Index

1. Analog recurrent neural network simulation, Θ(log2n) unordered search with an optically-inspired model of computation

2. Index • Continuous Space Machine Structure • Analog Recurrent Neural Network Simulation and Complexity Result • Unordered Search Algorithm

3. The Continuous Space Machine(CSM) • Definition: : grid dimensions : address of sta, a, and b : addresses of the k input images : the r programming symbols and their addresses : addresses of l output images

4. Instructions of CSM • h and v : h gives the 1-D Fourier transformation in the x-direction, and v gives the 1-D Fourier transformation in the y-direction.

5. Instructions of CSM (II) • * : * gives the complex conjucate of its argument image. where f* is the complex conjucate of f.

6. Instructions of CSM (III) • ∙and +: ∙gives the pointwise complex product of its two argument images, + gives the pointwise complex sum of its two argument images.

7. Instructions of CSM (IV) • ρ: ρ performs amplitude thresholding on its first image argument using its other two real-valued image arguments as lower and upper amplitude thresholds, respectively.

8. Instructions of CSM (V) • ld and st ld parameters p1 to p4 to image at well-known address a. st copies the image at well-known address a to a ‘rectangle’ of images specified by the st parameters p1 to p4.

9. Instructions of CSM (VI) • br and hlt br gives the unconditional jump to the address that the parameter indicates. hlt gives the program termination.

10. Instructions of CSM (Review)

11. The relation betweenimages and data • Complex-valued image A complex-valued image is a function , where [0, 1] is the real unit interval. • Zero Image An image that has value 0 everywhere represents 0.

12. The relation betweenimages and data (II) • Binary symbol image The symbol ψ  is represented by the binary symbol image fψ • Real number image The real number r  R is represented by the real number image fr

13. Two kinds of Binary words • Stack images ld and st instead of push and pop. • List images Load all images at once.

14. Matrix image for ARNN simulation • RC matrix image The RC matrix A with real-valued components aij, is represented by the RC matrix image fA

15. Complexity measure • Time The number of instructions executed in the program. • Space The total space needed to execute the program. • Resolution The maximum resolution of the grid images in the Computation sequences • Range The maximum amplitude precision needed.

16. ARNN ARNNs are finite size feedback first order neural networks wirh real weights. The state of each neuron xi at time t + 1 is given by an update equation of the form: We can take p neurons of xi for output.

17. ARNN (II) • The CSM model can simulate the ARNN The pseudo code is as below

18. ARNN (III) • Complexity If ARNN being simulated is defined for time t = 1, 2, 3, … has M input, N neurons, and k is the number of stacked image elements used to encode the active input to the simulator, the four complexity are Time = O((N + M + 1)t + 1), Space = O(1), Resolution = Max(2k+M-1, 22N-2, 2N+M-2, 2t+N-1), Range = Infinity. (Real value needs infinite bits.)

19. ARNN Conclusion • Because ARNN can be simulated by CSM, the computation power of CSM is at least as strong as TM.

20. Unordered Search(Needle in the haystack problem) L = {w: w  0*10*}, ω L be written as ω = ω0ω1…ωn-1. • Input: ω • Output: Binary representation of i, where ωi=1.

21. Solve NIH in other model • In the classic model, this may be solved in O(n) time naïvely, and it seems that the naive method might have the best performance in this model. • In the quantum computer, this may be solved in Ω( ) with Grover’s work.

22. NIH in the CSM model • Thinking… Use a binary list image to represent ω, and a binary stack image to represent n with log2n bits. Because the ωhas only one non-zero point, we can use some convenient instructions in CSM to solve this problem in shorter time…

23. Pseudo Code ofθ(log2n) unordered search