1. Generalization of the Ashenhurst-Curtis decomposition model to MV functions

2. This kind of tables known from Rough Sets, Decision Trees, etc Data Mining

3. At every step many decompositions exist
Decomposition is hierarchical
At every step many decompositions exist

4. Decomposition Algorithm
Find a set of partitions (Ai, Bi) of input variables (X) into free variables (A) and bound variables (B)
For each partitioning, find decomposition F(X) = Hi(Gi(Bi), Ai) such that column multiplicity is minimal, and calculate DFC
Repeat the process for all partitioning until the decomposition with minimum DFC is found.

5. Algorithm Requirements
Since the process is iterative, it is of high importance that minimization of the column multiplicity index is done as fast as possible.
At the same time, for a given partitioning, it is important that the value of the column multiplicity is as close to the absolute minimum value

6. Generalizations of the Ashenhurst-Curtis decomposition model to MV relations

7. Compatibility of columns for Relations is not transitive !
C0, C1 and C2 are pairwise compatible but not compatible as a group
This is an important difference between decomposing functions and relations

8. Four Extensions of Ashenhurst Decomposition

9. Fist Extension: Dealing with noise in data

10. Compatibility graph construction for data with noise
Numbers treated as symbols
Incompatible in 3 of 4 cases
compatible
Kmap
Compatibility Graph for Threshold 0.75
Compatibility Graph for Threshold 0.25
Figure 12:
Create a graph with edges labeled by percent of incompatibility
Remove nodes with values above the selected threshold
Create a graph by disregarding labels.
Use this graph for coloring, follow the standard procedure
The method can be repeated for various values of thresholds
All edges with values above 0.25 are removed

11. Second Extension: Dealing with numerical (metric) versus symbolic (nominal) data

12. Compatibility graph for metric data
Numbers treated as metric (numbers)
Compatibility Graph for nominal (symbolic) data
Compatibility Graph for metric data
Figure 13:
Kmap
If the difference is only 1 then nodes are treated as compatible
The method can be repeated for various values of thresholds of differences

13. Third Extension: Dealing with data in contingency tables

14. MV relations can be created from contingency tables
THRESHOLD 50
THRESHOLD 70
Number of cases for the given combination of inputs
The method can be repeated for various values of thresholds

15. Fourth Extension: Removing some assumption of AC method: Column multiplicity

16. Example of decomposing a “Curtis non-decomposable” function.
Function f in Kmap has a multiplicity of 3
Function f in Kmap cannot be decomposed according to Curtis decomposition
It can be encoded by two function values
One of them is AND , other is OR, they are both “cheap functions”. (see c)
Now we can create a new table (see d)
Next we can create circuit from e
Next we create table from f
Next we create circuit a

17. APPLICATIONS FPGA SYNTHESIS VLSI LAYOUT SYNTHESIS
DATA MINING AND KNOWLEDGE DISCOVERY
MEDICAL DATABASES
EPIDEMIOLOGY
ROBOTICS
FUZZY LOGIC DECOMPOSITION
CONTINUOUS FUNCTION DECOMPOSITION

18. Importance of Intepretability in decomposition methods
You do not only want to have high accuracy.
You want to be able to understand and interpret the results.

19. Example of an application
Knowledge discovery in data with no error

20. Michalski’s Trains
Figure 5: Trains problem

21. Michalski’s Trains Multiple-valued functions.
There are 10 trains, five going East, five going West, and the problem is to nd the simplest rule which, for a given train, would determine whether it is East or Westbound.
The best rules discovered at that time were:
1. If a train has a short closed car, then it Eastbound and otherwise Westbound.
2. If a train has two cars, or has a car with a jagged roof then it is Westbound and otherwise Eastbound.
Espresso format. MVGUD format.

22. Michalski’s Trains

23. Michalski’s Trains

24. Michalski’s Trains Attribute 33: Class attribute (east or west)
direction (east = 0, west = 1)
The number of cars vary between 3 and 5. Therefore, attributes referring to properties of cars that do not exist (such as the 5 attributes for the “5th" car when the train has fewer than 5 cars) are assigned a value of “-".
Applied to the trains problem our program discovered the following rules:
1. If a train has triangle next to triangle or rectangle next to triangle on adjacent cars then it is Eastbound and otherwise Westbound.
2. If the shape of car 1 (s1) is jagged top or open rectangle or u-shaped then it is Westbound and otherwise Eastbound.

25. Another examples of meaning of intermediate variables
Figure 5: Trains problem

26. MV benchmarks: zoo
Figure 9:

27. MV benchmarks: shuttle
Figure 6:

28. MV benchmarks: lenses
Figure 7:

29. MV benchmarks: breastc
Figure 8:

30. Evaluation of variants
Decomposition versus DNF versus Decision Tree

31. Evaluation of results for learning
1. Learning Error
2. Occam Razor , complexity

32. A machine learning approach versus several logic synthesis approaches
Decomposition has small error.
Decomposition needs less samples to learn

33. Finding the error, DFC, and time of the decomposer on the benchmark kdd5.

34. The average error over 54 benchmark functions.

35. Example of a application
Gait control of a robot puppet for Oregon Cyber Theatre

37. Model with a gripper

38. Model with an internet camera

39. camera camera stamp Universal Logic Machine radio DecStation radio
Spider I control
camera
Universal Logic Machine
radio
camera
DEC PERLE
DecStation
stamp
radio
radio
Turbochannel

40. The following formula describes the exact motion of the shaft of every servo.
Theta, the angle of the servo’s shaft, is a function of time.
Theta naught is a base value corresponding to the servo’s middle position. Theta naught will be the same for all the servos.
‘A’ is called the amplitude of the oscillation. It relates to how many degrees the shaft is able to rotate through.
Omega relates to how fast the servo’s shaft rotates back and forth. Currently, for all servos, there are only four possible value that omega may take
Phi is the relative phase angle.

41. And a familiar table again

42. Medical Applications
Ovulation prediction (Perkowski, Grygiel, Pierzchala, students)
Melanoma cancer (Anika, Lizzy Zhao)
Breast cancer (Clemen)
Drought in California (Robin)
Endangered species in Oregon (Nashita)
HIV (Leo)
Forest Fire in Oregon (Leo)
Poisonous spiders in Oregon
Poisonous Mushrooms
Robin on heart bits

43. Conclusion Stimulated by practical hard problems: Machine Learning,
Data Mining.
Robotics (hexapod gaits, face recognition),
Field Programmable Gate Arrays (FPGA),
Application Specific Integrated Circuits (ASIC)
High performance custom design (Intel)
Very Large Scale of Integration (VLSI) layout-driven synthesis for custom processors,

44. Conclusion Developed 1989-present
Intel, Washington County epidemiology office, Northwest Family Planning Services, Lattice Logic Corporation, Cypress Semiconductor, AbTech Corp., Air Force Office of Scientific Research, Wright Laboratories.
A set of tools for decomposition of binary and multi-valued functions and relations.
Extended to fuzzy logic, reconstructability analysis and real-valued functions.

45. Conclusion
Our recent software allows also for bi-decomposition, removal of vacuous variables and other preprocessing/postprocessing operations.
Variants of our software are used in several commercial companies.
The applications of the method are unlimited and it can be used whenever decision trees or artificial neural nets are used now.
The quality of learning was better than in the top decision tree creating program C4.5 and various neural nets.
The only problem that remains is speed in some applications.

46. Conclusion On our WWW page, http:// www.ee.pdx.edu/~cfiles/papers.html
the reader can find many benchmarks from various disciplines that can be used for comparison of machine learning and logic synthesis programs.
We plan to continue work on decomposition and its various practical applications such as epidemiology or robotics which generate large real-life benchmarks.
We work on FPGA-based reconfigurable hardware accelerator for decomposition to be used on a mobile robot.

47. Questions and Problems
What is Ashenhurst Decomposition?
What is Curtis Decomposition?
Give example of decomposition of MV data.
What is a difference of decomposing functions and relations?
Discuss one example of dealing with noise in data while using decomposition.
Discuss decomposition of numeric (metric) data.
Discuss how to use decomposition to deal with contingency data.

48. Questions and Problems
Discuss one generalization to Ashenhurst/Curtis Decomposition?
Explain the problem of intepretability of results.
Find a new solution to Michalski’s Trains Problem.
Discuss how to compare DNF, Decision Tree, Decomposition and other ML methods.
Discuss how to use decomposition to learn gaits for a hexapod robot.
Find a new application of A/C decomposition.