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1
Sequential Machine TheorySequential Machine Theory
Prof. K. J. HintzDepartment 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)?
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?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/SoftwareHardware/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/SoftwareHardware/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 LanguagesFormal Languages
• Unambiguous
• Can Be Finite or Infinite
• Can Be Rule-based or Enumerated
• Various Classes With Different Properties
7
Finite State MachinesFinite 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 lecturesGoal 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 HierarchyMachine/Mathematics Hierarchy
• AI Theory Intelligent Machines
• Computer Theory Computer Design
• Automata Theory Finite State Machine
• Boolean Algebra Combinational Logic
10
Combinational LogicCombinational 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 MachineFinite 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 DesignComputer 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 MachinesIntelligent Machines
• Ability to Learn
• Possibly Not Computable
14
Automata, aka FSMAutomata, aka FSM
• Concepts of Machines:– Mechanical– Computer programs– Political– Biological– Abstract mathematical
15
Abstract MathematicalAbstract 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
Set Theoretic Formulation of Finite State Machine
S I O, , , ,
• 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 FSMsTypes 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 FSMsTypes 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
ComputerComputer
• 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 TheoryReview of Set Theory
• Element: “a”, a single object with no special property
• Set: “A”, a collection of elements, i.e.,
– Enumerated Set:
– Finite Set:
a A
A
A
A
1
2 1 2 3
3
2 5 7 4
, , ,
, , ,a a a
Larry, Curly, Moe
A4 0 10 a a: , integer
21
SetsSets
– Infinite set
– Set of sets
A
A5
6
R
I
real numbers
integers
A A A7 3 6 ,
22
SubsetsSubsets
• 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
B A A3
Larry,
Curly,
Moe
A6
integers
A7
A A6 7
23
Proper SubsetProper 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
B A
24
Set EqualitySet Equality
• Set A is equal to set B
AB
BA
BA
and
iff
25
SetsSets
• Null Set– A set with no elements,
• Every set is a subset of itself
• Every set contains the null set
26
Operations on SetsOperations on Sets
• Intersection
• Union
C A B
C A B
a a a|
D A B
D A B
a a a|
27
Operations on SetsOperations on Sets
• Set Difference
• Cartesian Product, Direct Product
E A B
E A B
a a a|
BAF
BAF
yxyx |,
28
Special SetsSpecial Sets
• Powerset: set of all subsets of A
*no braces around the null set since the symbol represents the set
1,0,1,0,
then
1,0
let .,.
2 ,
A
A
A
P
ge
P ba
29
Special SetsSpecial Sets
• Disjoint sets: A and B are disjoint if
• Cover:
A B
ii
if
all
set another covers
sets, ofset A
BA
A
B,BB 2,1
30
Properties of Operations on SetsProperties of Operations on Sets
• Commutative, Abelian
• Associative
• Distributive
A B B A
A B B A
A B C A B C
A B C A B C
LHD
RHD
A B C A B A C
A B C A C B C
31
Partition of a SetPartition of a Set
• Properties
• pi are called “pi-blocks” or “-blocks” of PI
i
i
P
p
Ap
p
Ap|pA
c)
, b)
disjoint, are a)
and,
32
Relations Between SetsRelations 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:
R:A B
R A B
not necessarily a proper subset
a b, R
33
Domain of a RelationDomain of a Relation
BABA bbaa somefor ,|Dom RR :
a
A
B
b
Domain of R
R
34
Range of a RelationRange of a Relation
Range for some R: RA B B A b a b a| ,
a
A
B
bR
Range of R
35
Inverse Relation, R-1Inverse Relation, R-1
ABAB
BA
RR
R
baab ,|,
then
:
Given
1
a
bA
BR-1
36
Partial Function, MappingPartial Function, Mapping
• A single-valued relation such that
if
and
then
a b
a b
b b
,
,
R
R
a
AB
b
b’
R
a’ *
* can be many to one
37
Partial FunctionPartial 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
FunctionFunction
• 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
ABA : Dom R
39
Function ExampleFunction Example
wvvu
wvvu
wvu
,4,,3,,2,,1
or,
4,3,2,1
let then
,,4,3,2,1let
R
RRRR
BA
•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, OntoSurjective, 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
A a
R
41
Injective, One-to-OneInjective, 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.”
given ,
and ',
then
a b
a b
a a
R
R
'
42
Injective, One-to-OneInjective, One-to-One
a could map to b’
also if it were not at
least a partial function
which implies
single-valued
aa’
=
R
b
43
BijectiveBijective
• 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 ExamplesFunction Examples
• Monotonically increasing
if injective
• Not one-to-one,
but single-valued
A
B
B
A
b
a a’
45
Function ExamplesFunction Examples
• Multivalued, but one-to-one
A
B
a
b
b’
b’’
46
The End of the BeginningThe End of the Beginning