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)?
Some Things Cannot be Parallelized
Theory Leads to New Ways of Doing Things, has also practical applications in software and hardware (compiler design, controllers design, etc.)
Understand Fundamental FSM Limits
Minimize FSM Complexity and Size
Find the “Essence” of a Machine,
what does it mean that there is a machine for certain task?

3. Why Sequential Machine Theory?
Discuss FSM properties that are unencumbered by Implementation Issues:
Software
Hardware
FPGA/ASIC/Memory, etc.
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 (VHDL, Verilog, higher) 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
Give some simple examples
Can Be Rule-based or Enumerated
Various Classes With Different Properties

7. Finite State Machines FSMs are Equivalent to One Class of Languages
Prototypical Sequence Controller
Generator
acceptor
controller
Many Processes Have Temporal Dependencies and Cannot Be Parallelized,
the need some form of state machine.
FSM Costs
Hardware: More States More Hardware
Time: More States, Slower Operation
Technology dependent: how many CPLD chips?

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
Technology related, combinational design related

12. Computer Design: Turing Machines
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
Show example of a very simple Turing machine now:
x--> x+1

13. Intelligent Machines Some machines display an ability to learn
How a machine can learn?
Some problems are possibly not computable
What problems?
Why not computable?
Something must be infinite?

14. concepts in other areas
Automata, aka FSM
Concepts of Machines:
Mechanical
Counters, adders
Computer programs
Political
Towns, highways, social groups, parties, etc
Biological
Tissues, cells, genetic, neural, societies
Abstract mathematical
Functions, relations, graphs
You should be
able to use FSM
concepts in other areas
like robotics

15. FSM - Abstract mathematical concept of many types of behavior
Discrete
Continuous system can be discretized to any degree of resolution
Finite State:
finite alphabets for inputs, outputs and states.
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 FSM Mealy FSM Output is a function of state only
Output is a function of both the present state and the present input
Discuss timing differences, show examples and diagrams, discuss fast signaling and PLD realization

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 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 Every set is a subset of itself
A set with no elements, 
Every set is a subset of itself
Every set contains the null set

26. Operations on Sets
Intersection
Union
Logical AND
Logical OR

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:
We know set covering problem from 572. It was defined as a matrix problem

30. Properties of Operations on Sets
Commutative, Abelian
Associative
Distributive
Left hand 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
u v w
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 (called also Onto) relations
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
Not mapped A a
1234
s1s2s3

41. Injective, or One-to-One relations
“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’ = R b 12
abc

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 b
There are no two a’s which would have the same b, so it is one-to-one