MIT6_034-Lab 0 Solved

The purpose of this lab is to familiarize you with this term's lab system and to diagnose your programming ability and facility with Python. 6.034 uses Python for all of its labs, and you will be called on to understand the functioning of large systems, as well as to write significant pieces of code yourself.  

While coding is not, in itself, a focus of this class, artificial intelligence is a hard subject full of subtleties. As such, it is important that you be able to focus on the problems you are solving, rather than the mechanical code necessary to implement the solution.  

Python
There are a number of versions of Python available. This course will use standard Python ("CPython") from http://www.python.org/. If you are running Python on your own computer, you should download and install Python 2.5 or Python 2.6 from http://www.python.org/download/ . All the lab code will require at least version 2.3.  

Mac OS X comes with Python 2.3 pre-installed, but the version you can download from python.org has better support for external libraries and a better version of IDLE.  

Answering questions  
The main file of this lab is called lab0.py. Open that file in IDLE. The file contains a lot of incomplete statements that you need to fill in with your solutions.  

The first thing to fill in is a multiple choice question. The answer should be extremely easy. Many labs will begin with some simple multiple choice questions to make sure you're on the right track.  

Run the tester  
Every lab comes with a file called tester.py. This file checks your answers to the lab. For problems that ask you to provide a function, the tester will test your function with several different inputs and see if the output is correct. For multiple choice questions, the tester will tell you if your answer was right. Yes, that means that you never need to submit wrong answers to multiple choice questions.  

The tester has two modes: "offline" mode (the default), and "online" or "submit" mode. The former runs some basic, self-contained internal tests on your code. It can be run as many times as you would like.

The latter runs some more tests, some of which may be randomly generated, and uploads your code to the 6.034 grader for grading.  

You can run the online tester as many times as you want. If your code fails a test, you can submit it and try again. Because you always have the opportunity to fix your bugs, you can only get a 5 on a problem set if it passes all the tests. If your code fails a test, your grade will be 4 or below.  

Using IDLE  
If you are using IDLE, or if you do not have easy access to a command line (as on Windows), IDLE can run the tester.  

Open the tester.py file and run it using Run Module or F5. This will run the offline tests for you. When the offline tests pass (or when you're up against a deadline, or when you have questions for the staff) you can  

 test_online() 

to submit your code and run the online tests.  

In fact, it will run the submission and online test just as soon as you pass the offline tests, saving you a few keystrokes.  

You should run the tester (and submit!) early and often. Think of it as being like the "Check" button from 6.01. It makes sure you're not losing points unnecessarily. Submitting your code makes it easy for the staff to look at it and help you.  

Using the command line  
If you realize just how much emacs and/or the command line rock, then you can open your operating system's Terminal or Command Prompt, and cd to the directory containing the files for Lab 0. Then, run:  

python tester.py 

to run the offline tester, and  

python tester.py submit 

to submit your code and run the online tester.  

You should run the tester (and submit!) early and often. Think of it as being like the "Check" button from 6.01. It makes sure you're not losing points unnecessarily. Submitting your code makes it easy for the staff to look at it and help you.  

Python programming  

Now it's time to write some Python.  

Warm-up stretch  
Write the following functions:  

•        cube(n), which takes in a number and returns its cube. For example, cube(3) => 27.  

•        factorial(n), which takes in a non-negative integer n and returns n!, which is the product of the integers from 1 to n. (0! = 1 by definition.)  

We suggest that you should write your functions so that they raise nice clean errors instead of dying messily when the input is invalid. For example, it would be nice if factorial rejected negative inputs right away; otherwise, you might loop forever. You can signal an error like this: raise Exception, "factorial: input must not be negative"  

Error handling doesn't affect your lab grade, but on later problems it might save you some angst when you're trying to track down a bug.  

•        count_pattern(pattern lst), which counts the number of times a certain pattern of symbols appears in a list, including overlaps. So count_pattern( ('a', 'b'), ('a', 

'b', 'c', 'e', 'b', 'a', 'b', 'f')) should return 2, and count_pattern(('a', 'b', 'a'), ('g', 'a', 'b', 'a', 'b', 'a', 

'b', 'a')) should return 3.  

Expression depth  
One way to measure the complexity of a mathematical expression is the depth of the expression describing it in Python lists. Write a program that finds the depth of an expression.  

For example:  

•        depth('x') => 0  

•        depth(('expt', 'x', 2)) => 1  

•        depth(('+', ('expt', 'x', 2), ('expt', 'y', 2))) => 2  

•        depth(('/', ('expt', 'x', 5), ('expt', ('-', ('expt', 'x', 2), 1), ('/', 5, 2)))) => 4  

 

Note that you can use the built-in Python "isinstance()" function to determine whether a variable points to a list of some sort. "isinstance()" takes two arguments: the variable to test, and the type (or list of types) to compare it to. For example:  

>>> x = [1, 2, 3] 

>>> y = "hi!" 

>>> isinstance(x, (list, tuple)) 

True 

>>> isinstance(y, (list, tuple)) 

False 

Tree reference  
  

Your job is to write a procedure that is analogous to list referencing, but for trees. This "tree_ref" procedure will take a tree and an index, and return the part of the tree (a leaf or a subtree) at that index.

For trees, indices will have to be lists of integers. Consider the tree in Figure 1, represented by this

Python tuple: (((1, 2), 3), (4, (5, 6)), 7, (8, 9, 10))  

To select the element 9 out of it, we’d normally need to do something like tree[3][1]. Instead, we’d prefer to do tree_ref(tree, (3, 1)) (note that we’re using zero-based indexing, as in list-ref, and that the indices come in top-down order; so an index of (3, 1) means you should take the fourth branch of the main tree, and then the second branch of that subtree). As another example, the element 6 could be selected by tree_ref(tree, (1, 1, 1)).  

Note that it’s okay for the result to be a subtree, rather than a leaf. So tree_ref(tree, (0,)) should return ((1, 2), 3).  

Symbolic algebra  
Throughout the semester, you will need to understand, manipulate, and extend complex algorithms implemented in Python. You may also want to write more functions than we provide in the skeleton file for a lab.  

In this problem, you will complete a simple computer algebra system that reduces nested expressions made of sums and products into a single sum of products. For example, it turns the expression (2 * 

(x + 1) * (y + 3)) into ((2 * x * y) + (2 * x * 3) + (2 * 1 * y) + (2 * 

1 * 3)). You could choose to simplify further, such as to ((2 * x * y) + (6 * x) + (2 * y) + 6)), but it is not necessary.  

This procedure would be one small part of a symbolic math system, such as the integrator presented in Monday's lecture. You may want to keep in mind the principle of reducing a complex problem to a simpler one.  

An algebraic expression can be simplified in this way by repeatedly applying the associative law and the distributive law.  

Associative law  

((a + b) + c) = (a + (b + c)) = (a + b + c)  

((a * b) * c) = (a * (b * c)) = (a * b * c)  

Distributive law  

((a + b) * (c + d)) = ((a * c) + (a * d) + (b * c) + (b * d))  

The code for this part of the lab is in algebra.py. It defines an abstract Expression class, Sum

and Product expressions, and a method called Expression.simplify(). This method starts by applying the associative law for you, but it will fail to perform the distributive law. For that it delegates to a function called do_multiply that you need to write. Read the documentation in the code for more details.  

This exercise is meant to get you familiar with Python and using it to solve an interesting problem. It is intended to be algorithmically straightforward. You should try to reason out the logic that you need for this function on your own.