1 Lecture 12 Computations –Formal Definition –Important Terms Computational Models –Formal...

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Lecture 12

• Computations– Formal Definition– Important Terms

• Computational Models– Formal Definition– Sequential access memory (tape)– Graphical representation

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Computations and Configurations

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Computations

• What is a computation?– Execution of a program P on an input x

• Is the computation just the output produced by running P on x?– No, we define a computation to be a trace of the

entire execution

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Exercise1 int x,y,r;

2 cin >> x;

3 cin >> y;

4 r = x % y;

5 If r = 0 goto 10;

6 x = y;

7 y = r;

8 r = x % y;

9 goto 5;

10 return y;

Execute the above program on input 657 282

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Definitions

• Configuration– Intuitive Definition

• Given the original program and the current configuration of a computation, someone should be able to complete the computation

– Contents of a configuration• current instruction to be executed• current value of all variables

• Computation– Complete sequence of configurations

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Computation 1

1 int x,y,r;

2 cin >> x;

3 cin >> y;

4 r = x % y;

5 If r = 0 goto 10;

6 x = y;

7 y = r;

8 r = x % y;

9 goto 5;

10 return y;

Input: 10 3

• Line 1, x=?,y=?,r=?

• Line 2, x=?, y=?,r=?

• Line 3, x=10, y=?, r=?

• Line 4, x=10, y=3, r=?

• Line 5, x= 10, y=3, r=1

• Line 6, x=10, y=3, r=1

• Line 7, x=3, y=3, r=1

• Line 8, x=3, y=1, r=1

• Line 9, x=3, y=1, r=0

• Line 5, x=3, y=1, r=0

• Line 10, x=3, y=1, r=0

• Output is 1

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Computation 2

1 int x,y,r;

2 cin >> x;

3 cin >> y;

4 r = x % y;

5 If r = 0 goto 10;

6 x = y;

7 y = r;

8 r = x % y;

9 goto 5;

10 return y;

Input: 53 10

• Line 1, x=?,y=?,r=?

• Line 2, x=?, y=?,r=?

• Line 3, x=53, y=?, r=?

• Line 4, x=53, y=10, r=?

• Line 5, x= 53, y=10, r=3

• Line 6, x=53, y=10, r=3

• Line 7, x=10, y=10, r=3

• Line 8, x=10, y=3, r=3

• Line 9, x=10, y=3, r=1

• Line 5, x=10, y=3, r=1

• ...

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Computations 1 and 2

• Line 1, x=?,y=?,r=?

• Line 2, x=?, y=?,r=?

• Line 3, x=53, y=?, r=?

• Line 4, x=53, y=10, r=?

• Line 5, x= 53, y=10, r=3

• Line 6, x=53, y=10, r=3

• Line 7, x=10, y=10, r=3

• Line 8, x=10, y=3, r=3

• Line 9, x=10, y=3, r=1

• Line 5, x=10, y=3, r=1

• ...

• Line 1, x=?,y=?,r=?

• Line 2, x=?, y=?,r=?

• Line 3, x=10, y=?, r=?

• Line 4, x=10, y=3, r=?

• Line 5, x= 10, y=3, r=1

• Line 6, x=10, y=3, r=1

• Line 7, x=3, y=3, r=1

• Line 8, x=3, y=1, r=1

• Line 9, x=3, y=1, r=0

• Line 5, x=3, y=1, r=0

• Line 10, x=3, y=1, r=0

• return 1

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Observation

1 int x,y,r;

2 cin >> x;

3 cin >> y;

4 r = x % y;

5 If r = 0 goto 10;

6 x = y;

7 y = r;

8 r = x % y;

9 goto 5;

10 return y;

• Line 5, x= 10, y=3, r=1– Enough information to

complete the computation

– What was the original input?• Uncertain

• Both previous inputs, 10 3 as well as 53 10 eventually reached above configuration

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New Computational Model

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Computational Models

• So far, we have worked with C++ as our computational model

• We now define some new computational models• We first restrict the computational model to

simplify their formalization– Note, these models will be just as powerful as C++

• We then restrict the power of the computational models in ways that limit their power

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Language Recognition Problems

• We define our computational models with language recognition problems as our focus– Decision problems are general– Decision problems can be formulated as

languages with language recognition problems

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Components of Computational Models

• Memory

• Data Types

• Operations

• Graphical Representation

• Formal Definition

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Memory

• Random Access Memory Model– We have been assuming a random access memory

computational model for C++

– Time to access next memory location is independent of previously accessed memory location

• Sequential Access Memory Model– Memory is stored on a tape broken up into cells– Tape head scans currently accessed cell

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Illustration

TapeHead

Time to access a particular cell is dependent on currenttape head location

......

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Data Types

• C++ has many data types– integers– reals– characters– allows definition of new data types

• Our models– just characters– Alphabet will typically be just {a,b} or {0,1}

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Illustration

...... a b a a b a b

TapeHead

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Operations

• C++ has a complex operation set– some operations change memory

– other operations control flow of execution

• Our operation set– Memory operations

• read a character

• write a character

• tape head movement

– simple control flow • switch statements

• goto statements

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Illustration

...... a b a a b a b

TapeHead

1 switch(current tape cell) {case a: write b; move tapehead left 1 cell; goto 7;case b: write a; move tapehead right 1 cell; goto 4;case EOF: return YES;

}2 switch(current tape cell) {...

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Sample Program1 switch(current tape cell) {

case a: write a; move tapehead right 1 cell; goto 2;

case b: write b; move tapehead right 1 cell; goto 2;

case EOF: return YES;

}

2 switch(current tape cell) {case a: write a; move tapehead right 1 cell; goto 1;


case EOF: return NO;

}

• Claim– This computational model is as powerful as C++

– Any language which can be decided by a C++ program can be decided by one of these programs

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Example1 switch(current tape cell) {

case a: write a; move tapehead right 1 cell; goto 2;



}

2 switch(current tape cell) {case a: write a; move tapehead right 1 cell; goto 1;



}

• Run this program on input aabbaa

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Computation

1 switch(current tape cell) {case a: write a; move tapehead right 1

cell; goto 2;



}


cell; goto 1;



}

Input: aabbaa

• Line 1, tape=aabbaa$







• Output yes– $ is EOF

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Configurations

• What needs to be recorded?– Current instruction

– Tape contents

– Tape head location








• Return yes– $ is EOF

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Graphical Representation


cell; goto 2;



}


cell; goto 1;



}

1 2

a;a;Rb;b;R

a;a;Rb;b;R

Color coded.Still missing one thing.

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