2026-09-20T08:57:11Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1501492023-04-01T03:06:00Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Demaine, Erik D.
Ani, Joshua
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2023-03-31T14:35:50Z
2023-03-31T14:35:50Z
2023-02
2023-02-27T18:43:21.420Z
https://hdl.handle.net/1721.1/150149
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.
M.Eng.
Massachusetts Institute of Technology
In Copyright - Educational Use Permitted
Copyright MIT
http://rightsstatements.org/page/InC-EDU/1.0/
Unsimulability, Universality, and Undecidability in the Gizmo Framework
Thesis
application/pdf
Master
Master of Engineering in Electrical Engineering and Computer Science
Publication
unknown
unknown
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<Abstract>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.</Abstract>
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