The purpose of this assignment is to help you get acquainted with pure object-oriented programming. The assignment is divided into two parts.
You will find a uSmalltalk interpreter in
This interpreter treats the variable
&trace specially; by defining it to be an integer n with
val &trace n,
you can trace message sends and answers to depth
It is an invaluable aid to debugging.
You will find useful sources are in the git repository, which you can clone by
git clone linux.cs.tufts.edu:/comp/105/build-prove-compare
examples directory includes
copies of the initial basis, collection classes, financial history,
and other examples from the textbook.
localProtocolmethods which are defined on every class, as shown in Figure 10.5 on page 427.
To solve these problems, you shouldn't need to add or change more than 10 lines of code in total.
Natural class is over 100 lines of uSmalltalk code;
my large-integer classes are 22 lines apiece.
My modifications to predefined number classes are about 25 lines.
Estimated difficulty: ****
You will find bignums and the bignum algorithms discussed at some length in Dave Hanson's book (which should be free online to Tufts students) and in the article by Per Brinch Hansen. Be aware that your assignment below differs significantly from the implementation in Hanson's book.
Natural. I recommend that you use the
digit:put:methods to hide the 1-based nature of the underlying representation.
decimalmethod must return a list of decimal digits, even if base 10 is not what is used in the representation. Suppress leading zeroes unless the value of
Naturalis itself zero.
XP_adddoes add with carry.
XP_subdoes subtract with borrow.
z := z + x * y, which is useful, but is not what we want unless
zis zero initially. Moreover, Hanson has to pass all the lengths explicitly.
Naturalis an immutable type. Your methods must not mutate existing natural numbers; you can mutate only a newly allocated number that you are sure has not been seen by any client.
digit:method carefully, you'll have to worry about sizes only when you allocate new results.
SmallInteger. In order to make your solution work with an unmodified usmalltalk, you must use this technique.
<fact.smt>= (define factorial (n) (if (strictlyPositive n) [(* n (value factorial (- n 1)))] )) (class Factorial Object () (classMethod printUpto: (limit) (locals n nfac) (begin (set n 1) (set nfac 1) (while [(<= n limit)] [(print n) (print #!) (print space) (print #=) (print space) (println nfac) (set n (+ n 1)) (set nfac (* n nfac))]))))
You might find it useful to test your implementation with the following table of factorials:
1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40320 9! = 362880 10! = 3628800 11! = 39916800 12! = 479001600 13! = 6227020800 14! = 87178291200 15! = 1307674368000 16! = 20922789888000 17! = 355687428096000 18! = 6402373705728000 19! = 121645100408832000 20! = 2432902008176640000 21! = 51090942171709440000 22! = 1124000727777607680000 23! = 25852016738884976640000 24! = 620448401733239439360000 25! = 15511210043330985984000000Be warned that this test by itself is inadequate. You will want other tests. Here is some advice
Naturalby generating a long, random string of digits, then computing the corresponding number using a combination of addition and multiplication by 10.
(set p n) ; p == n (set p (+ p p)) ; p == 2n (set p (+ p p)) ; p == 4n (set p (+ p n)) ; p == 5n (set p (+ p p)) ; p == 10nThis idea will test only your addition; if you have bugs there, fix them before you go on.
Listthat prints just the digits, with no spaces.
timesRepeat:method is defined on any integer.
definesyntax is slightly different from what we use in uScheme.) We recommend you use ghci on the command line; standard infix syntax works. If you want something more elaborate, use Standard ML of New Jersey (command sml), which has an
IntInfmodule that implements bignums.
Your bigtests.smt file should be formatted as follows:
; Summary: .........(Your summary should be a simple English phrase that describes your test. Examples might be ``Ackermann's function of (1, 1),'' ``sequence of powers of 2,'' or ``combinations of +, *, and - on random numbers.'')
Test105with a class method
runthat actually runs the test.
runmethod to the
Test105class, and nothing else. If the output is a single line, write a one-line comment. If the output takes multiple lines, put each line of output in a comment on its own line.
(run Test105)must take less than 2 CPU seconds to evaluate.
<example bigtests.smt>= ; Summary: 10 to the tenth power (class Test105 Object () (class-method run () (locals n 10-to-the-n) (set n 0) (set 10-to-the-n 1) (whileTrue: [(< n 10)] [(set n (+ n 1)) (set 10-to-the-n (* 10 10-to-the-n))]) 10-to-the-n) ) ; 10000000000
If this test is run in an unmodified interpreter, it breaks with an arithmetic overflow and a stack trace.
b = 2and sometimes
b = 10, but when we want bignums, the choice of
bis hard to make in the general case:
b= 10, then converting to decimal representation is trivial, but storing bignums requires lots of memory.
bis, the less memory is required, and the more efficient everything is.
bis a power of 10, converting to decimal is relatively easy and is very efficient. Otherwise it requires (possibly long) division.
(b-1) * (b-1)fits in a machine word, than you can implement multiplication in high-level languages without difficulty. (Serious implementations pick the largest
a[i]is guaranteed to fit in a machine word, e.g.,
2^64on modern machines. Unfortunately, to work with such large values of
brequires special machine instructions to support ``add with carry'' and 128-bit multiply, so serious implementations have to be written in assembly language.)
bis a power of 2, bit-shift can be very efficient, but conversion to decimal is expensive. Fast bit-shift can be important in cryptographic and communications applications.
b's-complement. Knuth volume 2 is pretty informative about these topics.
For extra credit,
try the following variations on your implementation of class
baseand not any other code.) Measure the time needed to compute the first 50 factorials. Note both your measurements and your argument in your README file.
Naturaland for large integers. If this changes your argument for the largest possible base, explain how.
Change the base to the largest reasonable base, not necessarily a
power of 10.
You will have to re-implement
decimal using long division.
Measure the time needed to compute and print the first 50 factorials.
Does the smaller number of digits recoup the higher cost of converting
Make sure comparisons work, even with mixed kinds of integers.
So for example, make sure comparisons such as
(< 5 (* 1000000 1000000)) produce sensible answers.
to keep track of the size of numbers, and measure the
space cost of the different bases.
Estimate the difference in garbage-collection overhead for computing
with the different bases, given a fixed-size heap.
Pi (hard). Use a power series to compute the first 100 digits of pi (the ratio of a circle's circumference to its diameter). Be sure to cite your sources for the proper series approximation and its convergence properties. Hint: I vaguely remember that there's a faster convergence for pi over 4. Check with a numerical analyst.
finhist.smtshowing your solution to Exercise 9.
basis.smtshowing whatever changes you had to make to the initial basis to do Exercise 29a. Please identify your solutions using conspicuous comments, e.g.,
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;; ;;;; solution to Exercise 4 (class Array ... )
READMEfile that should include some indication of how you tested your bignum code for Part II. Of course we also want the usual stuff about your collaborators and your time spent.
bignum.smtshowing your solutions to Exercises 31, 32, and 33. This file must work with an unmodified
usmalltalkinterpreter. Therefore, if for example you use results from Exercises 4, 29a, or any other problem (e.g., the class method
Arrayclass), you will need to duplicate those results in
bignum.smtas well as in
bigtests.smtshowing your solutions to Exercise T.
Submit code using submit105-small-solo and submit105-small-pair.
• The base used for natural numbers appears in exactly one place, and all code that depends on it consults that place.
• Or, the base used for natural numbers appears in exactly one place, and code that depends on either consults that place or assumes that the base is some power of 10
• No matter how many bits are used to represent a machine integer, overflow is detected by using appropriate primitive methods, not by comparing against particular integers.
• Code uses method dispatch instead of conditionals.
• Mixed operations on different classes of numbers are implemented using double dispatch.
• Or, mixed operations on different classes of numbers are implemented by arranging for the classes to share a common protocol.
• Or, mixed operations on different classes of numbers are implemented by arranging for unconditional coercions.
• Code deals with exceptional or unusual conditions by passing a suitable
• Code achieves new functionality by reusing existing methods, e.g., by sending messages to
• Or, code achieves new functionality by adding new methods to old classes to respond to an existing protocol.
• An object's behavior is controlled by dispatching (or double dispatching) to an appropriate method of its class.
• The base used for natural numbers appears in exactly one place, but code that depends on it knows what it is, and that code will break if the base is changed in any way.
• Overflow is detected only by assuming the number of bits used to represent a machine integer, but the number of bits is explicit in the code.
• Code contains one avoidable conditional.
• Mixed operations on different classes of integers involve explicit conditionals.
• Code protects itself against exceptional or unusual conditions by using Booleans.
• Code contains methods that appear to have been copied and modified.
• An object's behavior is influenced by interrogating it to learn something about its class.
• The base used for natural numbers appears in multiple places.
• Overflow is detected only by assuming the number of bits used to represent a machine integer, and the number of bits is implicit in the value of some frightening decimal literal.
• Code contains more than one avoidable conditional.
• Mixed operations on different classes of integers are implemented by interrogating objects about their classes.
• Code copies methods instead of arranging to invoke the originals.
• Code contains case analysis or a conditional that depends on the class of an object.
• Course staff see no more code than is needed to solve the problem.
• Course staff see how the structure of the code follows from the structure of the problem.
• Course staff see somewhat more code than is needed to solve the problem.
• Course staff can relate the structure of the code to the structure of the problem, but there are parts they don't understand.
• Course staff roughly twice as much code as is needed to solve the problem.
• Course staff cannot follow the code and relate its structure to the structure of the problem.
• Private methods are documented with contracts.
• Or, private methods use exactly the contracts suggested in the bignums handout.
• Private methods are neither documented nor consistent with the bignums handout.
• Code that works with collections works with any
• Code that is supposed to works with all collections works only with some subclasses of collections.