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When we refer to the computability of a problem on a computer, what question are we trying to answer?
We want to know what the limitations of the problem-solving capabilities of computers.
When we refer to the complexity of solving a problem with a computer, what are we referring to?
We want to know what kinds of problems can be solved on a computer in a reasonable time, using a reasonable amount of memory.
What are the three main components of a Turing machine?
A Turing machine consists of a tape (that extends infinitely in both directions), a read/write head that moves at the direction of the control unit, and a control unit that keeps track of what "state" the machine is in, directs the tape to move (and in which direction), and whether and what the head should write.
How does the control unit decide what the machine should do?
- The control unit keeps track of the state of the machine (what it is currently doing), then evaluates the current symbol under the read/write head to make 3 determinations:
- 1) determine which symbol to place in the current cell
- 2) determine whether to move the read/write head one cell to the left or right, and
- 3) determine the next state of the machine
A Turing machine can be defined by a ____________________________.
state transition diagram
Review state transition diagram class activity.
What is the Church-Turing Thesis?
The Church-Turing Thesis states that the set of functions that can be calculated on a computer is exactly the same as the set of functions for which a Turing machine can be devised.
Give an example of a problem that has been proven to be non-computable.
The Halting Problem has been proven to be non-computable. It asks: given a program with a set of input values, does the program halt on that input, or does it get stuck in an infinite loop?
What does time complexity refer to?
It is a measure of how many steps are executed when the associated program is run. (Big O notation)
What does "Big O" notation provide information about?
- It provides information regarding the program's "order of complexity".
- O(1) -> execution time doesn't relate to the size of the number n.
- O(n) -> execution time increases linearly as n increases
- O(n*squared) -> execution time increases quadratically as n increases
Name three types of complexity an algorithm can have other than linear and quadratic (also not O(1)).
- *logarithmic - number of steps in its execution is bounded by a logarithmic function (k ln(n))
- *polynomial - number of steps in its execution is bounded by a polynomial function (ix^2+jx+k)
- *exponential - number of steps in its execution is bounded by an exponential function (k*2 to the n)
Why do we care about the complexity of an algorithm?
When you perform a small number of calculations, you can use an algorithm that is costly in time and not be penalized too much. But if you apply the same algorithm to a large number of calculations, the problem may take so long on your machine that it is effectively incalculable.
What is The Knapsack Problem?
You have a knapsack that can hold up to W weight; no more than that. Given a set of jewels, each of which has a weight w and a price p, pack the maximum worth of gems into the knapsack without exceeding its capacity.