Great Ideas of CS with Java
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Part 1 WWW & Computer programming in the language Java
Ch 1: The World Wide Web
Ch 2: Watch out: Here comes Java
Ch 3: Numerical computation & Function
Ch 4: Subroutines & Databases
Ch 5: Graphics
Ch 6: Simulation
Ch 7: Software engineering
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Part 2 Understanding what a computer is and how it works
Ch 8: Machine architecture
Ch 9: Language translation
Ch 10: Virtual Environment for Computing
Ch 11: Security, Privacy, and Wishful thinking
Ch 12: Computer Communication
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Part 3 Advanced topics
Ch 13: Program Execution Time
Ch 14: Parallel Computation
Ch 15: Noncomputability
Ch 16: Artificial intelligence
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OOPSLA Lab.
Ch 15. Noncomputability
Copyright © SNU OOPSLA Lab.
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OOPSLA Lab.
Table of Contents
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Noncomputable Functions
Programs that Read Programs
Solving the Halting Problem
Examples of non-computable problems
Proof of noncomputability
Summary
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Noncomputable Functions
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Certain Problems Not Amenable to Computer Solution
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Intuitively speaking
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Examples given here may seem strained and artificial
However, computers have very real limitation
A class of problems that are unsolvable by any computer within the
current paradigm of modern programming
A function is computable if a Java program exists that can compute it
We want to prove the existence of noncomputable functions
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Existence of Noncomputable Functions (1/5)
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Approach: Matching up Java Programs and Functions
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E.g., assume 4 functions (f1, f2, f3, f4) and only 3 java programs (p1,
p2, p3)
Without details, conclude that one function has no program
Figure 15.1
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Existence of Noncomputable Functions (4/5)
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Try to Draw Lines Between Functions and Programs
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Could draw lines from every program to every function (problem)
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Figure
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Existence of Noncomputable Functions (2/5)
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Consider all the Java programs that input an integer and
return an integer
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But as shown in chapter 3, the set of functions that accept
integers as input and yield positive numbers as output is
uncountable
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The set of such Java programs is a countable set
See page 426 of textbook
See page 125 of textbook
Therefore Some functions (problems) cannot have the
corresponding Java programs!
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Table of Contents
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Noncomputable Functions
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Programs that Read Programs
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Solving the Halting Problem
Examples of non-computable problems
Proof of noncomputability
Summary
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Programs that Read Programs (1/3)
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Express programs as a single string
We can easily write program to see if there is an IF statement in the
program.
How about, does the program halt?
Figure 15.3
Program that input other programs
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Programs that Read Programs (2/3)
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“no while statement in its input program”  halt,
otherwise  “Not known whether it halts”
Figure 15.4
String E (String P)
{
if (p.indexOf(“while”) >= 0)
{return “Not known whether it halts.”;}
else
{return “Halts on all inputs.”;}
}
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Programs that Read Programs (3/3)
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However, what we want is the following function!
Figure 15.5
We want F such that can halt after a finite time with the correct answer
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Table of Contents
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Noncomputable Functions
Programs that Read Programs
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Solving the Halting Problem
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Examples of non-computable problems
Proof of noncomputability
Summary
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Solving the Halting Problem (1)
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Does it halt for all input?
DoesItHalt.java, DoesItHalt.html
// input an integer value for k
while (k > 1)
if ((k/2) * 2 == k)
// is k even?
k = k / 2;
else
k = 3 * k + 1;
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public boolean action(Event e, Object arg)
{
int yea;
String mak, sty, col, own;
import awb.*;
import java.awt.*;
if (e.target == gStart)
{
j = 0;
mResults.setText("Trace of the integer value: \n");
k = gStart.getInt();
while (k > 1)
{
j = j + 1;
if ((k/2) * 2 == k)
// is k even?
{
k = k / 2;
}
else
{
k = 3 * k + 1;
}
public class DoesItHalt extends java.applet.Applet
{
TextField mInstruct;
IntField gStart;
TextArea mResults;
int j, k;
public void init()
{
mInstruct = new TextField(60);
mInstruct.setText("Enter starting integer:");
gStart = new IntField(10);
mResults = new TextArea(10, 50);
}
add(mInstruct);
add(gStart);
add(mResults);
mResults.appendText("k = " + k + "\n");
}
mResults.appendText(j + " steps before halting.\n");
return true;
}
}
}
return false;
DoesItHalt.java
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Solving the Halting Problem (2)
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Try It!
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e.g. 17: 52 26 13, 40 20 10 5, 16 8 4 2 1
No one knows whether this quits for all inputs.
Mathematicians have proven that no one, finite program can
check this property for all possible programs
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Table of Contents
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Noncomputable Functions
Programs that Read Programs
Solving the Halting Problem
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Examples of non-computable problems
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Proof of noncompuatbility
Summary
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Examples of non-computable problems (1/2)
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Equivalence: Define by same input > same output
Use variation of above program; not sure it ends
Cannot generally prove equivalence
Figure 15.6
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Examples of non-computable problems (2/2)
Figure 15.7
To check for property X in the behavior of all other programs is impossible
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Table of Contents
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Noncomputable Functions
Existence of Noncomputable Functions
Programs that Read Programs
Solving the Halting Problem
Examples of non-computable problems
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Proof of noncompuatbility
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Summary
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Proof (1/4)
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Proofs by Contradiction (Indirect Proof)
Proving non-computability
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Sketch of proof
Find more details in book
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Proof (2/4)
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Assume existence of function halt:
String halt(string p, string x);
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Inputs: p = program, x = input data
Returns: If halt  "Halts“ or Unless halt  "Does not halt"
Now write:
String selfhalt(string p)
{ String answer;
answer = halt (p, p);
if (answer.equals(“Halts.”)) {return (“Halts on self.”);}
else
{return (“Does not halt on self.”);}
}
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Proof (3/4)
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String contrary (String p)
{
String answer;
answer = selfhalt(p);
if (answer.equals(“Halts on self.”))
{while (true) // go into an infinite loop
{answer = “x”}
}
return “Halt.” // program halts;
}
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Now run contrary on itself
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Proof (4/4)
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We are running contrary on itself.
Paradox!
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Therefore the halt program cannot exist!
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If the halt program decides that it halts, it goes into infinite loop
and goes on forever.
If the halt program decides that it doesn't halt, it quits immediately.
If the halting problem is computable, then the halt program exist.
Since the halt program cannot exist, the halting problem is not
computable
In general, whole classes of problems on deciding program
behavior are non-computable.
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What Does It All Mean?
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Programs cannot do everything
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Table of Contents
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Noncomputable Functions
Solving the Halting Problem
Examples of non-computable problems
Proof of noncomputability
Summary
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Summary
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Limit of contemporary programming languages
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based on the Von Neumann Computer Architecture
There are uncomputable problems
We proved that the halting problem is uncompuatble
Also the cousins of the halting problem (regarding the
behavioral property of programs) are all uncomputable
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Ch15: Noncomputability
Text Review Time
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OOPSLA Lab.