This assignment is to be done either individually or in pairs. Do not show your code to any other group and do not look at any other group's code. Do not put your code in a public directory or otherwise make it public. However, you may get help from the mentors, TAs, or the instructor. You are encouraged to use the LMS Discussions page to post problems so that other students can also answer/see the answers.
The goal of this assignment is to write a lambda calculus interpreter in a functional programming language to reduce lambda calculus expressions in a call-by-value (applicative order) manner.
You are to use the following grammar for the lambda calculus:
<expression> ::= <atom> | "\" <atom> "." <expression> | "(" <expression> " " <expression> ")"
Your interpreter is expected to take each lambda calculus expression and repeatedly perform beta reduction until no longer possible (a value expression that can no longer be beta-reduced) and then eta reduction until no longer possible.
In the above grammar, <atom> is defined as a lower
case letter followed by a sequence of zero or more alphanumeric
characters, excluding Oz language keywords. A full listing of Oz
language keywords can be found on P. 839 Table C.8 of "Concepts,
Techniques, and Models of Computer Programming".
Your interpreter is to take lambda calculus expressions from a text
file (one expression per line) and reduce them sequentially.
To enable you to focus on the lambda calculus semantics, a parser is
provided in Haskell, and parsed expressions are provided in Oz.
Hints: You may define auxiliary procedures for alpha-renaming, beta-reduction, and eta-conversion. For beta reduction, you may want to write an auxiliary procedure that substitutes all occurrences of a variable in an expression for another expression. Be sure that the replacing expression does not include free variables that would become captured in the substitution. Remember that in call-by-value, the argument to a function is evaluated before the function is called.
Below are some lambda calculus interpretation test cases:
| Expression | Result | Comment |
|---|---|---|
(\x.\y.(y x) (y w)) | \z.(z (y w)) | Avoid capturing the free variable y in (y w) |
(\x.\y.(x y) (y w)) | (y w) | Avoid capturing the free variable y in (y w), and perform eta reduction |
(\x.x y) | y | Identity combinator |
\x.(y x) | y | Eta reduction |
((\y.\x.(y x) \x.(x x)) y) | (y y) | Application combinator |
(((\b.\t.\e.((b t) e) \x.\y.x) x) y) | x | If-then-else combinator |
\x.((\x.(y x) \x.(z x)) x) | (y z) | Eta reductions |
(\y.(\x.\y.(x y) y) (y w)) | (y w) | Alpha renaming, beta reduction and eta reduction all involved |
For your convenience, these have been given in a sample input file, where each line contains one lambda expression.
You should remove any other Oz installations and use Oz 1.4.0 downloadable from http://sourceforge.net/projects/mozart-oz/files/v1/ and Emacs downloadable from http://gnu.mirror.vexxhost.com/emacs/. To enable the Oz programming interface to find the proper emacs running command such as runemacs.exe, you need to create an environment variable OZEMACS and set its value to the install location. For example: F:\Programs\emacs-21.1\bin\runemacs.exe
Use this starter code to see how to interpret parsed lambda calculus expressions. You must include this code in your final submission. The answers should be printed in the Browser window with the Browse function, one line per expression. Lambda calculus expressions are parsed as atoms for variables, as two-element lambda tuples for functional abstractions, and as two-element lists for function applications. Browse statements have been provided for the 8 test cases detailed in the above table.
Use this parser to get a list of lambda
calculus expressions from an input file. See also the sample usage of the parser (in
particular, see the runProgram function.) Specifically, type
constructors for the Lexp datatype have been exported
from the module. This datatype is used to represent lambda calculus expressions
in Haskell, and the type constructors should be used to pattern match
a lambda expression. Your goal is to create a reducer
function that takes an Lexp value as input and returns a
Lexp value as output.
Note: Please name your main file 'main.hs'. It should take a filename as an argument (defaulting to 'input.lambda'); the file itself will consist of multiple lines, as described earlier, each with a single lambda expression on it. The provided code will parse this file. You should output the simplified lambda expression, as described in the sample interaction below.
$ runghc main.hs sample.lambda Please enter a filename containing lambda expressions: Input 1: (\x.x y) Result 1: y Input 2: (\x.\y.(x y) (y w)) Result 2: (y w)
If you get an error about Text.Parsec being undefined, run stack install parsec (or cabal install parsec)
Further Haskell Hints: It may be useful to consider Map and Set, which can be found in the Data.Map and Data.Set modules, respectively. It is also recommended to use Hoogle, a search engine for looking up Haskell documentation.
Grading: The assignment will be graded mostly on correctness, but code clarity / readability will also be a factor (comment, comment, comment!).
Submission Requirements: Please submit a ZIP file with your code, including a README file. Your ZIP file should be named with your LMS user name(s) as the filename underscore the language you chose. Examples: userid1_oz.zip, userid1_userid2_hs.zip, userid1_lisp.zip. Only submit one assignment per pair via LMS. In the README file, place the names of each group member (up to two). Your README file should also have a list of specific features/bugs in your solution.