Exact reconstruction of finite memory automata with the GSPS

1. Exact reconstruction of finite memory automata with the GSPS And a surprising application to the reconstruction of cellular automata James Nutaro nutarojj@ornl.gov

2. Reconstruction with the GSPS • Begin with one or more time series • Hypothesize a relationship between the variables in these time series • Visualized as a mask with squares for output and circles for input • Construct an input-output model from the mask

3. The reconstruction procedure, step #1 Input observation Output observation v2(t)=f(v1(t),v1(t-1)) B=f(A,A)

4. The reconstruction procedure, step #2 Input observation Output observation v2(t)=f(v1(t),v1(t-1)) B=f(A,B)

5. The reconstruction procedure, step #3 Input observation Output observation v2(t)=f(v1(t),v1(t-1)) A=f(B,A)

6. f may not be deterministic Input observation Output observation A=f(B,B) v2(t)=f(v1(t),v1(t-1)) B=f(B,B)

7. Simulation with the GSPS • Begin with first observation and observations of all data not generated by the model • Generate subsequent observations with the model

8. A simulation with the GSPS, step #1 • First observation is v1(t)=v1(t-1)=B • Outcome is A with 50% change and B with 50 % • A selected at random

9. A simulation with the GSPS, step #2 • Second observation is v1(t)=v1(t-1)=B • Outcome is A with 50% change and B with 50 % • B selected at random

10. Finite memory automata • Has a finite number of inputs , outputs , and states • State transition function and output function are such that a finite history of input and output yield the next outputs • Specifically, there is and and a function such that

11. Examples of finite memory automata 0/0 b a 1/1 0/0 1/1 1/0 b a 0/1 0/0 1/1

12. Not a finite memory automaton Consider the input string 1111111110. What is the outcome? We can’t know. 1/1 b a 0/1 0/0 1/1

13. GSPS and finite memory automata Given a complete set of observations of a finite memory automaton, there is a mask that can exactly reconstruct its input/output behavior. This mask is the one corresponding to the function The number of unique entries in a complete set of observations is at most

14. GSPS and stochastic, finite memory automata Given copies of a complete set of observations of a stochastic, finite memory automaton, the same mask can exactly reconstruct its input/output behavior as . 0.1 0.9 Example of a stochastic automaton with single input, single state, and two outputs. a 1/0 1/1

15. Cellular automata • The cells in a cellular automata are finite memory automata • Given inputs from neighbors, we have • Note that • Follows that each cell has finite memory • Same can be said for stochastic cellular automata

16. Wolfram’s rule #24 • Automaton copies the cell to its left • Assume cell-space wraps around the edges • Only 4 possible observations at each cell • Simulation of this automaton with four cells produces four time series • Two such series are shown on the right • Mask for this automaton is super-imposed on the table • This table has a two complete sets of observations

17. Reconstruction of Wolfram’s rule #24 Reconstruction Simulation

18. Activity in cellular automata • A cellular automaton appears to be highly active when it generate a large number of distinct patterns • The Game of Life is highly active • The trivial cellular automaton with does not appear to be very active

19. Activity and computational costs • The perception of a highly active automaton implies large , large , a long memory, stochasticity, or some combination of these. • The reconstruction problem grows rapidly in all of these dimensions • In the number of entries in the table for • In the amount of data that must be processed Binary cellular automata Rule #24 has entries in its table Game of Life has =1024 Multi-state cellular automata w/o memory m-state game of life has table entries Multi-state cellular automata with memory m-state game of life has table entries

20. Is exact reconstruction of highly active systems feasible? • Problem posed by highly active systems • The necessary data grows exponentially with the variety of input and output • Exponential growth factor increases with the memory • Can quickly reach peta- and exa- scale data • Taming activity: directions for research • Simplification • Preserve essential behaviors while reducing the level of activity • High performance computing • GSPS algorithms implemented for large-scale computing and storage systems

21. In conclusion…a curious example of simplification and HPC (a) Biologically based simulation (b) GSPS simulation based on data produced by (a) Simulated tumor growth at day 90 beginning from 5 occupied pixels on day 1. Expected error in size of the tumor’s bounding box at 90 days is 3 pixels. Simplification: GSPS model has c. 190,000 possible observations at each cell; biological model has millions. Computing: Divide and conquer type parallel algorithm for constructing the GSPS table; required c. 2 days of computing on four cores to process c. 250,000,000 time series. Software for this example @ http://sourceforge.net/projects/gsps/