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Hadi Hosseini CS 6805 Logic Synthesis Faculty of Computer Science Spring 2009

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Introduction Interaction and Switch gates Mirrors and delays N-variable gates OR gate Invert Interaction gate Perkowskis full adder Constraints NAND Algorithm Proposed half adder Toffoli gate Conclusion & future work BBM design2

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Introduced by Edward Fredkin & Thomas Toffoli Reversible method of computing using elastic collisions (non- dissipative) Rules: the exact laws used in Newtonian kinematics Interacting objects: Balls Walls two-dimensional grid and hard balls velocity of one unit of space per unit time interval Ball => 1 No ball => 0 3BBM design

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Interaction gate AB A B 4BBM design

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6 DelaySideway shift Nontrivial crossover

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Mirrors (walls) : changing the direction or path Synchronization No clock signal Timing: Inputs & outputs to each level of gate 7BBM design

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Attempts to build a BBM OR gate With just using walls & balls We can never redirect two balls into the same trajectory 9BBM design

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10BBM design ABAB 000000 010100 100010 111001 z1 z3 z2 z4 A B Only disjoint inputs are OR-ed

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11BBM design C AB A B Carry 0 Sum A B B (A+B)C AB+AXC +BXC

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Physical Constraints No fan-out : We cant split a ball back-feeds are not always possible Only disjoint inputs can be OR-ed 12BBM design

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Divide the circuit into different levels and mark the inputs and outputs of each level Where ever there is an AND gate, substitute it with the BBM interaction gate (AND gate) replace each gate with its NAND gate equivalent Simplify the system by deleting adjacent pairs of NOT gates (marked X above). Replace all the NAND gates with equal inputs with a BBM NOT gate in each level replace each NAND gate with a BBM NAND gate 13BBM design

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Conclusion Physical aspect of BBM Fan in/out Redirecting Space & time Keep them reversible Future Work: Toffoli gates Analyzing positive XOR representation & NAND representation General methodology to build BBM circuits and feasibility 17BBM design

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[1] Edward Fredkin and Tommaso Toffoli. Conservative logic. International Journal of Theoretical Physics, 21:219–253, 1982. [2] T. Toffoli, Reversible Computing,Tech. Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (1980). [3] T. Toffoli, Computation and Construction Universality of Reversible Cellular Automata,J. Comput. System Sci. 15 (1977), 213–231. [4] T. Toffoli, Reversible computing, Tech. Memo MIT/LCS/TM-151, Mit Lab. for Comp. Sci. (1980) [Toffoli gate, reversible automata].Reversible computing [5] N. Margolus. Universal cellular automata based on the collision of soft spheres. New constructions in Cellular Automata, Oxford Press (2003). [6] Quantum Computation by.Norman Margolus [7] http://en.wikipedia.org/wiki/Billiard-ball_computer (accessed 2009- 03-10)http://en.wikipedia.org/wiki/Billiard-ball_computer [8] Marek Perkowski, Portland state university 18BBM design

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