Folding Programs Erik Demaine CSAIL, MIT

Geometric Folding Algorithms Linkages Paper Polyhedra

Hinged Dissection Universality [Abbott, Abel, Charlton, Demaine, Demaine, Kominers 2008] Theorem: For any finite set of polygons of equal area, there is a hinged dissection that can fold into any of the polygons, continuously without self-intersection Number of pieces is pseudopolynomial (optimal) Working on pseudopolynomial, which is best you can hope for (even without hinging)

Folding Meets Computing Algorithms Computing Geometry Folding Conformal Computing Physics Folding Algorithms Conformal Computing

Mesoscale Hinged Dissection from http://pubs.acs.org/isubscribe/journals/jacsat/124/i49/figures/ja021043df00002.gif from http://pubs.acs.org/isubscribe/journals/jacsat/124/i49/figures/ja021043df00002.html from http://pubs.acs.org/cgi-bin/article.cgi/jacsat/2002/124/i49/html/ja021043d.html from http://dx.doi.org/10.1021/ja021043d [Mao, Thalladi, Wolfe, Whitesides, Whitesides 2002]

DNA Folding Synthetic DNA to fold into desired polygon [Rothemund — Nature 2006] http://www.msnbc.msn.com/id/11829347/ ~100nm

Self-Assembly & Nanomanufacturing Algorithms for programming matter into arbitrary shapes [Cheung, Demaine, Griffith, Jacobson et al. 2008]

Self-Assembly of Sorting Circuit

Self-Assembly of Sorting Circuit

Folding Programs Folding assembly process is useful for more than assembly Can reprogram an already-assembled device by “feeding” program (circuit)

Operating System High-level goals: Challenges: Load programs/circuits Destroy programs/circuits Communication channels between running programs/circuits Challenges: Embedding programs onto hardware Routing “reprogramming” instructions Feedback from program to OS (spawn) xxx figure

Asynchronous Logic Automaton 2 bits = which gate (4) 3 bits for input 1 (8 neighbors including diagonal) 3 bits for input 2 Bit = 0 wire and 1 wire Neither occupied => ready 0 wire occupied => 0 input 1 wire occupied => 1 input Both wires occupied: unused

Reprogrammable Asynchronous Logic Automaton Simple solution: Bigger rule table, more states, more wires Ideally: Simple rules Small number of additional states Use 0+1 for reprogramming; no extra wires Somewhere in the middle: von Neumann’s Universal Constructor (self-replicating cellular automaton)

One Solution Regular state (AND/OR/etc.) Reset state Forward state 0+1 switches to reset state Reset state Accept AND/OR/etc.; programs next regular state Then accept N/E/W/S forwarding direction Once programmed, auto-switch to forward state Forward state Send 0+1 to N/E/W/S to turn next cell to reset state Then pass on all input bits (0/1) to next cell 0+1 switches to ready state, which ignores 0/1 inputs Last node in chain sends back final 0+1 stream to switch ready state to regular state

Ongoing Work Converting data to programs Cleaner integration of logic & programming Hierarchical programming Simulation Design & develop full operating system within this framework