1 Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 1 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 TheoryIntelligent Machines Computer TheoryComputer Design Automata TheoryFinite State Machine Boolean AlgebraCombinational 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 A 3 Larry, Curly, Moe A 6 integers A7A7

23 Proper Subset All elements of B are elements of A and there is at least one element 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 p i 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 a A B b Domain of R R

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

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

36 Partial Function, Mapping A single-valued relation such that a A B b b’b’ R a’a’ * * 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 B Aa R

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

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 Function Examples Monotonically increasing if injective Not one-to-one, but single-valued A B B A b a a’

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

46 The End of the Beginning