Module 4: Formal Definition of Solvability

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Module 4 Formal Definition of Solvability

• Analysis of decision problems
• Two types of inputsyes 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

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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

• S The set of all finite length strings over
  finite alphabet S
• 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
• S for some finite alphabet S
• Yes input instances
• Some set L subset of S
• No input instances
• S - L
• When S 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 S
• Task
• Is x in some language L subset of S?

• 3 set formulation
• Input Universe
• S for a finite alphabet S
• Yes input instances
• Some set L subset of S
• No input instances
• S - L
• When S 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 S
• 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
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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 P. What does Ps 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 P. What does Ps
  declaration look like?
• bool main(string x)
• We will assume that the string declaration is
  correctly defined for the input alphabet S
• If S a,b, then string will define variables
  consisting of only as and bs
• If S 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
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 P if and only
  if
• The set of legal inputs for P is identical to the
  set of input instances of P
• Y(P) is the same as the set of yes input
  instances for P
• N(P) is the same as the set of no input instances
  for P
• Otherwise, program P does not solve problem P
• Note C(P) and I(P) must be empty in order for P
  to solve problem P

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Solvable Problem

• A decision problem P is solvable if and only if
  there exists some C program P which solves P
• 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 P is unsolvable if and only if
  all C programs P do not solve P
• Similar comment as above

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Illustration of Solvability
Inputs of Program P
Y(P)
N(P)
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Program half-solving a problem

• Formal Definition
• A program P half-solves problem P if and only if
• The set of legal inputs for P is identical to the
  set of input instances of P
• Y(P) is the same as the set of yes input
  instances for P
• N(P) union C(P) union I(P) is the same as the set
  of no input instances for P
• Otherwise, program P does not half-solve problem
  P
• Note C(P) and I(P) need not be empty

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Half-solvable Problem

• A decision problem P is half-solvable if and only
  if there exists some C program P which
  half-solves P
• When the decision problem is a language
  recognition problem for language L, we often say
  that L is half-solvable
• A decision problem P is not half-solvable if and
  only if all C programs P do not half-solve P

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Illustration of Half-Solvability
Inputs of Program P
Y(P)
N(P)
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Hierarchy of Decision Problems
All decision problems
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.
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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!