Inductive modeling of finite memory automata

Exact reconstruction of finite memory automata with the GSPSAnd a surprising application to the reconstruction of cellular automata

James Nutaronutarojj@ornl.govReconstruction with the GSPSBegin with one or more time seriesHypothesize a relationship between the variables in these time seriesVisualized as a mask with squares for output and circles for inputConstruct an input-output model from the masktv1v2v1v27ABAB6ABAB5BABB4ABBA3BBBA2BAAB1BBAAv1(t)v1(t-1)v2(t)%AAB100 (3/3)ABB100 (3/3)BAA100 (2/2)BBA50 (2/4)B50 (2/4)The reconstruction procedure, step #1tv1v2v1v27ABAB6ABAB5BABB4ABBA3BBBA2BAAB1BBAAv1(t)v1(t-1)v2(t)%AAB100 (1/1)Input observationOutput observationv2(t)=f(v1(t),v1(t-1))B=f(A,A)The reconstruction procedure, step #2tv1v2v1v27ABAB6ABAB5BABB4ABBA3BBBA2BAAB1BBAAv1(t)v1(t-1)v2(t)%AAB100 (1/1)ABB100 (1/1)Input observationOutput observationv2(t)=f(v1(t),v1(t-1))B=f(A,B)The reconstruction procedure, step #3tv1v2v1v27ABAB6ABAB5BABB4ABBA3BBBA2BAAB1BBAAv1(t)v1(t-1)v2(t)%AAB100 (1/1)ABB100 (1/1)BAA100 (1/1)Input observationOutput observationv2(t)=f(v1(t),v1(t-1))A=f(B,A)f may not be deterministictv1v2v1v27ABAB6ABAB5BABB4ABBA3BBBA2BAAB1BBAAv1(t)v1(t-1)v2(t)%AAB100 (1/1)ABB100 (1/1)BAA100 (1/1)BBA50 (1/2)B50 (1/2)Input observationOutput observationv2(t)=f(v1(t),v1(t-1))B=f(B,B)A=f(B,B)Simulation with the GSPSBegin with first observation and observations of all data not generated by the modelGenerate subsequent observations with the modeltv1v27A6B5A4A3BB2BA1BBv1(t)v1(t-1)v2(t)%AAB100 (3/3)ABB100 (3/3)BAA100 (2/2)BBA50 (2/4)B50 (2/4) A simulation with the GSPS, step #1First observation is v1(t)=v1(t-1)=BOutcome is A with 50% change and B with 50 %A selected at randomtv1v27A6B5A4A3B2BA1BBv1(t)v1(t-1)v2(t)%AAB100 (3/3)ABB100 (3/3)BAA100 (2/2)BBA50 (2/4)B50 (2/4) A simulation with the GSPS, step #2Second observation is v1(t)=v1(t-1)=BOutcome is A with 50% change and B with 50 %B selected at randomtv1v27A6B5A4A3BB2BA1BBv1(t)v1(t-1)v2(t)%AAB100 (3/3)ABB100 (3/3)BAA100 (2/2)BBA50 (2/4)B50 (2/4)Finite memory automataExamples of finite memory automataab0/01/11/10/0ab1/01/10/10/0Not a finite memory automatonab1/11/10/10/0Consider the input string 1111111110. What is the outcome? We cant know.GSPS and finite memory automatatxy76Given 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 mosttxy7GSPS and stochastic, finite memory automataa1/11/00.90.1Example of a stochastic automaton with single input, single state, and two outputs. Cellular automataWolframs rule #24 54321txyxy5WBBW4BBWW3BWWB2WWBB1WBBWReconstruction of Wolframs rule #24 txyxy5WBBW4BBWW3BWWB2WWBB1WBBWx(t-1)y(t-1)y(t)WWWWBWBWBBBB54321ReconstructionSimulationActivity in cellular automata

Activity and computational costsIs exact reconstruction of highly active systems feasible?Problem posed by highly active systemsThe necessary data grows exponentially with the variety of input and outputExponential growth factor increases with the memory Can quickly reach peta- and exa- scale data

Taming activity: directions for researchSimplificationPreserve essential behaviors while reducing the level of activityHigh performance computingGSPS algorithms implemented for large-scale computing and storage systemsIn conclusiona curious example of simplification and HPC

Simulated tumor growth at day 90 beginning from 5 occupied pixels on day 1. Expected error in size of the tumors 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/(a) Biologically based simulation(b) GSPS simulation based on data produced by (a)