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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

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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.

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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

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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?

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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

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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

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Illustration

All Inputs

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

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Analyzing Programs and Decision Problems

Distinguish the two carefully

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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)

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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

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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

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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)

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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

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Illustration of Half-Solvability *

Inputs of Program P

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

Inputs of Problem

Yes Inputs No Inputs

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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

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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

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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!