Sequential Machine Theory

1. Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 1 http://cpe.gmu.edu/~khintz Adaptation to this class and additional comments by Marek Perkowski

2. Why Sequential Machine Theory (SMT)? • Sequential Machine Theory – SMT • Some Things Cannot be Parallelized • Theory Leads to New Ways of Doing Things • Understand Fundamental FSM Limits • Minimize FSM Complexity and Size • Find the “Essence” of a Machine

3. Why Sequential Machine Theory? • Discuss FSM properties that are unencumbered by Implementation Issues • Technology is Changing Rapidly, the core of the theory remains forever. • Theory is a Framework within which to Understand and Integrate Practical Considerations

4. Hardware/Software • There Is an Equivalence Relation Between Hardware and Software • Anything that can be done in one can be done in the other…perhaps faster/slower • System design now done in hardware description languages without regard for realization method • Hardware/software/split decision deferred until later stage in design

5. Hardware/Software • Hardware/Software equivalence extends to formal languages • Different classes of computational machines are related to different classes of formal languages • Finite State Machines (FSM) can be equivalently represented by one class of languages

6. Formal Languages • Unambiguous • Can Be Finite or Infinite • Can Be Rule-based or Enumerated • Various Classes With Different Properties

7. Finite State Machines • Equivalent to One Class of Languages • Prototypical Sequence Controller • Many Processes Have Temporal Dependencies and Cannot Be Parallelized • FSM Costs • Hardware: More States More Hardware • Time: More States, Slower Operation

8. Goal of this set of lectures • Develop understanding of Hardware/Software/Language Equivalence • Understand Properties of FSM • Develop Ability to Convert FSM Specification Into Set-theoretic Formulation • Develop Ability to Partition Large Machine Into Greatest Number of Smallest Machines • This reduction is unique

9. Machine/Mathematics Hierarchy • AI Theory Intelligent Machines • Computer Theory Computer Design • Automata Theory Finite State Machine • Boolean Algebra Combinational Logic

10. Combinational Logic • Feedforward • Output Is Only a Function of Input • No Feedback • No memory • No temporal dependency • Two-Valued Function Minimization Techniques Well-known Minimization Techniques • Multi-valued Function Minimization Well-known Heuristics

11. Finite State Machine • Feedback • Behavior Depends Both on Present State and Present Input • State Minimization Well-known With Guaranteed Minimum • Realization Minimization • Unsolved problem of Digital Design

12. Computer Design • Defined by Turing Computability • Can compute anything that is “computable” • Some things are not computable • Assumed Infinite Memory • State Dependent Behavior • Elements: • Control Unit is specified and implemented as FSM • Tape infinite • Head • Head movements

13. Intelligent Machines • Ability to Learn • Possibly Not Computable

14. Automata, aka FSM • Concepts of Machines: • Mechanical • Computer programs • Political • Biological • Abstract mathematical

15. Abstract Mathematical • Discrete • Continuous system can be discretized to any degree of resolution • Finite State • Input/Output • Some cause, some result

16. Set Theoretic Formulation of Finite State Machine • S: Finite set of possible states • I: Finite set of possible inputs • O: Finite set of possible outputs • : Rule defining state change • : Rule determining outputs

17. Types of FSMs • Moore • Output is a function of state only • Mealy • Output is a function of both the present state and the present input

18. Types of FSMs • Finite State Acceptors, Language Recognizers • Start in a single, specified state • End in particular state(s) • Pushdown Automata • Not an FSM • Assumed infinite stack with access only to topmost element

19. Computer • Turing Machine • Assumed infinite read/write tape • FSM controls read/write/tape motion • Definition of computable function • Universal Turing machine reads FSM behavior from tape

20. Review of Set Theory • Element: “a”, a single object with no special property • Set: “A”, a collection of elements, i.e., • Enumerated Set: • Finite Set:

21. Sets • Infinite set • Set of sets

22. Subsets • All elements of B are elements of A and there may be one or more elements of A that is not an element of B A3 Larry, Curly, Moe A6 integers A7

23. Proper Subset • All elements of B are elements of A and there is at least oneelement of A that is not an element of B

24. Set Equality • Set A is equal to set B

25. Sets • Null Set • A set with no elements,  • Every set is a subset of itself • Every set contains the null set

26. Operations on Sets • Intersection • Union

27. Operations on Sets • Set Difference • Cartesian Product, Direct Product

28. Special Sets • Powerset: set of all subsets of A *no braces around the null set since the symbol represents the set

29. Special Sets • Disjoint sets: A and B are disjoint if • Cover:

30. Properties of Operations on Sets • Commutative, Abelian • Associative • Distributive

31. Partition of a Set • Properties • pi are called “pi-blocks” or “-blocks” of PI

32. Relations Between Sets • If A and B are sets, then the relation from A to B, is a subset of the Cartesian product of A and B, i.e., • R-related:

33. Domain of a Relation Domain of R R B a A b

34. Range of a Relation Range of R R b A a B

35. Inverse Relation, R-1 R-1 B a A b

36. Partial Function, Mapping • A single-valued relation such that R a b b’ a’ * A B * can be many to one

37. Partial Function • Also called the Image of a under R • Only one element of B for each element of A • Single-valued • Can be a many-to-one mapping

38. Function • A partial function with • A b corresponds to each a, but only one b for each a • Possibly many-to-one: multiple a’s could map to the same b

39. Function Example • Unique, one image for each element of A and no • more • Defined for each element of A, so a function, • not partial • Not one-to-one since 2 elements of A map to v

40. Surjective, Onto • Range of the relation is B • At least one a is related to each b • Does not imply • single-valued • one-to-one R B A a

41. Injective, One-to-One • “A relation between 2 sets such that pairs can be removed, one member from each set, until both sets have been simultaneously exhausted.”

42. a = a’ R b Injective, One-to-One a could map to b’ also if it were not at least a partial function which implies single-valued

43. Bijective • A function which is both Injective and Surjective is Bijective. • Also called “one-to-one” and “onto” • A bijective function has an inverse, R-1, and it is unique

44. B A b B aa’ A Function Examples • Monotonically increasing if injective • Not one-to-one, but single-valued

45. b b’ B b’’ a A Function Examples • Multivalued, but one-to-one

46. The End of the Beginning