1. Orderings and Bounds Parallel FSM Decomposition Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 10 Update and modified by Marek Perkowski

2. Orderings on Inclusion Relations Inclusion Relation –Reflexive –Anti-symmetric –Transitive Partial Ordering –An inclusion relationship for which some orderings are not specified

3. Orderings on Inclusion Relations Total Ordering Chain, simply ordered set An inclusion relationship for which all orderings are specified A chain is “connected” because

4. Partially Ordered Set (POSET) POSET: a set on which a partial ordering is specified, One way to show a partial ordering is with a Hasse Diagram –AKA directed graph

5. POSET Example* *Lee, 1.1.4

6. Least Element

7. Lower Bound

8. Lower Bound Example From the Hasse Diagram Example, e.g.,

9. Greatest Lower Bound Infimum of a Set, Greatest Lower Bound

10. Greatest Lower Bound Example From the Hasse Diagram Example, e.g.,

11. Greatest Lower Bound Example 10 Is the glb of { 50, 20 } Under the Inclusion Relation Integer Division Given the Set A. The glb of { x, y } Is Written As And Called the Meet of x and y The Meet Is Not the Boolean AND since we are only seeking the greatest value

12. Upper Bound

13. Upper Bound Example From the Hasse Diagram Example, e.g.,

14. Least Upper Bound Supremum of a Set, Least Upper Bound

15. Least Upper Bound Example From the Hasse Diagram Example, e.g., Least upper bound = lub

16. Least Upper Bound Example lub20 Is the lub of { 4, 10 } Under the Inclusion Relation Integer Division Given the Set A. lubThe lub of { x, y } Is Written As Join And Called the Join of x and y Is NotThe Join Is Not the Boolean OR since we are only seeking the greatest value

17. Bounds A Set May Have No Upper or Lower Bound or It May Have Many If the meet and join exist, they are unique Not every POSET has the property that each pair of elements possesses a glb or lub. lattice.If a POSET has a glb and lub for every pair of elements, then they form a lattice.

18. Lattice Example* *Hartsfield, Ringel, Pearls in Graph Theory

19. LatticeLattice Lattice Properties –Idémpotent –Commutative –Associative Lattice: a POSET, L, in Which Any two Elements, X and Y, Have Both a Meet (glb) and a Join (lub).Lattice: a POSET, L, in Which Any two Elements, X and Y, Have Both a Meet (glb) and a Join (lub).

20. Properties of Lattices –Absorptive –Isotone Check these properties on the example at the right

21. –Modular Inequality –Distributive Inequality Check these properties on the example at the right Properties of Lattices

22. Partial Orderings on Partitions Check these properties on the example at the right

23. Partial Orderings on Partitions

24. not necessarily

25. Trivial Partitions Top The partition consisting of a single pi-block containing all states

26. Trivial Partitions Bottom The partition consisting of one state per pi- block

27. Machine Lattice Example* *Lee, p. 287 Closed partitions

28. Machine Lattice Example

29. Operations on Partitions Product, or meet, of 2 Partitions –the intersection of all blocks of two partitions Sum, or join, of 2 Partitions –merging all blocks of two partitions which have one or more states in common

30. Meet of Two Partitions The Intersection of All Blocks of Two Partitions, e.g., Which Just Happens to Be the First Partition, but This Is Not Always the Case nor Is It Required.

31. Join of Two Partitions Merging All Blocks of Two Partitions Which Have One or More States in Common, e.g., which just happens to be the 2nd partition, but this isn’t always the case nor is it required

32. Partition Operation Properties

33. Parallel Decomposition

36. Parallel Decomposition Theorem

37. Parallel Decomp. Example* *Lee, p.291 Output compatible

38. Parallel Decomposition Example Does a parallel decomposition exist? Are the two partitions non-trivial?

39. Parallel Decomp. Example Are the partitions SP? Substitution property

40. Parallel Decomp. Example Are the partitions SP?

41. Parallel Decomp. Example Are the partitions SP?

42. Parallel Decomp. Example Are the partitions SP?

43. Parallel Decomp. Example Are the partitions SP?

44. Parallel Decomp. Example Are the partitions orthogonal?

45. Parallel Decomp. Example states Output states

46. Parallel Decomp. Example Construct an Image Machine (cont’d)

47. Parallel Decomp. Example Construct an Image Machine (cont’d) Copy inputs to columns

48. Parallel Decomp. Example Construct an Image Machine (cont’d) outputs

49. Parallel Decomp. Example Construct an Image Machine We create cartesian products

50. Parallel Decomp. Example Construct an Image Machine (cont’d)

51. Parallel Decomp. Example State Behavior Assignment of Parallel Decomposition

52. Parallel Decomp. Example State Behavior Assignment of Parallel Decomposition

53. Parallel Decomp. Example