strings and languages operations

1

Strings and Languages Operations

Concatenation

Exponentiation

Kleene Star

Regular Expressions

2

Strings and Language Operations

• Concatenation

• Exponentiation

• Kleene star• Pages 28-32 of the recommended text

• Regular expressions• Pages 85-90 of the recommended text

3

String Concatenation

• If x and y are strings over alphabet , the concatenation of x and y is the string xy formed by writing the symbols of x and the symbols of y consecutively.

• Suppose x = abb and y = ba• xy = abbba• yx = baabb

4

Properties of String Concatenation

• Suppose x, y, and z are strings.

• Concatenation is not commutative.• xy is not guaranteed to be equal to yx

• Concatenation is associative• (xy)z = x(yz) = xyz

• The empty string is the identity for concatenation• x/\ = /\x = x

5

Language Concatenation

• Suppose L1 and L2 are languages (sets of strings).• The concatenation of L1 and L2, denoted L1L2,is

defined as• L1L2 = { xy | x L1 and y L2 }

• Example,• Let L1 = { ab, bba } and L2 = { aa, b, ba }• What is L1L2?

• Solution• Let x1= ab, x2= bba, y1= aa, y2= b, y3= ba• L1L2 = { x1y1, x1y2, x1y3, x2y1, x2y2, x2y3 } =

{ abaa, abb, abba, bbaaa, bbab, bbaba}

6

Language Concatenation is not commutative

• Let L1 = { aa, bb, ba } and L2 = { /\, aba }

• Let x1= aa, x2= bb, x3=ba, y1= /\, y2= aba

• L1L2 = { x1y1, x1y2, x2y1, x2y2, x3y1, x3y2 } = { aa, aaaba, bb, bbaba, ba, baaba }

• L2L1 = { y1x1, y1x2, y1x3, y2x1, y2x2, y2x3 } = { aa, bb, ba, abaaa, ababb, ababa }

• L2L2 = { y1y1, y1y2, y2y1, y2y2 } = { /\, aba, aba, abaaba } = { /\, aba, abaaba } (dropped extra aba)

7

Associativity of Language Concatenation

• (L1L2)L3 = L1(L2L3) = L1L2L3

• Example• Let L1={a,b}, L2={c,d}, and L3={e,f}

• L1L2L3=({a,b}{c,d}){e,f} ={ac, ad, bc, bd}{e,f} ={ ace,acf,ade,aef,bce,bcf,bde,bdf }

• L1L2L3={a,b}({c,d}{e,f}) ={a,b}{ce, df, ce, df} ={ ace,acf,ade,aef,bce,bcf,bde,bdf }

8

Special Cases

• What language is the identity for language concatenation?• The set containing only the empty string /\: {/\}• Example

• {aab,ba,abc}{/\} = {/\}{aab,ba,abc} = {aab,ba,abc}

• What about {}?• For any language L, L {} = {} L = {}

• Thus {} for concatenation is like 0 for multiplication

• Example• {aab,ba,abc}{} = {}{aab,ba,abc} = {}• The intuitive reason is that we must choose a string from both

sets that are being concatenated, but there is nothing to choose from {}.

9

Exponentiation

• We use exponentiation to indicate the number of items being concatenated• Symbols• Strings• Set of symbols ( for example)• Set of strings (languages)

• a3 = aaa• x3 = xxx• 3 = = { x * | |x|=3 }• L3 = LLL

10

Examples of Exponentiation

• Let x=abb, ={a,b}, L={ab,b}

• a4 = aaaa

• x3 = (abb)(abb)(abb) = abbabbabb

• = = {a,b}{a,b}{a,b}={aaa,aab,aba,abb,baa,bab,bba,bbb}

• L3 = LLL = {ab,b}{ab,b}{ab,b}= {ababab,ababb,abbab,abbb, babab,babb,bbab,bbb}

11

Results of Exponentiation

• Exponentiation of a symbol or a string results in a string.

• Exponentiation of a set of symbols or a set of strings results in a set of strings• a symbol a string• a string a string• a set of symbols a set of strings• a set of strings a set of strings

12

Special Cases of Exponentiation

• a0 = /\

• x0 = /\

• 0 = { /\ }

• L0 = { /\ } for any language L• {aa,bb}0 = { /\ }• { a, aa, aaa, aaaa, …}0 = { /\ }• { /\ }0 = { /\ }0 = { }0 = { /\ }

13

Kleene Star

• Kleene * is a unary operation on languages.• Kleene * is not an operation on strings

• However, see the pages on regular expressions.

• L* represents any finite number of concatenations of L.

L* = Uk>0 Lk = L0 U L1 U L2 U …

• For any L, /\ is always an element of L*• because L0 = { /\ }

• Thus, for any L, L* !=

14

Example of Kleene Star

• Let L={aa}• L0={ /\ }• L1=L={aa }• L2={ aaaa }• L3= …

• L* = L0 L1 L2 L3 …

• = { /\, aa, aaaa, aaaaaa, … }• = set of all strings that can be obtained by

concatenating 0 or more copies of aa

15

Example of Kleene Star

• Let L={aa, b}

• L0={ /\ }

• L1=L={aa,b}

• L2= LL={ aaaa, aab, baa, bb}

• L3= …

• L* = L0 L1 L2 L3 …

• = set of all strings that can be obtained by concatenating 0 or more copies of aa and b

16

Regular Languages

• Regular languages are languages that can be obtained from the very simple languages over , using only• Union• Concatenation• Kleene Star

17

Examples of Regular Languages

• {aab} (i.e. {a}{a}{b} )

• {aa,b} (i.e. {a}{a} {b} )

• {a,b}* language of strings that can be obtained by concatenating any number of a’s and b’s

• {bb}{a,b}* language of strings that begin with bb (followed by any number of a’s and b’s)

• {a}*{bb,/\} language of strings that begin with any number of a’s and end with an optional bb.

• {a}*{b}* language of strings that consist of only a’s or only b’s and /\.

18

Regular Expressions

• We can simplify the formula for regular languages slightly by• leaving out the set brackets { } and• replacing with +

• The results are called regular expressions.

19

Examples of Regular Expressions

Set notation Regular Expressions

{aab} aab

{aa,b} = {aa}{b} aa+b

{a,b}* = ({a}{b})* (a+b)*

{bb}{a,b}* = {bb}({a}{b})* bb(a+b)*

{a}*{bb,/\} = {a}*({bb}{/\}) a*(bb+/\)

{a}*{b}* a*+b*

20

String or Language?

• Consider the regular expression a*(bb+/\)

• a*(bb+/\) is a string over alphabet {a, b, *, +, /\, (, ), }

• a*(bb+/\) represents a language over alphabet {a, b}• It represents the language of strings over {a,b} that begin with any

number of a’s and end with an optional bb.

• Some regular expressions look just like strings over alphabet {a,b}• Regular expression aaba represents the language {aaba}

• Regular expression /\ represents the language {/\}

• It should be clear from the context whether a sequence of symbols is a regular expression or just a string.

21

Module 1: Course Overview

• Course: CSE 460

• Instructor: Dr. Eric Torng

• Grader: Yi Liu

22

What is this course?

• Philosophy of computing course• We take a step back to think about computing

in broader terms

• Science of computing course• We study fundamental ideas/results that shape

the field of computer science

• “Applied” computing course• We learn study a broad range of material with

relevance to computing today

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Philosophy

• Phil. of life• What is the purpose of

life?

• What are we capable of accomplishing in life?

• Are there limits to what we can do in life?

• Why do we drive on parkways and park on driveways?

• Phil. of computing• What is the purpose of

programming?

• What can we achieve through programming?

• Are there limits to what we can do with programs?

• Why don’t debuggers actually debug programs?

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Science

• Physics• Study of fundamental

physical laws and phenomenon like gravity and electricity

• Engineering• Governed by physical

laws

• Our material• Study of fundamental

computational laws and phenomenon like undecidability and universal computers

• Programming• Governed by

computational laws

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Applied computing

• Applications are not immediately obvious• In some cases, seeing the applicability of this

material requires advanced abstraction skills• Every year, there are people who leave this course

unable to see the applicability of the material

• Others require more material in order to completely understand their application

• for example, to understand how regular expressions and context-free grammars are applied to the design of compilers, you need to take a compilers course

26

Some applications

• Important programming languages• regular expressions (perl)

• finite state automata (used in hardware design)

• context-free grammars

• Proofs of program correctness• Subroutines

• Using them to prove problems are unsolvable

• String searching/Pattern matching• Algorithm design concepts such as recursion

27

Fundamental Theme *

• What are the capabilities and limitations of computers and computer programs?• What can we do with computers/programs?• Are there things we cannot do with

computers/programs?

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Module 2: Fundamental Concepts• Problems

• Programs• Programming languages

29

Problems

We view solving problems as the main application for computer

programs

30

Inputs Outputs

(4,2,3,1)(3,1,2,4)

(7,5,1)

(1,2,3)

(1,2,3,4)

(1,5,7)

(1,2,3)

Definition

• A problem is a mapping or function between a set of inputs and a set of outputs

• Example Problem: Sorting

31

How to specify a problem

• Input• Describe what an input instance looks like

• Output• Describe what task should be performed on the

input• In particular, describe what output should be

produced

32

Example Problem Specifications*

• Sorting problem• Input

– Integers n1, n2, ..., nk

• Output– n1, n2, ..., nk in nondecreasing order

• Find element problem• Input

– Integers n1, n2, …, nk

– Search key S

• Output– yes if S is in n1, n2, …, nk, no otherwise

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Programs

Programs solve problems

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Purpose

• Why do we write programs?

• One answer• To solve problems• What does it mean to solve a problem?

• Informal answer: For every legal input, a correct output is produced.

• Formal answer: To be given later

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Programming Language

• Definition• A programming language defines what

constitutes a legal program• Example: a pseudocode program may not be a

legal C++ program which may not be a legal C program

• A programming language is typically referred to as a “computational model” in a course like this.

36

C++

• Our programming language will be C++ with minor modifications• Main procedure will use input parameters in a

fashion similar to other procedures• no argc/argv

• Output will be returned• type specified by main function type

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Maximum Element Problem

• Input• integer n ≥ 1

• List of n integers

• Output• The largest of the n integers

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C++ Program which solves the Maximum Element Problem*

int main(int A[], int n) {int i, max;

if (n < 1)

return (“Illegal Input”);

max = A[0];

for (i = 1; i < n; i++)if (A[i] > max)

max = A[i];

return (max);

}

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Fundamental Theme

Exploring capabilities and limitations of C++ programs

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Restating the Fundamental Theme *

• We will study the capabilities and limits of C++ programs

• Specifically, we will try and identify• What problems can be solved by C++ programs• What problems cannot be solved by C++

programs

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Question

• Is C++ general enough?

• Or is it possible that there exists some problem such that• can be solved by some program P in some

other reasonable programming language • but cannot be solved by any C++ program?

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Church’s Thesis (modified)

• We have no proof of an answer, but it is commonly accepted that the answer is no.

• Church’s Thesis (three identical statements)• C++ is a general model of computation• Any algorithm can be expressed as a C++

program• If some algorithm cannot be expressed by a C+

+ program, it cannot be expressed in any reasonable programming language

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Summary *

• Problems• When we talk about what programs can or

cannot “DO”, we mean what PROBLEMS can or cannot be solved

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Module 3: Classifying Problems

• One of the main themes of this course will be to classify problems in various ways• By solvability

• Solvable, “half-solvable”, unsolvable

• We will focus our study on decision problems• function (one correct answer for every input)

• finite range (yes or no is the correct output)

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Classification Process

• Take some set of problems and partition it into two or more subsets of problems where membership in a subset is based on some shared problem characteristic

Set of Problems

Subset 1 Subset 2 Subset 3

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Classify by Solvability

• Criteria used is whether or not the problem is solvable• that is, does there exist a C++ program which

solves the problem?Set of All Problems

Solvable Problems Unsolvable Problems

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Function Problems

• We will focus on problems where the mapping from input to output is a function

Set of All ProblemsNon-Function Problems Function Problems

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General (Relation) Problem

• the mapping is a relation• that is, more than one output is possible for a

given input

Inputs Outputs

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Criteria for Function Problems

• mapping is a function• unique output for each input

Inputs Outputs

50

Example Non-Function Problem

• Divisor Problem• Input: Positive integer n• Output: A positive integral divisor of n

Inputs Outputs

9 13

9

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Example Function Problems

• Sorting

• Multiplication Problem• Input: 2 integers x and y• Output: xy

Inputs Outputs

2,5 10

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Another Example *

• Maximum divisor problem• Input: Positive integer n• Output: size of maximum divisor of n smaller

than n

Inputs Outputs

9 13

9

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Decision Problems

• We will focus on function problems where the correct answer is always yes or no

Set of Function ProblemsNon-Decision Problems Decision Problems

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Criteria for Decision Problems

• Output is yes or no• range = {Yes, No}

• Note, problem must be a function problem• only one of Yes/No is correct

Inputs Outputs

Yes

No

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Example

• Decision sorting• Input: list of integers• Yes/No question: Is the list in nondecreasing

order?

Inputs Outputs

Yes

No

(1,3,2,4)

(1,2,3,4)

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Another Example

• Decision multiplication• Input: Three integers x, y, z• Yes/No question: Is xy = z?

Inputs Outputs

Yes

No

(3,5,14)

(3,5,15)

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A Third Example *

• Decision Divisor Problem• Input: Two integers x and y• Yes/No question: Is y a divisor of x?

Inputs Outputs

Yes

No

(14,5)

(14,7)

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Focus on Decision Problems

Set of All ProblemsSolvable Problems Unsolvable Problems

DecisionProblems

OtherProbs

• When studying solvability, we are going to focus specifically on decision problems• There is no loss of generality, but we will not explore

that here

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Finite Domain Problems

• These problems have only a finite number of inputs

Set of All ProblemsFinite Domain Problems Infinite Domain Problems

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Lack of Generality

Set of All ProblemsSolvable Problems Unsolvable Problems

• All finite domain problems can be solved using “table lookup” idea

FiniteDomain

InfiniteDomain

Empty

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Table Lookup Program

int main(string x) {

switch x {

case “Bill”: return(3);

case “Judy”: return(25);

case “Tom”: return(30);

default: cerr << “Illegal input\n”;

}

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Key Concepts

• Classification Theme

• Decision Problems• Important subset of problems• We can focus our attention on decision

problems without loss of generality• Same is not true for finite domain problems

• Table lookup

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Module 4: Formal Definition of Solvability

• Analysis of decision problems• Two types of inputs:yes inputs and no inputs

• Language recognition problem

• Analysis of programs which solve decision problems• Four types of inputs: yes, no, crash, loop inputs

• Solving and not solving decision problems

• Classifying Decision Problems• Formal definition of solvable and unsolvable decision

problems

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Analyzing Decision Problems

Can be defined by two sets

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Decision Problems and Sets

• Decision problems consist of 3 sets• The set of legal input instances (or universe of

input instances)• The set of “yes” input instances • The set of “no” input instances

Yes Inputs No InputsSet of All Legal Inputs

66

Redundancy *

• Only two of these sets are needed; the third is redundant• Given

• The set of legal input instances (or universe of input instances)

– This is given by the description of a typical input instance

• The set of “yes” input instances – This is given by the yes/no question

• We can compute• The set of “no” input instances

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Typical Input Universes

• *: The set of all finite length strings over finite alphabet • Examples

• {a}*: {/\, a, aa, aaa, aaaa, aaaaa, … }• {a,b}*: {/\, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, … }• {0,1}*: {/\, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, … }

• The set of all integers• If the input universe is understood, a decision

problem can be specified by just giving the set of yes input instances

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Language Recognition Problem

• Input Universe• * for some finite alphabet

• Yes input instances• Some set L subset of *

• No input instances• * - L

• When is understood, a language recognition problem can be specified by just stating what L is.

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Language Recognition Problem *

• Traditional Formulation• Input

• A string x over some finite alphabet

• Task• Is x in some language L

subset of ?

• 3 set formulation• Input Universe

• * for a finite alphabet • Yes input instances

• Some set L subset of *

• No input instances• * - L

• When is understood, a language recognition problem can be specified by just stating what L is.

70

Equivalence of Decision Problems and Languages

• All decision problems can be formulated as language recognition problems• Simply develop an encoding scheme for

representing all inputs of the decision problem as strings over some fixed alphabet

• The corresponding language is just the set of strings encoding yes input instances

• In what follows, we will often use decision problems and languages interchangeably

71

Visualization *

Yes Inputs

Original DecisionProblem

No Inputs

EncodingScheme overalphabet

Language L

* - L

CorrespondingLanguage RecognitionProblem

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Analyzing Programs which Solve Decision Problems

Four possible outcomes

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Program Declaration

• Suppose a program P is designed to solve some decision problem . What does P’s declaration look like?

• What should P return on a yes input instance?

• What should P return on a no input instance?

74

Program Declaration II

• Suppose a program P is designed to solve a language recognition problem . What does P’s declaration look like?• bool main(string x) {

• We will assume that the string declaration is correctly defined for the input alphabet

– If = {a,b}, then string will define variables consisting of only a’s and b’s

– If = {a, b, …, z, A, …, Z}, then string will define variables consisting of any string of alphabet characters

75

Programs and Inputs

• Notation• P denotes a program

• x denotes an input for program P

• 4 possible outcomes of running P on x• P halts and says yes: P accepts input x

• P halts and says no: P rejects input x

• P halts without saying yes or no: P crashes on input x• We typically ignore this case as it can be combined with rejects

• P never halts: P infinite loops on input x

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Programs and the Set of Legal Inputs

• Based on the 4 possible outcomes of running P on x, P partitions the set of legal inputs into 4 groups• Y(P): The set of inputs P accepts

• When the problem is a language recognition problem, Y(P) is often represented as L(P)

• N(P): The set of inputs P rejects

• C(P): The set of inputs P crashes on

• I(P): The set of inputs P infinite loops on• Because L(P) is often used in place of Y(P) as described above, we

use notation I(P) to represent this set

77

Illustration

All Inputs

Y(P) N(P) C(P) I(P)

78

Analyzing Programs and Decision Problems

Distinguish the two carefully

79

Program solving a decision problem

• Formal Definition:• A program P solves decision problem if and only if

• The set of legal inputs for P is identical to the set of input instances of

• Y(P) is the same as the set of yes input instances for • N(P) is the same as the set of no input instances for

• Otherwise, program P does not solve problem • Note C(P) and I(P) must be empty in order for P to solve

problem

Y(P) N(P) C(P) I(P)

80

Solvable Problem

• A decision problem is solvable if and only if there exists some C++ program P which solves • When the decision problem is a language

recognition problem for language L, we often say that L is solvable or L is decidable

• A decision problem is unsolvable if and only if all C++ programs P do not solve • Similar comment as above

81

Illustration of Solvability

Inputs of Program P

Y(P) N(P) C(P) I(P)C(P) I(P)

Inputs of Problem

Yes Inputs No Inputs

82

Program half-solving a problem

• Formal Definition:• A program P half-solves problem if and only if

• The set of legal inputs for P is identical to the set of input instances of

• Y(P) is the same as the set of yes input instances for • N(P) union C(P) union I(P) is the same as the set of no

input instances for

• Otherwise, program P does not half-solve problem • Note C(P) and I(P) need not be empty

Y(P) N(P) C(P) I(P)

83

Half-solvable Problem

• A decision problem is half-solvable if and only if there exists some C++ program P which half-solves • When the decision problem is a language

recognition problem for language L, we often say that L is half-solvable

• A decision problem is not half-solvable if and only if all C++ programs P do not half-solve

84

Illustration of Half-Solvability *

Inputs of Program P

Y(P) N(P) C(P) I(P)

Inputs of Problem

Yes Inputs No Inputs

85

Hierarchy of Decision Problems

Solvable

The set of half-solvable decision problems is a proper subset of the set of all decision problems

The set of solvable decision problems is a proper subset of the set of half-solvable decision problems.

Half-solvable

All decision problems

86

Why study half-solvable problems?

• A correct program must halt on all inputs

• Why then do we define and study half-solvable problems?

• One Answer: the set of half-solvable problems is the natural class of problems associated with general computational models like C++ • Every program half-solves some decision problem

• Some programs do not solve any decision problem

• In particular, programs which do not halt do not solve their corresponding decision problems

87

Key Concepts

• Four possible outcomes of running a program on an input

• The four subsets every program divides its set of legal inputs into

• Formal definition of• a program solving (half-solving) a decision problem• a problem being solvable (half-solvable)

• Be precise: with the above two statements!

88

Module 5

• Topics• Proof of the existence of unsolvable problems

• Proof Technique– There are more problems/languages than there are

programs/algorithms

– Countable and uncountable infinities

89

Overview

• We will show that there are more problems than programs• Actually more problems than programs in any

computational model (programming language)

• Implication• Some problems are not solvable

90

Preliminaries

Define set of problems

Observation about programs

91

Define set of problems

• We will restrict the set of problems to be the set of language recognition problems over the alphabet {a}.

• That is• Universe: {a}*• Yes Inputs: Some language L subset of {a}*• No Inputs: {a}* - L

92

Set of Problems *

• The number of distinct problems is given by the number of languages L subset of {a}*• 2{a}* is our shorthand for this set of subset languages

• Examples of languages L subset of {a}*• 0 elements: { }

• 1 element: {/\}, {a}, {aa}, {aaa}, {aaaa}, …

• 2 elements: {/\, a}, {/\, aa}, {a, aa}, …

• Infinite # of elements: {an | n is even}, {an | n is prime}, {an | n is a perfect square}

93

Infinity and {a}*

• All strings in {a}* have finite length

• The number of strings in {a}* is infinite

• The number of languages L in 2{a}* is infinite

• The number of strings in a language L in 2{a}* may be finite or infinite

94

Define set of programs

• The set of programs we will consider are the set of legal C++ programs as defined in earlier lectures

• Key Observation• Each C++ program can be thought of as a finite

length string over alphabet P

• P = {a, …, z, A, …, Z, 0, …, 9, white space, punctuation}

95

Example *

int main(int A[], int n){ {26 characters including newline}

int i, max; {13 characters including initial tab}

{1 character: newline}

if (n < 1) {12 characters}

return (“Illegal Input”); {28 characters including 2 tabs}

max = A[0]; {13 characters}

for (i = 1; i < n; i++) {25 characters}

if (A[i] > max) {18 characters}max = A[i]; {15 characters}

return (max); {15 characters}

} {2 characters including newline}

96

Number of programs

• The set of legal C++ programs is clearly infinite

• It is also no more than |P*|


97

Goal

• Show that the number of languages L in 2{a}* is greater than the number of strings in P*


• Problem• Both are infinite

98

How do we compare the relative sizes of infinite sets?

Bijection (yes)

Proper subset (no)

99

Bijections

• Two sets have EQUAL size if there exists a bijection between them• bijection is a 1-1 and onto function between

two sets

• Examples• Set {1, 2, 3} and Set {A, B, C}• Positive even numbers and positive integers

100

Bijection Example

• Positive Integers Positive Even Integers 1 2

2 4

3 6

... ...

i 2i

… ...

101

Proper subset

• Finite sets• S1 proper subset of S2 implies S2 is strictly

bigger than S1• Example

– women proper subset of people

– number of women less than number of people

• Infinite sets• Counterexample

• even numbers and integers

102

Two sizes of infinity

Countable

Uncountable

103

Countably infinite set S *

• Definition 1• S is equal in size (bijection) to N

• N is the set of natural numbers {1, 2, 3, …}

• Definition 2 (Key property)• There exists a way to list all the elements of set

S (enumerate S) such that the following is true• Every element appears at a finite position in the

infinite list

104

Uncountable infinity *

• Any set which is not countably infinite

• Examples• Set of real numbers• 2{a}*, the set of all languages L which are a

subset of {a}*

• Further gradations within this set, but we ignore them

105

Proof

106

(1) The set of all legal C++ programs is countably infinite

• Every C++ program is a finite string

• Thus, the set of all legal C++ programs is a language LC

• This language LC is a subset of P*

107

For any alphabet , * is countably infinite

• Enumeration ordering• All length 0 strings

• ||0 = 1 string:

• All length 1 strings• || strings

• All length 2 strings• ||2 strings

• …

• Thus, P* is countably infinite

108

Example with alphabet {a,b} *

• Length 0 strings• 0 and

• Length 1 strings• 1 and a, 2 and b

• Length 2 strings• 3 and aa, 4 and ab, 5 and ba, 6 and bb, ...

• Question• write a program that takes a number as input and

computes the corresponding string as output

109

(2) The set of languages in 2{a}* is uncountably infinite

• Diagonalization proof technique • “Algorithmic” proof• Typically presented as a proof by contradiction

110

Algorithm Overview *

• To prove this set is uncountably infinite, we construct an algorithm D that behaves as follows:• Input

• A countably infinite list of languages L[] subset of {a}*

• Output• A language D(L[]) which is a subset of {a}* that is not

on list L[]

111

Visualizing D

List L[]

L[0]

L[1]

L[2]

L[3]

...

LanguageD(L[]) not inlist L[]

Algorithm D

112

Why existence of D implies result

• If the number of languages in 2{a}* is countably infinite, there exists a list L[] s.t.• L[] is complete

• it contains every language in 2{a}*

• L[] is countably infinite

• The existence of algorithm D implies that no list of languages in 2{a}* is both complete and countably infinite• Specifically, the existence of D shows that any

countably infinite list of languages is not complete

113

Visualizing One Possible L[ ] *

L[0]

L[1]

L[2]

L[3]

L[4]...

a aa aaa aaaa ...

IN ININININ

OUTIN IN INOUT

OUT OUT OUT OUT OUT

IN INOUT OUT OUT

ININOUT OUTOUT

•#Rows is countably infinite

•Given

•#Cols is countably infinite

• {a}* is countably infinite

• Consider each string to be a feature• A set contains or does not contain each string

114

Constructing D(L[ ]) *

• We construct D(L[]) by using a unique feature (string) to differentiate D(L[]) from L[i]• Typically use ith string for language L[i]• Thus the name diagonalization

L[0]

L[1]

L[2]

L[3]

L[4]...

a aa aaa aaaa ...

IN ININININ

OUTIN IN INOUT

OUT OUT OUT OUT OUT

IN INOUT OUT OUT

ININOUT OUTOUT

OUT

IN

IN

IN

OUT

D(L[])

115

D(L[]) cannot be any language L[i]

• D(L[]) cannot be language L[0] because D(L[]) differs from L[0] with respect to string a0 = /\.• If L[0] contains /\, then D(L[]) does not, and vice versa.

• D(L[]) cannot be language L[1] because D(L[]) differs from L[1] with respect to string a1 = a.• If L[1] contains a, then D(L[]) does not, and vice versa.

• D(L[]) cannot be language L[2] because D(L[]) differs from L[2] with respect to string a2 = aa.• If L[2] contains aa, then D(L[]) does not, and vice versa.

• …

116

Questions

• Do we need to use the diagonal?• Every other column and every row?• Every other row and every column?

• What properties are needed to construct D(L[])?

L[0]

L[1]

L[2]

L[3]

L[4]...

a aa aaa aaaa ...

IN ININININ

OUTIN IN INOUT

OUT OUT OUT OUT OUT

IN INOUT OUT OUT

ININOUT OUTOUT

117

Visualization

Solvable Problems

The set of solvable problems is a proper subset of the set of all problems.

All problems

118

Summary

• Equal size infinite sets: bijections• Countable and uncountable infinities

• More languages than algorithms• Number of algorithms countably infinite• Number of languages uncountably infinite• Diagonalization technique

• Construct D(L[]) using infinite set of features

• The set of solvable problems is a proper subset of the set of all problems

119

Module 6

• Topics• Program behavior problems

• Input of problem is a program/algorithm

• Definition of type program

• Program correctness• Testing versus Proving

120

Number Theory Problems

• These are problems where we investigate properties of numbers• Primality

• Input: Positive integer n

• Yes/No Question: Is n a prime number?

• Divisor• Input: Integers m,n

• Yes/No question: Is m a divisor of n?

121

Graph Theory Problems

• These are problems where we investigate properties of graphs• Connected

• Input: Graph G

• Yes/No Question: Is G a connected graph?

• Subgraph• Input: Graphs G1 and G2

• Yes/No question: Is G1 a subgraph of G2?

122

Program Behavior Problems

• These are problems where we investigate properties of programs and how they behave

• Give an example problem with one input program P

• Give an example problem with two input programs P1 and P2

123

Program Representation

• Program variables• Abstractly, we define the type “program”

• graph G, program P

• More concretely, we define type program to be a string over the program alphabet P = {a, …, z, A, …, Z, 0, …, 9, punctuation, white space}

• Note, many strings over P are not legal programs

• We consider them to be programs that always crash

• Possible declaration of main procedure• bool main(program P)

124

Program correctness

• How do we determine whether or not a program P we have written is correct?

• What are some weaknesses of this approach?

• What might be a better approach?

125

Testing versus Analyzing

ProgramP

Test Inputsx1

x2

x3

...

OutputsP(x1)P(x2)P(x3)

...

AnalyzerProgram P Analysis ofProgram P

126

2 Program Behavior Problems *

• Correctness• Input

• Program P

• Yes/No Question• Does P correctly solve the primality problem?

• Functional Equivalence• Input

• Programs P1, P2

• Yes/No Question• Is program P1 functionally equivalent to program P2

127

Module 7

• Halting Problem• Fundamental program behavior problem• A specific unsolvable problem• Diagonalization technique revisited

• Proof more complex

128

Definition

• Input• Program P

• Assume the input to program P is a single nonnegative integer– This assumption is not necessary, but it simplifies the following

unsolvability proof– To see the full generality of the halting problem, remove this

assumption

• Nonnegative integer x, an input for program P

• Yes/No Question• Does P halt when run on x?

• Notation• Use H as shorthand for halting problem when space is a

constraint

129

Example Input *

• Program with one input of type unsignedbool main(unsigned Q) {

int i=2;

if ((Q = = 0) || (Q= = 1)) return false;

while (i<Q) {

if (Q%i = = 0) return (false);

i++;

}

return (true);

}

• Input x4

130

Three key definitions

131

Definition of list L *

• P* is countably infinite where P = {characters, digits, white

space, punctuation}• Type program will be type string with P as the alphabet• Define L to be the strings in P

* listed in enumeration order• length 0 strings first• length 1 strings next• …

• Every program is a string in P • For simplicity, consider only programs that have

• one input• the type of this input is an unsigned

• Consider strings in P* that are not legal programs to be

programs that always crash (and thus halt on all inputs)

132

Definition of PH *

• If H is solvable, some program must solve H

• Let PH be a procedure which solves H

• We declare it as a procedure because we will use PH as a subroutine

• Declaration of PH

• bool PH(program P, unsigned x)• In general, the type of x should be the type of the input to P

• Comments• We do not know how PH works

• However, if H is solvable, we can build programs which call PH as a subroutine

133

bool main(unsigned y) /* main for program D */{ program P = generate(y);

if (PH(P,y)) while (1>0); else return (yes); }

/* generate the yth string in P* in enumeration order */

program generate(unsigned y) /* code for program of slide 21 from module 5 did this for

{a,b}* */

bool PH(program P, unsigned x)/* how PH solves H is unknown */

Definition of program D

134

Generating Py from y *

• We won’t go into this in detail here• This was the basis of the question at the bottom of slide 21 of

lecture 5 (alphabet for that problem was {a,b} instead of P).• This is the main place where our assumption about the input type for

program P is important• for other input types, how to do this would vary

• Specification• 0 maps to program • 1 maps to program a• 2 maps to program b• 3 maps to program c• …• 26 maps to program z• 27 maps to program A• …

135

Proof that H is not solvable

136

Argument Overview *

H is solvable

D is NOT on list L

PH exists

Definition ofSolvability

D exists

D’s code

D is on list L

L is list ofall programs

D does NOT exist

PH does NOT exist

H is NOT solvable

p → q is logically equivalent to (not q) → (not p)

137

Proving D is not on list L

• Use list L to specify a program behavior B that is distinct from all real program behaviors (for programs with one input of type unsigned)• Diagonalization argument similar to the one for proving the

number of languages over {a} is uncountably infinite

• No program P exists that exhibits program behavior B

• Argue that D exhibits program behavior B• Thus D cannot exist and thus is not on list L

138

Non-existent program behavior B

139

Visualizing List L *

P0

P1

P2

P3

P4...

1 2 3 4 ...

H HHHH

NHH H HNH

NH NH NH NH NH

H H H H H

HHNH NHNH

•#Rows is countably infinite

• p* is countably infinite

•#Cols is countably infinite

• Set of nnnegative integers is countably infinite

• Consider each number to be a feature• A program halts or doesn’t halt on each integer• We have a fixed L this time

140

Diagonalization to specify B *

P0

P1

P2

P3

P4...

1 2 3 4 ...

H HHHH

NHH H HNH

NH NH NH NH NH

H H H H H

HHNH NHNH

•We specify a non-existent program behavior B by using a unique feature (number) to differentiate B from Pi

NH

H

H

H

NH

B

141

Arguing D exhibits program behavior B

142

bool main(unsigned y) /* main for program D */{ program P = generate(y);

if (PH(P,y)) while (1>0); else return (yes); }

/* generate the yth string in P* in enumeration order */

program generate(unsigned y) /* code for extra credit program of slide 21 from lecture 5 did

this for {a,b}* */

bool PH(program P, unsigned x)/* how PH solves H is unknown */

Code for D

143

Visualization of D in action on input y

• Program D with input y• (type for y: unsigned)

Given input y, generate the program (string) Py

Run PH on Py and y

• Guaranteed to halt since PH solves H

IF (PH(Py,y)) while (1>0); else return (yes);

P0

P1

P2

Py...

1 2 ...

HH

H H

NH NH

H

NH

NH

D ... y

...

HNH

144

D cannot be any program on list L

• D cannot be program P0 because D behaves differently on 0 than P0 does on 0.• If P0 halts on 0, then D does not, and vice versa.



• …

145

Alternate Proof

146

Alternate Proof Overview

• For every program Py, there is a number y that we associate with it

• The number we use to distinguish program Py from D is this number y

• Using this idea, we can arrive at a contradiction without explicitly using the table L• The diagonalization is hidden

147

H is not solvable, proof II

• Assume H is solvable• Let PH be the program which solves H

• Use PH to construct program D which cannot exist

• Contradiction• This means program PH cannot exist.

• This implies H is not solvable

• D is the same as before

148

Arguing D cannot exist

• If D is a program, it must have an associated number y

• What does D do on this number y?

• 2 cases• D halts on y

• This means PH(D,y) = NO

– Definition of D

• This means D does not halt on y

– PH solves H

• Contradiction

• This case is not possible

149

Continued

• D does not halt on this number y

• This means PH(D,y) = YES

– Definition of D

• This means D halts on y– PH solves H

• Contradiction

• This case is not possible

• Both cases are not possible, but one must be for D to exist

• Thus D cannot exist

150

Implications *

• The Halting Problem is one of the simplest problems we can formulate about program behavior

• We can use the fact that it is unsolvable to show that other problems about program behavior are also unsolvable

• This has important implications restricting what we can do in the field of software engineering• In particular, “perfect” debuggers/testers do not exist

• We are forced to “test” programs for correctness even though this approach has many flaws

151

Summary

• Halting Problem definition• Basic problem about program behavior

• Halting Problem is unsolvable• We have identified a specific unsolvable

problem• Diagonalization technique

• Proof more complicated because we actually need to construct D, not just give a specification B

152

Module 8

• Closure Properties• Definition• Language class definition

• set of languages

• Closure properties and first-order logic statements

• For all, there exists

153

Closure Properties

• A set is closed under an operation if applying the operation to elements of the set produces another element of the set

• Example/Counterexample• set of integers and addition• set of integers and division

154

Integers and Addition

Integers

2

5

7

155

Integers and Division

Integers

2

5

.4

156

Language Classes

• We will be interested in closure properties of language classes

• A language class is a set of languages

• Thus, the elements of a language class (set of languages) are languages which are sets themselves

• Crucial Observation• When we say that a language class is closed under

some set operation, we apply the set operation to the languages (elements of the language classes) rather than the language classes themselves

157

Example Language Classes *

• In all these examples, we do not explicitly state what the underlying alphabet is

• Finite languages• Languages with a finite number of strings

• CARD-3• Languages with at most 3 strings

158

Finite Sets and Set Union *

Finite Sets

{0,1,00}

{0,1,11}

{0,1,00,11}

159

CARD-3 and Set Union

CARD-3

{0,1,00}

{0,1,11}

{0,1,00,11}

CARD-3: sets with at most 3 elements

160

Finite Sets and Set Complement

Finite Sets

{0,1,01}

{/\,00,10,11,000,...}

161

Infinite Number of Facts

• A closure property often represents an infinite number of facts

• Example: The set of finite languages is closed under the set union operation• {} union {} is a finite language• {} union {} is a finite language• {} union {0} is a finite language• ...• {} union {} is a finite language• ...

162

First-order logic and closure properties *

• A way to formally write (not prove) a closure property• L1, ...,Lk in LC, op (L1, ... Lk) in LC

• Only one expression is needed because of the for all quantifier

• Number of languages k is determined by arity of the operation op

163

Example F-O logic statements *

• L1,L2 in FINITE, L1 union L2 in FINITE

• L1,L2 in CARD-3, L1 union L2 in CARD-3

• L in FINITE, Lc in FINITE

• L in CARD-3, Lc in CARD-3

164

Stating a closure property is false

• What is true if a set is not closed under some k-ary operator?• There exist k elements of that set which, when

combined together under the given operator, produce an element not in the set

• L1, ...,Lk in LC, op (L1, …, Lk) not in LC

• Example• Finite sets and set complement

165

Complementing a F-O logic statement

• Complement “L1,L2 in CARD-3, L1 union L2 in CARD-3”

• not (L1,L2 in CARD-3, L1 union L2 in CARD-3)

• L1,L2 in CARD-3, not (L1 union L2 in CARD-3)

• L1,L2 in CARD-3, L1 union L2 not in CARD-3

166

Proving/Disproving

• Which is easier and why?• Proving a closure property is true• Proving a closure property is false

167

Module 9

• Recursive and r.e. language classes• representing solvable and half-solvable problems

• Proofs of closure properties • for the set of recursive (solvable) languages• for the set of r.e. (half-solvable) languages

• Generic element/template proof technique• Relationship between RE and REC

• pseudoclosure property

168

RE and REC language classes

• REC• A solvable language is commonly referred to as

a recursive language for historical reasons• REC is defined to be the set of solvable or

recursive languages

• RE• A half-solvable language is commonly referred

to as a recursively enumerable or r.e. language• RE is defined to be the set of r.e. or half-

solvable languages

169

Why study closure properties of RE and REC?

• It tests how well we really understand the concepts we encounter• language classes, REC, solvability, half-

solvability

• It highlights the concept of subroutines and how we can build on previous algorithms to construct new algorithms• we don’t have to build our algorithms from

scratch every time

170

Example Application

• Setting• I have two programs which can solve the

language recognition problems for L1 and L2

• I want a program which solves the language recognition problem for L1 intersect L2

• Question• Do I need to develop a new program from scratch

or can I use the existing programs to help?• Does this depend on which languages L1 and L2 I am

working with?

171

Closure Properties of REC *

• We now prove REC is closed under two set operations• Set Complement• Set Intersection

• In these proofs, we try to highlight intuition and common sense

172

Set Complement Example

• Even: the set of even length strings over {0,1}• Complement of Even?

• Odd: the set of odd length strings over {0,1}

• Is Odd recursive (solvable)?• How is the program P’ that solves Odd related to

the program P that solves Even?

173

Set Complement Lemma

• If L is a solvable language, then L complement is a solvable language

• Proof• Let L be an arbitrary solvable language

• First line comes from For all L in REC

• Let P be the C++ program which solves L• P exists by definition of REC

174

• Modify P to form P’ as follows• Identical except at very end

• Complement answer – Yes → No

– No → Yes

• Program P’ solves L complement• Halts on all inputs

• Answers correctly

• Thus L complement is solvable• Definition of solvable

proof continued

175

P’ Illustration

PInput x

YES

No

P’

YES

No

176

Code for P’

bool main(string y)

{

if (P (y)) return no; else return yes;

}

bool P (string y)

/* details deleted; key fact is P is guaranteed to halt on all inputs */

177

Set Intersection Example

• Even: the set of even length strings over {0,1}• Mod-5: the set of strings of length a multiple of 5

over {0,1}• What is Even intersection Mod-5?

• Mod-10: the set of strings of length a multiple of 10 over {0,1}

• How is the program P3 (Mod-10) related to programs P1 (Even) and P2 (Mod-5)

178

Set Intersection Lemma

• If L1 and L2 are solvable languages, then L1 intersection L2 is a solvable language

• Proof• Let L1 and L2 be arbitrary solvable languages

• Let P1 and P2 be programs which solve L1 and L2, respectively

179

• Construct program P3 from P1 and P2 as follows

• P3 runs both P1 and P2 on the input string

• If both say yes, P3 says yes

• Otherwise, P3 says no

• P3 solves L1 intersection L2 • Halts on all inputs

• Answers correctly

• L1 intersection L2 is a solvable language

proof continued

180

P3 Illustration

P1

P2

Yes/No

Yes/No

ANDYes/No

P3

181

Code for P3

bool main(string y)

{

if (P1(y) && P2(y)) return yes;

else return no;

}bool P1(string y) /* details deleted; key fact is P1 always halts. */

bool P2(string y) /* details deleted; key fact is P2 always halts. */

182

Other Closure Properties

• Unary Operations• Language Reversal• Kleene Star

• Binary Operations• Set Union• Set Difference• Symmetric Difference• Concatenation

183

Closure Properties of RE *

• We now try to prove RE is closed under the same two set operations• Set Intersection • Set Complement

• In these proofs• We define a more formal proof methodology• We gain more intuition about the differences

between solvable and half-solvable problems

184

RE Closed Under Set Intersection• Expressing this closure property as an

infinite set of facts• Let Li denote the ith r.e. language

• L1 intersect L1 is in RE


• ...


• ...

185

Generic Element or Template Proofs

• Since there are an infinite number of facts to prove, we cannot prove them all individually

• Instead, we create a single proof that proves each fact simultaneously

• I like to call these proofs generic element or template proofs

186

Basic Proof Ideas

• Name your generic objects• In this case, we use L1 and L2

• Only use facts which apply to any relevant objects• We will only use the fact that there must exist P1 and

P2 which half-solve L1 and L2

• Work from both ends of the proof• The first and last lines are usually obvious, and we can

often work our way in

187

Set Intersection Example *

• Let L1 and L2 be arbitrary r.e. languages

• L1 intersection L2 is an r.e. language

• There exist P1 and P2 s.t. Y(P1)=L1 and Y(P2)=L2

• By definition of half-solvable languages

• There exists a program P which half-solves L1 intersection L2

• Construct program P3 from P1 and P2

• Note, we can assume very little about P1 and P2

• Prove Program P3 half-solves L1 intersection L2

188

Constructing P3 *

• Run P1 and P2 in parallel

• One instruction of P1, then one instruction of P2, and so on

• If both halt and say yes, halt and say yes

• If both halt but both do not say yes, halt and say no

189

P3 Illustration

P1

P2

Yes/No/-

Yes/No/-

ANDYes/No/-

P3

Input

190

Code for P3

bool main(string y){

parallel-execute(P1(y), P2(y)) until both return;

if ((P1(y) && P2(y)) return yes;else return no;

}

bool P1(string y) /* key fact is P1 only guaranteed to halt on yes input instances */

bool P2(string y) /* key fact is P2 only guaranteed to halt on yes input instances */

191

Proving P3 Is Correct

• 2 steps to showing P3 half-solves L1 intersection L2

• For all x in L1 intersection L2, must show P3 • accepts x

– halts and says yes

• For all x not in L1 intersection L2, must show P3 does what?

192

Part 1 of Correctness Proof

• P3 accepts x in L1 intersection L2

• Let x be an arbitrary string in L1 intersection L2

• Note, this subproof is a generic element proof

• P1 accepts x

• L1 intersection L2 is a subset of L1

• P1 accepts all strings in L1

• P2 accepts x

• P3 accepts x• We reach the AND gate because of the 2 previous facts

• Since both P1 and P2 accept, AND evaluates to YES

193

Part 2 of Correctness Proof

• P3 does not accept x not in L1 intersection L2

• Let x be an arbitrary string not in L1 intersection L2

• By definition of intersection, this means x is not in L1 or L2

• Case 1: x is not in L1 • 2 possibilities

• P1 rejects (or crashes on) x

– One input to AND gate is No– Output cannot be yes

– P3 does not accept x

• P1 loops on x

– One input never reaches AND gate– No output

– P3 loops on x

• P3 does not accept x when x is not in L1

• Case 2: x is not in L2

• Essentially identical analysis

• P3 does not accept x not in L1 intersection L2

194

RE closed under set complement?• First-order logic formulation?

• What this really means• Let Li denote the ith r.e. language

• L1 complement is in RE

• L2 complement is in RE

• ...

195

Set complement proof overview

• Let L be an arbitrary r.e. language

• L complement is an r.e. language

• There exists P s.t. Y(P)=L• By definition of r.e. languages

• There exists a program P’ which half-solves L complement

• Construct program P’ from P• Note, we can assume very little about P

• Prove Program P’ half-solves L complement

196

Constructing P’

• What did we do in recursive case?• Run P and then just complement answer at end

• Accept → Reject• Reject → Accept

• Does this work in this case?• No. Why not?

• Does this prove that RE is not closed under set complement?

197

Other closure properties

• Unary Operations• Language reversal• Kleene Closure

• Binary operations• union (on practice hw)• concatenation

• Not closed• Set difference (on practice hw)

198

Closure Property Applications

• How can we use closure properties to prove a language LT is r.e. or recursive?

• Unary operator op (e.g. complement)• 1) Find a known r.e. or recursive language L

• 2) Show LT = L op

• Binary operator op (e.g. intersection)• 1) Find 2 known r.e or recursive languages L1 and L2

• 2) Show LT = L1 op L2

199

Closure Property Applications *

• How can we use closure properties to prove a language LT is not r.e. or recursive?

• Unary operator op (e.g. complement)• 1) Find a known not r.e. or non-recursive language L

• 2) Show LT op = L

• Binary operator op (e.g. intersection)• 1) Find a known r.e. or recursive language L1

• 2) Find a known not r.e. or non-recursive language L2

• 3) Show L2 = L1 op LT

200

Example

• Looping Problem• Input

• Program P

• Input x for program P

• Yes/No Question• Does P loop on x?

• Looping Problem is unsolvable• Looping Problem complement = H

201

Closure Property Applications

• Proving a new closure property

• Theorem: Unsolvable languages are closed under set complement• Let L be an arbitrary unsolvable language

• If Lc is solvable, then L is solvable• (Lc)c = L

• Solvable languages closed under complement

• However, we are assuming that L is unsolvable

• Therefore, we can conclude that Lc is unsolvable

• Thus, unsolvable languages are closed under complement

202

Pseudo Closure Property

• Lemma: If L and Lc are half-solvable, then L is solvable.

• Question: What about Lc?

203

High Level Proof

• Let L be an arbitrary language where L and Lc are both half-solvable

• Let P1 and P2 be the programs which half-solve L and Lc, respectively

• Construct program P3 from P1 and P2

• Argue P3 solves L

• L is solvable

204

Constructing P3

• Problem• Both P1 and P2 may loop on some input strings,

and we need P3 to halt on all input strings

• Key Observation• On all input strings, one of P1 and P2 is

guaranteed to halt. Why?

205

Illustration

*

L

P1 halts

Lc

P2 halts

206

Construction and Proof

• P3’s Operation

• Run P1 and P2 in parallel on the input string x until one accepts x

• Guaranteed to occur given previous argument

• Also, only one program will accept any string x

• IF P1 is the accepting machine THEN yes ELSE no

207

P3 Illustration

P1

P2

Yes

Yes

P3

Input

Yes

No

208

Code for P3 *

bool main(string y)

{

parallel-execute(P1(y), P2(y)) until one returns yes;

if (P1(y)) return yes;

if (P2(Y)) return no;

}bool P1(string y) /* guaranteed to halt on strings in L*/

bool P2(string y) /* guaranteed to halt on strings in Lc */

209

RE and REC

REC

RE

All Languages

L Lc

210

RE and REC

REC

RE

All Languages

LLc

Lc

Lc

Are there any languages L in RE - REC?

211

Module 10

• Universal Algorithms• moving beyond one problem at a time• operating system/general purpose computer

212

Observation

• So far, each program solves one specific problem• Divisor• Sorting• Multiplication• Language L

213

Universal Problem/Program

• Universal Problem (nonstandard term)• Input

• Program P

• Input x to program P

• Task• Compute P(x)

• Univeral Program• Program which solves universal problem

• Universal Turing machine

214

Example Input *

int main(A[6]) { Inputint i,temp;for (i=1;i<=3;i++) A[1] = 6

if (A[i] > A[i+3]) { A[2] = 4temp = A[i+3]; A[3] = 2

A[i+3] = A[i]; A[4] = 3A[i] = temp; A[5] = 5

} A[6] = 1

for (i=1; i<=5; i++)for (j=i+1;j<=6;j++)

if (A[j-1] > A[j]) {temp = A[j];A[j] = A[j-1];A[j-1] = temp;

}

}

215

Organization

Universal Program’s Memory

Program P

Program P’s Memory

Program P

int main(A[6]){

int i,temp;

for (i=1;i<=3;i++)if (A[i] > A[i+3]) {

temp = A[i+3];

A[i+3] = A[i];

A[i] = temp;

}

for (i=1; i<=5; i++)

for (j=i+1;j<=6;j++)if (A[j-1] > A[j]) {

temp = A[j];

A[j] = A[j-1];

A[j-1] = temp;

}

}Program Counter

int A[6],i,temp;

642531

Line 1

216

Description of Universal Program

• Basic Loop• Find current line of program P• Execute current line of program P

• Update program P’s memory

• Update program counter• Return to Top of Loop

217

Past, Present, Future• Turing came up with the concept of a universal program (Universal

Turing machine) in the 1930’s• This is well before the invention of the general purpose computer

• People were still thinking of computing devices as special-purpose devices (calculators, etc.)

• Turing helped move people beyond this narrow perspective

• Turing/Von Neumann perspective• Computers are general purpose/universal algorithms

• Focused on computation• Stand-alone

• Today, we are moving beyond this view• Computation, communication, cyberspace

• However, results in Turing perspective still relevant

218

Halting Problem Revisited *

• Halting Problem is half-solvable• Modified Universal Program (MUP) half-

solves HRun P on xOutput yes

– This step only executed if first step halts

• Behavior• What does MUP do on all yes instances of H?• What does MUP do on all no inputs of H?

219

Debuggers

• How are debugger’s like gdb or ddd related to universal programs?

• How do debuggers simplify the debugging process?

220

RE and REC

• We now have a problem that is half-solvable but not solvable

• What do we now know about the complement of the Halting Problem?

• What additional fact about RE and set complement does this prove?

221

RE and REC

REC

RE

ll Languages

HHc

222

Summary

• Universal Programs• 1930’s, Turing• Introduces general purpose computing concept• Not a super intelligent program, merely a

precise follower of instructions

• Halting Problem half-solvable but not solvable• RE not closed under set complement

223

Module 11

• Proving more specific problems are not solvable

• Input transformation technique• Use subroutine theme to show that if one

problem is unsolvable, so is a second problem• Need to clearly differentiate between

• use of program as a subroutine and

• a program being an input to another program

224

Basic Idea/Technique

225

Proving a problem L is unsolvable

• Assume PL is a procedure that solves problem L• We have no idea how PL solves L

• Construct a program PH that solves H using PL as a subroutine• We use PL as a black box• (We could use any unsolvable problem in place of H)

• Argue PH solves H• Conclude that L is unsolvable

• Otherwise PL would exist and then H would be solvable• L will be a problem about program behavior

226

Focusing on H

• In this module, we will typically use H, the Halting Problem, as our known unsolvable problem

• The technique generalizes to using any unsolvable problem L’ in place of H.• You would need to change the proofs to work with L’

instead of H, but in general it can be done

• The technique also can be applied to solvable problems to derive alternative consequences

• We focus on H to simplify the explanation

227

Constructing PH using PL

Answer-preserving input transformations and Program PT

228

PH has two subroutines

• There are many ways to construct PH using program PL that solves L

• We focus on one method in which PH consists of two subroutines• Procedure PL that solves L• Procedure PT which computes a function f that I

call an answer-preserving (or answer-reversing) input transformation

• More about this in a moment

229

Pictoral Representation of PH *

PH

x Yes/NoPLY/NPT

PT(x)

230

Answer-preserving input transformation PT

• Input• An input to H

• Output• An input to L such that

• yes inputs of H map to yes inputs of L• no inputs of H map to no inputs of L

• Note, PT must not loop when given any legal input to H

231

Why this works *

PH

PLPT

yes input to H yes input to L yes

no input to H no input to L no

We have assumed that PL solves L

232

Answer-reversing input transformation PT

• Input• An input to H

• Output• An input to L such that

• yes inputs of H map to no inputs of L• no inputs of H map to yes inputs of L

• Note, PT must not loop when given any legal input to H

233

Why this works

PH

PLPT

yes input to H no input to L yes

no input to H yes input to L no

We have assumed that PL solves L

no

yes

234

Yes->Yes and No->No

Domain of H

Yes inputsfor H

No inputsfor H

Yes inputsfor L

No inputsfor L

Domain of L

PLPT

PH

x PT(x) Yes/No

235

Notation and Terminology

• If there is such an answer-preserving (or answer-reversing) input transformation f (and the corresponding program PT), we say that H transforms to (many-one reduces to) L

• NotationH ≤ L

Domain of H

Yes inputs No inputs

Yes inputs No inputsDomain of L

236

Examples not involving the Halting Problem

237

Generalization

• As noted earlier, while we focus on transforming H to other problems, the concept of transformation generalizes beyond H and beyond unsolvable program behavior problems

• We work with some solvable, language recognition problems to illustrate some aspects of the transformation process in the next few slides

238

Example 1

• L1 is the set of even length strings over {0,1}• What are the set of legal input instances and no inputs for

the L1 LRP?

• L2 is the set of odd length strings over {0,1}• Same question as above

• Tasks• Give an answer-preserving input transformation f that

shows that L1 LRP ≤ L2 LRP

• Give a corresponding program PT that computes f

Domain of L1


Yes inputs No inputsDomain of L2

239

Program PT

string main(string x)

{

return(x concatenate “0”);

}

240

Example 2

• L1 is the set of all strings over {0,1}

• What is the set of all inputs, yes inputs, no inputs for the L1 LRP?

• L2 is {0}• Same question as above

• Tasks• Give an answer-preserving input transformation f which

shows that the L1 LRP ≤ L2 LRP

• Give a corresponding program PT which computes f

Domain of L1



241

Program PT

string main(string x)

{

return( “0”);

}

242

Example 3

• L1 • Input: Java program P that takes as input an unsigned int• Yes/No Question: Does P halt on all legal inputs

• L2

• Input: C++ program P that takes as input an unsigned int• Yes/No Question: Does P halt on all legal inputs

• Tasks• Describe what an answer-preserving input transformation f

that shows that L1 ≤ L2 would be/do?

Domain of L1



243

Proving a program behavior problem L is unsolvable

244

Problem Definitions *

• Halting Problem H• Input

• Program QH that has one input of type unsigned int

• non-negative integer y that is input to program QH

• Yes/No Question• Does QH halt on y?

• Target Problem L• Input

• Program QL that has one input of type string

• Yes/No question• Does Y(QL) = the set of

even length strings?

• Assume program PL solves L

245

Construction review

PH

x Yes/No

•We are building a program PH to solve the halting problem H

PTPT(x)

•PH will use PT as a subroutine, and we must explicitly construct PT using specific properties of H and L

PLY/N

•PH will use PL as a subroutine, and we have no idea how PL accomplishes its task

246

P’s and Q’s

• Programs which are PART of program PH and thus “executed” when PH executes• Program PT, an actual program we construct• Program PL, an assumed program which solves

problem L

• Programs which are INPUTS/OUTPUTS of programs PH, PL, and PT and which are not “executed” when PH executes• Programs QH, QL, and QYL

• code for QYL is available to PT

247

Two inputs for L *


• Program Q that has one input of type string

• Yes/No question• Does Y(Q) = the set of


• Program PL

• Solves L• We don’t know how

• Consider the following program Q1

bool main(string z)

{while (1>0) ;}

• What does PL output when given Q1 as input?

• Consider the following program Q2

bool main(string z)

{ if ((z.length %2) = = 0) return (yes)

else return (no); }

• What does PL output when given Q2 as input?

248

Another input for L *


• Program Q that has one input of type string

• Yes/No question• Does Y(Q) = the set of


• Program PL

• Solves L• We don’t know how

• Consider the following program QL with 2 procedures Q1 and QYL

bool main(string z) {

Q1(5); /* ignore return value */

return(QYL(z));}

bool Q1(unsigned x) {if (x > 3) return (no); else loop;

}

bool QYL(string y) {if ((y.length( ) % 2) = = 0) return (yes);

else return(no);}

• What does PL output when given QL as input?

249

Input and Output of PT *

• Input of PT

• (Also Input of H)• Program QH

• one input of type unsigned int

• Non-negative integer y

• Program QL that is the output of PT

• (Also input of L)bool main(string z) {

QH(y); /* QH and y come left-hand side */ /* ignore return value */

return(QYL(z));}

bool QH(unsigned x) {/* comes from left-hand side

}

bool QYL(string y) {if ((y.length( ) % 2) = = 0) return (yes);

else return(no);}

QH,y PT QL

250

Example 1 *

Input to PT

Program QH

bool main(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}

Input y5

Output of PT

Program QL

bool QH(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}bool QYL(string z) { if ((z.length % 2) == 0) return (yes) else return (no);}bool main(string z) { unsigned y = 5; QH(y); return (QYL(z));}

QH,y PT QL

251

Example 2

Input to PT

Program QH

bool main(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}

Input y3

Output of PT

Program QL

bool QH(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}bool QYL(string z) { if ((z.length % 2) == 0) return (yes) else return (no);}bool main(string z) { unsigned y = 3; QH(y); return (QYL(z));}

QH,y PT QL

252

PT in more detail

253

Declaration of PT

• What is the return type of PT?• Type program1 (with one input of type string)

• What are the input parameters of PT?• The same as the input parameters to H; in this case,

• type program2 (with one input of type unsigned int)• unsigned int (input type to program2)

program1 main(program2 QH, unsigned y)

PLPT

PH

QH,y QL Yes/No

254

program1 main(program2 P, unsigned y) {

/* Will be viewing types program1 and program2 as STRINGS over the program alphabet P */

program1 QL = replace-main-with-QH(P);/* Insert line break */

QL += “\n”;

/* Insert QYL */

QL += “bool QYL(string z) {\n \t if ((z.length % 2) == 0) return (yes) else return (no);\n }”;

/* Add main routine of QL */

QL += “bool main(string z) {\n\t”; /* determined by L */

QL += “unsigned y =”

QL += convert-to-string(y);

QL += “;\n\t QH(y)\n\t return(QYL(z));\n}”;

return(QL);}

program1 replace-main-with-QH(program2 P) /* Details hidden */string convert-to-string(unsigned y) /* Details hidden */

Code for PT PLPT

PH

QH,y QL Yes/No

255

PT in action

PT

code for QYL

QH

unsigned ystart QYL

Y/Nz

Y/N

QL

haltQH

y

Program QHbool main(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}

Input y5

Program QLbool QH(unsigned y) { if (y ==5) return yes; else if (y ==4) return no; else while (1>0) {};}bool QYL(string z) { if ((z.length % 2) == 0) return (yes) else return (no);}bool main(string z) { unsigned y = 5; QH(y); return (QYL(z));}

PT

QYL

PLPT

PH

QH,y QL Yes/No

256

Constructing QL (and thus PT)

How to choose QYL or QNL

257

Start with no input for H

• If QH, y is a no input to the Halting problem

• Program QL bool main(string z) {

QH(y); /* ignore return value */

return(Q?L(z)); /* yes or no? */}


}

bool Q?L(string y) {

}

– Thus Y(QL) = {}

– QH loops on y

– Determine if this makes QL a no or yes input instance to L

258

Answer-preserving input transformation


– Thus Y(QL) = {}

– QH loops on y




return(QYL(z)); /* yes */}


}

bool QYL(string y) {

}

– Now choose a QYL (or QNL) that is a yes (or no) input instance to L

259

Make yes for H map to yes for L


– Thus Y(QL) = {}

– QH loops on y







}

bool QYL(string y) {if ((y.length( ) % 2) = = 0) return (yes);else return (no);

}

260

Possible shortcut

Program QL bool main(string z) {

QH(y); /* ignore return value */if ((z.length( ) % 2) = = 0) return (yes);else return (no);

}


}

Program QL bool main(string z) {




}


}

261

Another Example

262

Problem Definitions

• Halting Problem H• Input

• Program QH that has one input of type unsigned int

• non-negative integer y that is input to program QH

• Yes/No Question• Does QH halt on y?


• Program QL that has one input of type string

• Yes/No question• Is Y(QL) finite?

• Assume program PL solves L

263

Start with no input for H




return(Q?L(z)); /* yes or no? */}


}

bool Q?L(string y) {

}

– Thus Y(QL) = {}

– QH loops on y


264

Answer-reversing input transformation




return(QNL(z)); /* no */}


}

bool QNL(string y) {

}

– Thus Y(QL) = {}

– QH loops on y



265

Make yes for H map to no for L




return(QNL(z)); /* no */}


}

bool QNL(string y) {if ((y.length( ) % 2) = = 0) return(yes);else return(no);

}

– Thus Y(QL) = {}

– QH loops on y



266

Analyzing proposed transformations

4 possibilities

267

Problem Setup

• Input of Transformation• Program QH, unsigned x

• Output of Transformation• Program QL bool main(string z) {


return(QYL(z)); /* yes or no */}

bool QH(unsigned x) {}


}

• Problem L• Input: Program P• Yes/No Question: Is Y(P) =

{aa}?

• Question: Is the transformation on the left an answer-preserving or answer-reversing input transformation from H to problem L?

268

Key Step







}


{aa}?

• The output of the transformation is the input to the problem.• Plug QL in for

program P above• Is Y(QL) = {aa}?

269

Is Y(QL) = {aa}?


{aa}?• Analysis

• If QH loops on x, Y(QL)={}• No input to H creates a QL that

is a no input for L• If QH halts on x, Y(QL) =

{even length strings}• Yes input to H creates a QL

that is a no input for L• Transformation does not work

• All inputs map to no inputs







}

270

Three other problems

• Problem L1• Input: Program P• Yes/No Question: Is Y(P)

infinite?


finite?

• Problem L3• Input: Program P• Yes/No Question: Is Y(P) =

{} or is Y(P) infinite?







}

271

Is Y(QL) infinite?


infinite?• Analysis

• If QH loops on x, Y(QL)={}• No input to H creates a QL

that is a no input for L• If QH halts on x, Y(QL) =


that is a yes input for L• Transformation works

• Answer-preserving







}

272

Is Y(QL) finite?


finite?• Analysis


that is a yes input for L• If QH halts on x, Y(QL) =


that is a no input for L• Transformation works

• Answer-reversing







}

273

Is Y(QL) = {} or is Y(QL) infinite?

• Problem L3• Input: Program P• Yes/No Question: Is Y(P) =

{} or is Y(P) infinite?• Analysis


that is a yes input for L• If QH halts on x, Y(QL) =


that is a yes input for L• Transformation does not work

• All inputs map to yes inputs







}

strings and languages operations

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