1. Lecture 2: Fundamental Concepts
Problems
Programs
Programming languages

2. Studying the Theme
How do we prove something CAN be done by SOME program?
We write a specific program which does it.
How do we prove something CANNOT be done by ANY program?
We have to develop an argument which takes into account EVERY program that could be written.

3. We view solving problems as the main application for computer programs

4. Definition
A problem is a mapping or function between a set of inputs and a set of outputs
Example Problem: Sorting
Inputs
Outputs
(4,2,3,1)
(1,2,3,4)
(3,1,2,4)
(1,5,7)
(7,5,1)
(1,2,3)
(1,2,3)

5. 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

6. Example Problem Specifications
Sorting problem
Input
Integers n1, n2, ..., nk
Output
n1, n2, ..., nk in nondecreasing order
Find element problem
Integers n1, n2, …, nk
Search key S
yes if S is in n1, n2, …, nk, no otherwise

7. Programs solve problems

8. 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

9. 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.

10. 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

11. Maximum Element Problem
Input
integer n >= 1
List of n integers
Output
The largest of the n integers

12. 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); }

13. Exploring capabilities and limitations of C++ programs
Fundamental Theme
Exploring capabilities and limitations of C++ programs

14. 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

15. Studying the Theme
How do we prove a problem CAN be solved by SOME C++ program?
We write a specific C++ program which solves it.
How do we prove something CANNOT be solved by ANY C++ program?
We have to develop an argument which takes into account EVERY C++ program that could be written.

16. 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 programming language
but P cannot be solved by any C++ program?

17. Church’s Thesis (modified)
We have no proof of an answer, but it is commonly accepted that the answer is no.
Church’s Thesis (modified)
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

18. Summary Problems Questions
When we talk about what programs can or cannot “DO”, we mean what PROBLEMS can or cannot be solved
Questions
For any problem, can there be more than one program which solves it?
For any program, can it solve more than one problem?
Which is harder: proving a problem solvable or unsolvable?