;DR

This work constructs a partial two-party boolean function with randomized communication complexity whose every total completion has randomized complexity , establishing an exponential separation between randomized and pseudodeterministic communication protocols. The authors extend Gavinsky’s lower bound from parity decision trees to the full communication model via a white-box lifting approach using the Inner Product gadget, and develop a robust protocol-transformation and density-restoring framework. Central to the argument are the Stage Lemma and Closeness Lemma, which track a deficit/entropy potential across iterative stages and ensure that sprinkling fixed-1 gadgets remains undetectable by shallow protocols. The construction and analysis yield explicit insights into the structure of pseudodeterministic search in communication, with open directions including tighter bounds for GapHD, TFNP-like, and BPP-verifiable variants, and connections to monochromatic-rectangle phenomena. The results demonstrate that FP_s is strictly smaller than FBPP in the partial-function setting, highlighting fundamental limits of pseudodeterminism in efficient communication.

Abstract

We exhibit an -bit partial function with randomized communication complexity but such that any completion of this function into a total one requires randomized communication complexity . In particular, this shows an exponential separation between randomized and \emph{pseudodeterministic} communication protocols. Previously, Gavinsky (2025) showed an analogous separation in the weaker model of parity decision trees. We use lifting techniques to extend his proof idea to communication complexity.

Paper Structure

This paper contains 28 sections, 26 theorems, 89 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our techniques
  3. History and Motivations
  4. Open Problems
  5. Warm-Up and Proof Overview
  6. Warm-Up: Lower Bound for Decision Trees
  7. Proof sketch.
  8. Full proof (skippable on first reading).
  9. Proof Overview
  10. Stage Lemma:
  11. Closeness Lemma:
  12. Proof of the Main Theorem
  13. Proving the
  14. Proof of the \ref{['lemma:safe-stage-lemma']}
  15. Proof of the \ref{['lemma:unsafe-stage-lemma']}
  16. ...and 13 more sections

Key Result

Theorem 1

There is a partial communication problem $\{0,1\}^n\times\{0,1\}^n\to\{0,1,*\}$ with randomized communication complexity $O(\log n)$ but pseudodeterministic communication complexity $\Omega(\sqrt{n})$.

Figures (3)

  • Figure 1: Decision tree stages
  • Figure 2: Communication stages. The second stage is unsafe, we assign too many coordinates
  • Figure 3: Outline of the proof of the Stage Lemma.

Theorems & Definitions (52)

  • Theorem 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4: CLV19HHH23
  • Theorem 5: Simplified version of Gavinsky2025
  • Lemma 6: Stage Lemma for Decision Trees
  • proof : Proof of \ref{['thm:query-main']} given \ref{['lemma:dt-stage']}
  • Lemma 7: Closeness Lemma for Decision Trees
  • proof
  • proof : Proof of \ref{['lemma:dt-stage']}.
  • ...and 42 more