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Morphisms of State Machines

This lecture covers various types of relations, properties, and operations related to semi-groups, homomorphisms, isomorphisms, and machine equivalences in sequential machine theory. Concepts such as single-valued systems, groupoids, monoids, and closed binary operations are explained.

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Morphisms of State Machines

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  1. Morphisms of State Machines Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 8 Updated and adapted by Marek Perkowski

  2. Notation

  3. Free SemiGroup

  4. String or Word

  5. Concatenation

  6. Partition of a Set • Properties • pi are called “pi-blocks” of a partition, (A)

  7. Types of Relations • Partial, Binary, Single-Valued System • Groupoid • SemiGroup • Monoid • Group

  8. Partial Binary Single-Valued

  9. Groupoid • Closed Binary Operation • Partial, Binary, Single-Valued System with • It is defined on all elements of S x S • Not necessarily surjective

  10. SemiGroup • An Associative Groupoid • Binary operation, e.g., multiplication • Closure • Associative • Can be defined for various operations, so sometimes written as

  11. Closed Binary Operation • Division Is Not a Closed Binary Operation on the Set of Counting Numbers 6/3 = 2 = counting number 2/6 = ? = not a counting number • Division Is Closed Over the Set of Real Numbers.

  12. Monoid Semigroup With an Identity Element, e.

  13. Group Monoid With an Inverse

  14. ‘Morphisms’ Homomorphism (J&J) “A correspondence of a set D (the domain) with a set R (the range) such that each element of D determines a unique element of R [single-valued] and each element of R is the correspondent of at least one element of D.“ and...

  15. Homomorphism “If operations such as multiplication, addition, or multiplication by scalars are defined for D and R, it is required that these correspond...” and...

  16. Homomorphism “If D and R are groups (or semigroups) with the operation denoted by * and x corresponds to x’ and y corresponds to y’ then x * y must correspond to x’ * y’ “ • Product of Correspondence = Correspondence of product

  17. Homomorphism

  18. Homomorphism • Correspondence must be • Single-valued: therefore at least a partial function • Surjective: each y in the R has at least one x in the D • Non-Injective: not one-to-one else isomorphism

  19. Endomorphism • A ‘morphism’ which maps back onto itself • The range, R, is the same set as the domain, D, e.g., the real numbers. ‘morphism’ R=D

  20. SemiGroup Homomorphism

  21. SemiGroup Homomorphism

  22. SmGp. HmMphsm. Example* *Larsen, Intro to Modern Algebraic Concepts, p. 53

  23. SmGp. HmMphsm. Example* Is the relation • single-valued? • Each symbol of D maps to only one symbol of R • surjective? • Each symbol of R has a corresponding element in D • not-injective? • e and g4 correspond to the same symbol, 0

  24. SmGp. HmMphsm. Example* Do the results of operations correspond? same

  25. Monoid Homomorphism

  26. Isomorphism • An Isomorphism Is a Homomorphism Which Is Injective • Injective: One-to-One Correspondence • A relation between two sets such that pairs can be removed, one member from each set until both sets have been simultaneously exhausted

  27. SemiGroup Isomorphism Injective Homomorphism

  28. Isomorphism Example* • Define two groupoids • non-associative semigroups • groups without an inverse or identity element • SG1: A1 = { positive real numbers } *1 = multiplication = * • SG2: A2 = { positive real numbers } *2 = addition = + *Ginzberg, pg 10

  29. Isomorphism Example

  30. SemiGroup Isomorphism

  31. Machine Isomorphisms • Input-output isomorphism, but usually abbreviated to just isomorphism • An I/O isomorphism exists between two machines, M1 and M2 if there exists a triple

  32. Machine Isomorphisms

  33. Machine Isomorphisms Interpret

  34. Machine State Isomorphism

  35. Machine Output Isomorphism

  36. Homo- vice Iso- Morphism Reduction Homomorphism • Shows behavioral equivalence between machines of different sizes • Allows us to only concern ourselves with minimized machines (not yet decomposed, but fewest states in single machine) • If we can find one, we can make a minimum state machine

  37. Homo- vice Iso- Morphism Isomorphism • Shows equivalence of machines of identical, but not necessarily minimal, size • Shows equivalence between machines with different labels for the inputs, states, and/or outputs

  38. Block Diagram Isomorphism I1 I2 O2 O1 M2 O1 M1 I1

  39. Block Diagram Isomorphism

  40. Block Diagram Isomorphism which is the same as the preceding state diagram and block diagram definitions therefore M1 and M2 are Isomorphic to each other

  41. Machine Information • Since the Inputs and Outputs Can Be Mapped Through Isomorphisms Which Are Independent of the State Transitions, All of the State Change Information Is Maintained in the Isomorphic Machine • Isomorphic Machines Produce Identical Outputs

  42. Output Equivalence

  43. Identity Machine Isomorphism

  44. Inverse Machine Isomorphism

  45. Machine Equivalence

  46. Machine Homomorphism

  47. Machine Homomorphism • If alpha is injective, then have isomorphism • “State Behavior” assignment, • “Realization” of M1 • If alpha not injective • “Reduction Homomorphism”

  48. Behavioral Equivalence

  49. Behavioral Equivalence

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