The Computational Complexity of Evil Hangman

TL;DR

This paper investigates the computational difficulty of playing optimally in Evil Hangman, showing that it is computationally hard even with perfect knowledge, through reductions from known hard problems.

Contribution

It proves that optimal Evil Hangman gameplay is computationally difficult across various language classes, using reductions from established NP-hard problems.

Findings

Greedy strategies can be arbitrarily suboptimal.

Optimal play in Evil Hangman is computationally hard.

Hardness holds for multiple language classes, including Turing computable.

Abstract

The game of Hangman is a classical asymmetric two player game in which one player, the setter, chooses a secret word from a language, that the other player, the guesser, tries to discover through single letter matching queries, answered by all occurrences of this letter if any. In the Evil Hangman variant, the setter can change the secret word during the game, as long as the new choice is consistent with the information already given to the guesser. We show that a greedy strategy for Evil Hangman can perform arbitrarily far from optimal, and most importantly, that playing optimally as an Evil Hangman setter is computationally difficult. The latter result holds even assuming perfect knowledge of the language, for several classes of languages, ranging from Finite to Turing Computable. The proofs are based on reductions to Dominating Set on 3-regular graphs and to the Membership problem, combinatorial problems already known to be computationally hard.