# Unordered CNF Games

2021

Dissertation

## Degree Name

Doctor of Philosophy

Computer Science

THOMAS WATSON

MAX GARZON

NIRMAN KUMAR

VINHTHUY PHAN

## Abstract

The classic TQBF problem is to determine who has a winning strategy in a game played on a given CNF formula, where the two players alternate turns picking truth values for the variables in a given order, and the winner is determined by whether the CNF gets satisfied. We study variants of this game in which the variables may be played in any order, and each turn consists of picking a remaining variable and a truth value for it. For the version where the set of variables is partitioned into two halves and each player may only pick variables from his/her half, we prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for unbounded-width CNFs (Schaefer, STOC 1976). For the general unordered version where each variable can be picked by either player, we also prove that the problem is PSPACE-complete for 5-CNFs, in P for 3-CNFs with some restrictions, and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for 6-CNFs (Ahlroth and Orponen, MFCS 2012). For the positive unordered version (where the CNF can only have positive literals), we also prove that the problem is PSPACE-complete for positive 6-CNFs. Previously, it was known to be PSPACE-complete for positive 11-CNFs (Schaefer, STOC 1976). The ErdHos-Selfridge Theorem says that the least number of clauses, in the positive unordered version with k-uniform-CNF, where the player who wants the CNF to be falsified has a winning strategy, is $\Theta(2^k)$. We prove bounds for the general unordered version. It is $\Theta(\sqrt{2}\,^k)$ when the player who wants the CNF to be falsified plays last, but $\Omega(1.5^k)$ and $O(r^k)$ when the player who wants the CNF to be satisfied plays last, where $r=(1+\sqrt{5})/2\approx 1.618$ is the golden ratio.

Data is provided by the student.”

## Library Comment

Dissertation or thesis originally submitted to ProQuest.

Open Access

COinS