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Sequential Machine Theory

Prof. K. J. HintzDepartment of Electrical and Computer

Engineering

Lecture 1http://cpe.gmu.edu/~khintz

Adaptation to this class and additional comments by Marek Perkowski

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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

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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

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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

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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

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Formal Languages

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

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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

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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

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Machine/Mathematics Hierarchy

• AI Theory Intelligent Machines

• Computer Theory Computer Design

• Automata Theory Finite State Machine

• Boolean Algebra Combinational Logic

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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

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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

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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

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Intelligent Machines

• Ability to Learn• Possibly Not Computable

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Automata, aka FSM

• Concepts of Machines:– Mechanical– Computer programs– Political– Biological– Abstract mathematical

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Abstract Mathematical

• Discrete– Continuous system can be discretized to any

degree of resolution• Finite State• Input/Output

– Some cause, some result

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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

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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

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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

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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

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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:

a A

AAA

1

2 1 2 3

3

2 5 7 4

, , ,, , ,a a a

Larry, Curly, Moe

A4 0 10 a a: , integer

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Sets

– Infinite set

– Set of sets

AA

5

6

RI

real numbers integers

A A A7 3 6 ,

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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

B A A3

Larry,

Curly,

Moe

A6

integers

A7

A A6 7

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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

B A

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Set Equality

• Set A is equal to set B

AB

BA

BA

and

iff

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Sets

• Null Set– A set with no elements,

• Every set is a subset of itself

• Every set contains the null set

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Operations on Sets

• Intersection

• Union

C A BC A B

a a a|

D A BD A B

a a a|

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Operations on Sets

• Set Difference

• Cartesian Product, Direct Product

E A BE A B

a a a|

BAFBAF

yxyx |,

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Special 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

1,0let .,.

2 ,

A

A

A

P

geP ba

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Special Sets

• Disjoint sets: A and B are disjoint if

• Cover:

A B

ii

if

all

set another covers

sets, ofset A

BA

AB,BB 2,1

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Properties of Operations on Sets

• Commutative, Abelian

• Associative

• Distributive

A B B AA B B A

A B C A B CA B C A B C

LHD RHD

A B C A B A CA B C A C B C

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Partition of a Set

• Properties

• pi are called “pi-blocks” or “-blocks” of PI

i

i

P

p

App

Ap|pA

c)

, b)

disjoint, are a)and,

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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:

R:A B

R A Bnot necessarily a proper subset

a b, R

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Domain of a Relation

BABA bbaa somefor ,|Dom RR :

a

A

B

b

Domain of R

R

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Range of a Relation

Range for some R: RA B B A b a b a| ,

aA

B

bR

Range of R

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Inverse Relation, R-1

ABAB

BA

RR

R

baab ,|,

then:

Given

1

a

bA

BR-1

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Partial Function, Mapping

• A single-valued relation such that

if

and

then

a b

a b

b b

,

,

R

Ra

AB

b

b’

R

a’ *

* can be many to one

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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

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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

ABA : Dom R

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Function Example

wvvu

wvvu

wvu

,4,,3,,2,,1or,

4,3,2,1let 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

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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

A a

R

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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.”

given ,and ',then

a ba b

a a

RR

'

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Injective, 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

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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

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Function Examples

• Monotonically increasing if injective

• Not one-to-one, but single-valued

A

B

B

A

b

a a’

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Function Examples

• Multivalued, but one-to-one

A

B

a

b

b’

b’’

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The End of the Beginning