Module 2: Fundamental Concepts

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Module 2 Fundamental Concepts

• Problems
• Programs
• Programming languages

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Problems

• We view solving problems as the main application
  for computer programs

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Definition

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

(4,2,3,1)
(1,2,3,4)
(3,1,2,4)
(1,5,7)
(7,5,1)
(1,2,3)
(1,2,3)
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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

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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.

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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 gt 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 lt 1)
• return (Illegal Input)
• max A0
• for (i 1 i lt n i)
• if (Ai gt max)
• max Ai
• 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
  P such that
• P can be solved by some program P in some other
  reasonable programming language
• but P cannot be solved by any C program?

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Churchs Thesis (modified)

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