Proving helltaker is PSPACE-complete, because why not lmao.

Todo

• prove containment in PSPACE
• prove pspace-hardness
• prove pspace-completeness

Containment in PSPACE

Proof: The states of The Helltaker and every entity can be described by a polynomial amount of space, since entities are confined to a polynomial-sized area. Thus, an algorithm which guesses The Helltaker's inputs on every frame and simulates Helltaker, until The Helltaker either reaches the goal location or a state is repeated, requires only polynomial to function. Thus, by savage's theorem, since NPSPACE=PSPACE, we know Helltaker is contained in PSPACE.

Containment in PSPACE-Hardness & PSPACE-completness

Via the planar motion problem, we need to construct a gadget that simulates any one door. Since it's generally easy to build hallways and branching hallways in helltaker, as we are working with an n x n grid, this is generally easy to see. You can very reasonably construct an pen-close traverse door to show PSPACE-hardness. From this gadget, PSPACE-completeless also follows.

Citations

• Planar Motion Problem: erikdemaine.org/papers/Doors_FUN2020/paper.pdf
• arxiv.org/pdf/2005.03192.pdf
• Celeste is PSPACE-hard: arxiv.org/pdf/2211.11839.pdf
• Block pushing: erikdemaine.org/papers/GadgetsChecked_FUN2022/paper.pdf
• erikdemaine.org/papers/Celeste_TJM/paper.pdf