Download Session 13 - Computer Science

Ethics of Computing
MONT 113G, Spring 2012
Session 13
Limits of Computer Science
1
Limitations of Computer
Science
In practice, we cannot solve every problem with a program
because...
1) The execution time of a program may be too long.
2) The problem may be non-computable.
3) We may not know how to solve the problem.
E.g. Computer Vision
Artificial Intelligence
Modeling complex systems such as weather
etc.
2
Execution Time
The execution time of a program can be a limitation in what
we can compute.
Examples:
Taxes: How long would it take to sort taxpayers by
Social Security number?
Electricity: What is the most efficient routing of power
lines through a region or neighborhood?
Tractable problems can be solved in a reasonable amount of
time.
Intractable problems require huge amounts of time to solve.
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A Tractable Problem
Suppose we have a list of patients and an associated list of patient
weights.
We want to write a program that prints out the names of patients
with weight above a given value.
The names are stored in a list of strings.
The weights are stored in a parallel list of real numbers.
name
Amy Ted Ann Tom . . .
1 2
3
4
5 ...
weight
115 145 132 224 . . .
1 2
3
4
5 ...
n
n
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The solution and its running
time
Python solution:
for i in range(n):
if weight[i] > targetWeight:
print name[i]
The program examines each patient in turn.
The running time is proportional to the number of patients in
the list (n).
time
n (# patients)
2500
5000
7500
10000
time (seconds)
1.275
2.550
3.825
5.100
5.1
10000 n
5
Linear Running time
The graph of the running time of the previous problem is a line.
t = 5.1 x 10-4 n
We say the the running time is linear.
Problems that have linear running times can be solved in a
reasonable amount of time. These problems are tractable.
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Another example
Suppose we want to sort the patients by their weight. I.e. we
would like to rearrange the list so that the weights go from
lowest to highest.
Many sorting algorithms run as a function of n2. For example:
t = 3.2 x 10-3 n2
Others run as a function of n log2 n. For example:
t = 2.1 x 10-3 n log2 n
Logarithms: If 2x = n, then log2 n = x
23= 8, log2 8 = 3
7
Graphing the running time for
a sorting function
# people (n)
2500
5000
7500
10000
t (seconds)
59.261
129.021
202.745
279.042
time
n2
n log n
n
n
Sorting a list takes more time than searching a list, but it still
can be done in a reasonable amount of time.
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Polynomial and
Polylogarithmic running times
In general, tractable problems are problems that have running
times that are a polynomial function of n (the size of the
input) or a polylogarithmic function of n.
Examples:
t = 4n3 + 5n2 -3
t = n2(log2(n))3
t = 17 n6 - log2 n + 6n
9
Intractable Computations
A computation is intractable if the running time increases
faster than a polynomial of the input size (n).
For example:
Running times that increase exponentially:
t = 3n
t = 2n + n3
Running times that increase like the factorial of n:
t = n!
t = 4n3 + 3n!
(Recall that n! = 1x2x3x4 ...x(n-1)xn )
10
Example of an intractable
problem:
Find all possible orderings (permutations) of n objects.
n = 3; A, B, C (we will work this out in class)
ABC BAC CAB
ACB BCA CBA
Total of 6 possibilities.
In general, for n objects, their are n! possible orderings.
n
5
6
7
8
9
10
time
< 1 sec
3.19 sec
22.57 sec
3.05 min
27.87 min
4.64 hours
n
20
21
22
time
Can you guess?
??
??
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The Traveling Salesman
problem
Suppose a salesperson wants to visit a series of n cities,
beginning and ending with the home city. He wants to visit
each city only once (except the home city where he begins
and ends).
What order of cities will give him the shortest possible trip?
B
15
A
43
27
16
E
21
14
19 45
32
52
37
12
F
15
13
C
23
D
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Solving Traveling Salesman
problem
To solve the traveling salesman problem, we must compute
the distance for every possible path:
ABCDEFA
ACBDEFA
ADBDEFA
...
How many possibilities are there?
(n - 1)!
Therefore, this is an intractable problem.
13
Alternative solutions for
intractable problems
Many intractable problems involve solving for the best
solution.
We can often speed up the solution if we are willing to accept a
good solution.
For example, in the traveling salesman problem:
1) Find the shortest path from the current city to a city that
has not yet been visited.
2) If no cities are still unvisited, go home. Otherwise,
repeat step 1.
This will lead to a good solution (e.g. a reasonably short path),
but not necessarily the best solution (i.e. the shortest path). 14
Limitations of Computer
Science
In practice, we cannot solve every problem with a program
because...
1) The execution time of a program may be too long.
2) The problem may be non-computable.
3) We may not know how to solve the problem.
E.g. Computer Vision
Artificial Intelligence
Modeling complex systems such as weather
etc.
15
The Halting Problem
One problem that frequently arises when programming is
when the program contains a loop that never terminates.
These are called infinite loops.
It would be a nice compiler feature if the compiler could
detect the presence of an infinite loop in a program and
inform the programmer that one exists.
The Halting Problem: Can we write a program that will
read in other programs and output a message saying whether
or not the program will halt on all inputs (i.e. it does not have
any infinite loops)?
16
Proof by Contradiction
•
•
The halting problem assertion is as follows:
No finite program can be written that will check other
programs and halt in a finite time giving a solution to
the halting problem.
We can prove the halting problem assertion using a
technique known as proof by contradiction.
The idea is as follows:
1) Assume that a program solving the halting problem exists.
2) Show that this assumption leads to impossible
consequences.
3) This implies that the assumption must be false and
therefore the program cannot exist.
17
Other non-computable
problems
Almost every problem related to the behavior of a program is
non-computable. For example:
1) Showing that 2 programs are equivalent (for each input
you always get the same output)
2) Determining whether a given output will occur
3) Determining the correctness of a program
etc.
18
Limitations in integer
representation
The circuitry of computers leads to limitations in what they can
compute.
Integer storage limitations:
Because we use a limited number of bits to store integers, there
is a maximum integer that can be represented:
16 bits: Maximum = 32,767 (if storing both positive and
negative integers)
32 bits: Maximum = 2 x 109 (approximately)
Is this big enough?
19
Limits of Real number
representation
The format of real numbers leads to limitations because of:
1) Limited precision of the representation.
2) Rounding error
3) Limits in the smallest number that can be represented
(underflow)
4) Limits in the largest number that can be represented
(overflow).
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Other hardware limitations
1. Components can fail (disk crash, computer crash, etc).
2. Components can have design flaws so they don't work as
they should (Intel pentium chip, 1994)
3. Electronic noise can change 1's to 0's and vice versa when
information is sent over wires or cables (or through the air
for wireless).
There are error checking and error correcting techniques,
but these are not 100% accurate.
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