Researcher ORCID Identifier
Open Access Senior Thesis
Bachelor of Science
Arthur T. Benjamin
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
@2022 Kye W Shi
In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in 𝐏 are thought of as 0-turn games, 𝐍𝐏 as 1-turn “puzzle” games, and in general 𝚺ₖ𝐏 as 𝑘-turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of 𝑘-turn CIRCUIT SATISFIABILITY games, whose 𝚺ₖ𝐏-completeness is assumed from prior work (we briefly justify this assumption on intuitive grounds, but no proof is given).
In the second half, we introduce and explore the properties of a novel family of games called the 𝑘-turn GRAPH 3-COLORABILITY games. By embedding boolean circuits in proper graph 3-colorings, we construct reductions from 𝑘-turn CIRCUIT SATISFIABILITY games to 𝑘-turn 3-COLORABILITY games, thereby showing that 𝑘-turn 3-COLORABILITY is 𝚺ₖ𝐏-complete
Finally, we conclude by discussing possible future generalizations of this work, vis-à-vis extending arbitrary 𝐍𝐏-complete puzzles to interesting 𝚺ₖ𝐏-complete games.
Shi, Kye, "Games for One, Games for Two: Computationally Complex Fun for Polynomial-Hierarchical Families" (2022). HMC Senior Theses. 259.