Unsimulability, Universality, and Undecidability in the Gizmo Framework

The gizmo framework is a recent development of the gadget framework used for proving computational complexity results of videogames and other motion planning problems. This thesis explores three aspects of the gizmo framework: unsimulability (the inability of one gizmo to simulate another gizmo), universality (the ability of a gizmo to simulate all gizmos in its simulability class), and undecidability (the inability to decide whether a maze made of a gizmo is solvable). We give a proof that the 1- toggle cannot simulate the 2-toggle, as it contains important techniques. We explore a class of gizmos called dicrumbler variants, and give partial results for which ones simulate which others. We give universal gizmos for simulability classes Reg and DAG, and explore the concept of finding all the gizmos that simulate a particular gizmo, with partial results given for the dicrumbler. We show that reachability for a gizmo representing a counter in a counter machine is undecidable, and show several gizmo simulations. We give a proof that generalized New Super Mario Bros. is undecidable using one of the undecidable gizmos.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2023

Author dc:creator

• Ani, Joshua

Advisor dc:contributor.advisor

• Demaine, Erik D.

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright MIT

Licence dc:rights.uri

• http://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/150149

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/150149