Lecture 1: A Formal Model of Computation

Lecture 1: A Formal Model of Computation

虞台文大同大學資工所智慧型多媒體研究室

Content

Formalizing the Idea of Programmable Computer.

Program Defined Machine Defined Computation Defined A Simple Machine


Formalizing the Idea of Programmable

Computer

大同大學資工所智慧型多媒體研究室

A Simple Programmable Device PC

PC Pocket Calculator

What is inside?

Inside PC Memory Set

– Two registers with unbounded capacity

Machine Instructions– Operations & tests

, , 0m n mM n

, ,m n m n ne.g.,




, , 0m n mM n

, ,m n m n ne.g.,

How about if n = 0?How about if n = 0?

Undefined if n = 0.Undefined if n = 0.




, , 0m n mM n

, ,m n m n ne.g.,

Partial FunctionsPartial

FunctionsPartial

PredicatesPartial

Predicates

Partial Functions

Let A and B be two sets, and

( ,, )A B a b a A Bf b is a set of ordered pairs.

We say that f is a partial function if( , )

( , )

a bb b

a b

f

f

Hence, (a, b)f is written as:

If, aA, there is no bB such that f(a) = b, then we say that f(a) is undefined, and denote this condition as: ( )f a

( )f a b

Partial Functions is reserved to denote undefined object. f() = . ( ) ( ) and Domain a a A f af

( ) , ( )st Range b b B a A a bf f

There Cases:

1. Totally undefined

2. Partially defined

3. Totally defined

( )Domain f

( )Domain f

( )Domain f A

Operations for PC

Each operation of PC would be defined by some partial function over M.

Example: define the division operation of PC as a partial function as follows:

if

if

, 0,

0

m n n nm n

n

Partial Predicates

: , ,p M true false which does not change the information environment.

Example: | ?m n

if

if not

i

|

, |

0f

true m n

m n false m n

n

Computable Functions

By a computable function, we mean a function that can be algorithmically specified.

When a algorithm is applied to an element outside its domain, it may– not terminated; or– the result B.

Whenever an algorithm is such that it computation does not always terminate, then the algorithm defines a computable partial function.

Exercises

Define “algorithm”.

Show that ,

where f, g, and h are partial functions.

( ) ( )f g h f g h


Program Defined


Programs vs. Flowcharts

Essentially, we take programs to be flowcharts constituting the following components.

START

HALT

Operation

Predicate (Test)

Flowchart

START

HALT

F0

P1

F1

F2truefalse

Labeled Statements

START

HALT

F0

P1

F1

F2truefalse

L0L1

L2

L3

L4

START: GOTO L3

L0: DO F2 GOTO L3

L1: IF P1 THEN GOTO L0 ELSE GOTO L3

L2: DO F1 GOTO L1

L3: DO F0 GOTO L2

L4: HALT

Instructions

Instructions are

made up from:

L : a set of labels;

F : a set of operations;

P : a set of predicates.

START: GOTO L3

L0: DO F2 GOTO L3

L1: IF P1 THEN GOTO L0 ELSE GOTO L3

L2: DO F1 GOTO L1

L3: DO F0 GOTO L2

L4: HALT

Four Types of Instructions

1. Start Instruction

2. Operation Instruction

3. Test Instruction

4. Halt Instruction

START: GOTO L

L: DO F GOTO L’

L: IF P1 THEN GOTO L’ ELSE GOTO L’’

L: HALT-Loop:

L: DO F GOTO L

Program Schema

A program schema is a set of instructions that contains:

Exactly one start instruction; No repeat on label.

Exercise

while P1 do begin W1; while P2 do

W2; end.

p1 := P1;p2 := false;while p1 or p2 do begin if p2 then W2 else W1; p2 := P2; p1 := P1; end.

Translate the above two Pascal Programs into labeled statements and prove their equivalence, using computation sequence in any convenient way.


Machine Defined


Machines

M-Machine

A machine is a function, say, M defined on

• the instruction set F P

• for which there exists a memory set M

such that

• MF is a partial function over M for each FF

• MP is a partial predicate over M for each PP.

M-Program

An M-program or a program for M is a program

in which no instruction, except perhaps -loops,

makes use an operation or test name that is

interpreted as the totally undefined function by

M.

Without Syntax ErrorWithout Syntax Error


Computation Defined


Complete Computation

A completed computation by a program on a machine M is a finite computation sequence: 0 0 1 1 1 1, , , , , , , , , , ,i i i i n nL m L m L m L m L m

Label ofstart instruction

Label of ahalt instruction

Complete Computation

A completed computation by a program on a machine M is a finite computation sequence: 0 0 1 1 1 1, , , , , , , , , , ,i i i i n nL m L m L m L m L m

Case 1:

Case 2:

: DO GO O TiL F L

: IF THEN GOT O ELSE GOTO iL P L L

1 1, ( )i i F iL L m m M

1 1

( ),

( )P i

i i iP i

L m truem m L

L m false

M

M

Without ambiguity at each step. Without ambiguity at each step.

Uncompletable Computation

The reasons for a computation being uncompletable:– Never reach halt instruction;

– Instruction Li: DO F GOTO L’ causes

– Instruction Li: IF P THEN GOTO L’ ELSE GOTO L’’

causes– Reaches a -loop.

( )F im M

( )P im M

Partial Function

The (partial) function

Computed by a program on a machine M is

such that

is a completed computation.

: M M M

0( ) nm m M

0 0 1 1, , , , , ,n nL m L m L m

0( ) nm completedm

othewise

M what a computer does.


A Simple Machine


Definition of Machine PC

Memory set:

Instruction set: Operations & Predicates

N N

F P

Operations of PC

1x x

1x x

1y y

1y y

y x y

y x y

, 1,m n m n

, 1, if 0

, 0, if 0

m n m n m

m n n m

, , 1m n m n

, , 1 if 0

, ,0 if 0

m n m n n

m n m n

, ,m n m m n

, ,m n m mn

other

Predicates of PC

0?x if 0

,if 0

true mm n

false m

0?y if 0

,if 0

true nm n

false n

?x yif

,if

true m nm n

false m n

other

A PC-Program

START

HALTtrue

false

1x x

1y y

y x y

0?x

GOTO

DO GOTO

IF THEN GOTO E

START: L0

L0: 1 L1

L1: 0? L4 L2

L2: L3

L3:

LSE GOTO

DO GOTO

DO GOT1 L1

L4: HALT

O

y y

x

y x y

x x

L0

L1

L2

L3

L4

Computation Sequence

START

HALTtrue

false

1x x

1y y

y x y

0?x

GOTO

DO GOTO

IF THEN GOTO E

START: L0

L0: 1 L1

L1: 0? L4 L2

L2: L3

L3:

LSE GOTO

DO GOTO

DO GOT1 L1

L4: HALT

O

y y

x

y x y

x x

L0

L1

L2

L3

L4

0 3,0m

0 0, 3,L 1 ,1, 3L 2 ,1, 3L

3 ,3, 3L 1 ,3, 2L 2 ,3, 2L

3 ,6, 2L 1 ,6, 1L 2 ,6, 1L

3 ,6, 1L 1 ,6, 0L 4 ,6, 0L

PC=?

START

HALTtrue

false

1x x

1y y

y x y

0?x

L0

L1

L2

L3

L4

: , 0, ( 1) !m n n m PC

0 0, 3,L 1 ,1, 3L 2 ,1, 3L

3 ,3, 3L 1 ,3, 2L 2 ,3, 2L

3 ,6, 2L 1 ,6, 1L 2 ,6, 1L

3 ,6, 1L 1 ,6, 0L 4 ,6, 0L

Encoding/Decoding

: , 0, ( 1) !m n n m PC

Encoding

Decoding

:e X M

:d M Y

,0k k

,m n n

by

by

PC

k k!

0, !k,0ke d

Encoding/Decoding

: !d e k kPC

PC

k k!

0, !k,0ke d

Man-Machine Interaction

M a machine an M-program

:e X M:d M Y

an encoding function a decoding function

Me d

input output

What Computed?

Me d

input output

d eM A partial function

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Lecture 1: A Formal Model of Computation