An Optical Approach to Computation Sama Goliaei Tarbiat Modares University, Tehran, Iran
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An Optical Approach to Computation Sama Goliaei Tarbiat Modares University, Tehran, Iran 1 Content Optical computing Optical 3-satisfiability Optical graph 3-colorability Future works 2 OPTICAL COMPUTING 3 Light Properties • Special physical properties of light – High parallel nature – High speed – Splitting abilities – Many different wavelengths in a single ray 4 Optical Computing In Different Areas • Data transmission • Data storage • Data processing – Optical logic gates – Processors based on light properties 5 Optical Data Processing • Continuous space machine • Images as memory cells • Optical operations – Copy, Fourier transformation,… • Contributions on solving NPC problems • Optical non-deterministic Turing machine • Light rays pass different computational path • Split light rays in decision points • Contributions on solving NPC problems 6 OPTICAL GRAPH 3-COLORABILITY 7 The 3-Sat Problem • Is the given Boolean formula satisfiable? – Conjunctive of some clauses • Clause: disjunction of some literals • Literal: variable or negation of a variable Literal (π₯1 ∨ π₯3 ∨ π₯4 ) ∧ (π₯2 ∨ π₯3 ∨ π₯5 ) ∧ (π₯4 ∨ π₯5 ∨ π₯1 ) ∧ (π₯2 ∨ π₯1 ∨ π₯5 ) Clause 8 Wavelngths as Value-Assignments 9 Wavelngths as Value-Assignments π₯1 π₯2 π₯3 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Divide a wavelength spectrum into 2π sections. Consider each section as a possible value-assignment. 10 Filters ππΉ π drops wavelengths not satisfying F from a given light ray π π ππΉ ππΉ (π ) 11 Literal Selector ο³πx2 π Punch an opaque ribbon where the literal is satisfied 12 Simple Clause Selectors ππΆ Punch an opaque ribbon where the clause is satisfied. 13 Combined Clause Selectors πΆ = π1 ∨ π2 ∨ π3 π π π π ππ1 ππ1 (π ) ππ2 ππ2 (π ) ππ3 ππ3 (π ) ππΆ (π ) 14 CNF Formula Selector ππΆ1∧πΆ2∧β―∧πΆπ ππΆ1 ππΆ2 β― ππΆπ Final Answer Drop wavelengths not satisfying clauses each after other. At the end, remaining wavelengths indicate that the formula is satisfiable. 15 Complexity • Preprocessing – Simple clause selectors • π(π3 2π ) time • π(π3 ) filters – Combined clause selectors • π(ππ2π ) time • π(ππ) filters 16 Complexity (cont’d) • Each problem instance – π π time – π(π) optical devices • π(2π ) long filters – 1.3 π long filters for π = 15 • π(2π ) different wavelengths 17 OPTICAL GRAPH 3-COLORABILITY 18 3-Colorability Problem • Is the given graph 3-colorable? – Graph coloring • Assign a color to each vertex of a graph – Proper 3-coloring • Using at most 3 colors • Different colors for adjacent vertices 19 Graph Colorings as Binary Sequences Write the color of vertices in binary format each after other 01 10 π£1 π£2 01 π£5 π£3 10 π£4 π£6 10 11 1 0 0 11 1 1 0 1 0 0 1 2π bits 20 Light Rays as Binary Numbers • Consider a wide ribbon of light – Divide the ribbon into 22π sections – Assign each section to binary number 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 21 Filter • An opaque ribbon – Punched in some places – Use to drop some binary numbers 22 Filters to Drop Improper Colorings • A valid color for each vertex – π vertex filters • ππ drops rays where 00 is assigned to π£π • Different colors for adjacent vertices – π edge filters • π(π,π) drops light rays where the same color is assigned to π£π and π£π Filters are created in preprocessing phase 23 Is the Graph 3-Colorable? vertex filters Final Answer edge filters 24 Complexity • Preprocessing – π vertex filters • π(π22π ) time π – edge filters 2 2 2π • π(π 2 ) time • Each problem instance – π(π + π) time • π 22π long filters – 1.3 π long filters for π = 15 (square shape filters) • π(22π ) number of photons 25 FUTURE WORKS 26 Future Works • Computational power • Polynomial time solutions for NPC problems • Reduce filter sizes 27 ? ? ? 28
