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A simple Lambda Calculus expression interpreter in Java
If you already understand the Lambda Calculus, you can skip this section and head straight to The Application.
<expression> = <name> | <function> | <application>
<function> = λ<name>.<expression>
<application> = <expression> <expression>
After running the application you will be presented with a short introductory message followed by the prompt into which you can enter lambda expressions or terms:
λ-CALCULUS: Type 'help' for more information
λ>
We can enter lambda expressions at the λ> prompt, which the application will attempt to beta-reduce:
λ-CALCULUS: Type 'help' for more information
λ> λx.x
β> λx.x
In the above we entered the expression λx.x which cannot be reduced so we see the β> prompt followed by our initial expression.
We can also create terms and then apply them:
λ> id=λx.x
β> id = λx.x
λ> id y
β> y
In the above we create a term called id and then apply the symbol y to it, which gives y.
If you enter help you will receive the following information:
λ> help
HELP: Help information
HELP: Commands: help - this screen
HELP: terms - display all known terms
HELP: debug - toggle debug mode
HELP: load <filename> - loads file
HELP: alpha - alpha-equivalence
It is possible to load files which contain terms or expressions by using the load command. For example, the application comes with a booleans.lbd file which contains the following:
# True and false values
true = \x.\y.x
false = \x.\y.y
# Bunch of normal operators for booleans
and = \a.\b.a b a
or = \a.\b.a a b
not = \a.a false true
xor = \x.\y. x (not y) y
In the above file we have used the \ symbol for λ character. Either one is fine, just make that if you are consistent and if using greek letters, that you save the file as UTF-8.
Also note that any line beginning with a # symbol will be treated as a comment and ignored.
We can load the file easily enough:
λ> load booleans.lbd
LOADING FILE: booleans.lbd
LOADING FILE: booleans.lbd - 6 term(s) and 0 expression(s) parsed
Having loaded the file, we can now perform boolean expressions
λ> not true
β> false
λ> or false true
β> true
Some files may require terms which exist in other files. In these cases we can start the file with a $ symbol followed by the names of other files we need to include.
For example maths.lbd file contains the following:
$ booleans.lbd
zero = \f.\a.a
a
two = \f.\a.f (f a)
three = \f.\a.f (f (f a))
four = \f.\a.f (f (f (f a)))
five = \f.\a.f (f (f (f (f a))))
six = \f.\a.f (f (f (f (f (f a)))))
seven = \f.\a.f (f (f (f (f (f (f a))))))
eight = \f.\a.f (f (f (f (f (f (f (f a)))))))
nine = \f.\a.f (f (f (f (f (f (f (f (f a))))))))
ten = \f.\a.f (f (f (f (f (f (f (f (f (f a)))))))))
succ = \n.\f.\a.f (n f a)
add = \m.\n.\f.\a.m f (n f a)
mult = \m.\n.\f.m (n f)
pow = \m.\n.n m
pred = \n.\f.\a.n (\g.\h.h (g f)) (\u.a) (\u.u)
sub = \m.\n.n pred m
is_zero = \n.n (\m.false) true
When we attempt to load the file we will see the following:
λ> load maths.lbd
LOADING FILE: maths.lbd
LOADING FILE: booleans.lbd
LOADING FILE: booleans.lbd - 6 term(s) and 0 expression(s) parsed
LOADING FILE: maths.lbd - 18 term(s) and 0 expression(s) parsed
As soon as we start loading the terms in maths.lbd, the application finds the line $ booleans.lbd and knows it needs to import the terms from that file otherwise there would be errors when we encounter terms like true and false which are used in maths.lbd.
Finally, it is also possible to load files on startup by passing filenames as parameters to the .class file. For more information on building the project, see the Building Notes and Problems section.
You can enter debug to toggle debug-mode:
λ> debug
DEBUG: Debug mode ON
λ> debug
DEBUG: Debug mode OFF
Debug-mode can be useful for showing the intermediate steps during beta-reduction. Below is the result of beta-reducing or false true with debug-mode turned on:
λ> debug
DEBUG: Debug mode ON
λ> or false true
β> ((λa.λb.(a(λx.λy.x))b)(λx.λy.y))(λx.λy.x)
β> (λb.((λx.λy.y)(λx.λy.x))b)(λx.λy.x)
β> ((λx.λy.y)(λx.λy.x))(λx.λy.x)
β> (λy.y)(λx.λy.x)
β> λx.λy.x
β> true
You can enter alpha to switch into alpha-equivalence mode. Simply enter a number of expressions, each on a separate line and then a blank line:
λ> alpha
α-EQUIVALENCE: Enter a number of expressions, each on a separate line, and then an empty line to begin comparison
α1> λa.b a
α2> λc.d c
α3> λb.a b
α4>
α> TRUE
In the above example we see TRUE reported because λa.b a, λc.d c and λb.a b are all equivalent to each other.
However, consider the following example:
λ> alpha
α-EQUIVALENCE: Enter a number of expressions, each on a separate line, and then an empty line to begin comparison
α1> λa.b a
α2> λc.d c
α3> λb.a a
α4>
α> FALSE
This the above output, we see the result FALSE. This is because although λa.b a and λc.d c are equivalent, λb.a a is not equivalent to either of them.
Finally, you can terminate the program by entering quit, exit or pressing the ESC key.
λ> quit
QUITTING: Bye!
As already mentioned, the application comes with a number of library files which contain definitions for boolean operators, maths, conditionals, etc:
Now that we have a full understanding of Lambda-Calculus and of how the application works, we can start writing some programs.
After loading booleans.lbd we can perform and, or and not operations:
λ> load booleans.lbd
λ> not true
β> false
λ> and true false
β> false
λ> or true false
β> true
After loading maths.lbd we can perform basic mathematical operations:
λ> load maths.lbd
λ> add one two
β> three
λ> mult two two
β> four
λ> succ one
β> two
λ> succ (succ one)
β> three
Note that because maths.lbd only contains term definitions for the first ten numbers, mathematical operations which return larger numbers will be displayed in lambda-syntax:
λ> load maths.lbd
λ> add nine nine
β> λf.λa.f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f a)))))))))))))))))
After loading tuples.lbd we can create pairs of values:
λ> load maths.lbd
λ> load tuples.lbd
λ> first (pair one two)
β> one
λ> second (pair one two)
β> two
We can even have pairs of pairs:
λ> second (pair one (pair two three))
β> λf.(f(λf.λa.f(f a)))(λf.λa.f(f(f a)))
The application is unable to resolve multiple-terms, but if we check what the expected answer (pair two three) is in lambda-syntax, we can see that the lambda sequences matches:
λ> pair two three
β> λf.(f(λf.λa.f(f a)))(λf.λa.f(f(f a)))
After loading conditionals.lbd we can check for equality with if_then_else:
λ> if_then_else true one two
β> one
λ> if_then_else false one two
β> two
The building instructions can be found runlambda.bat and are also included below:
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A fair amount of work went into getting the console I/O to work and display nicely. Since Java is fussy about supporting single-keypress-input, displaying Unicode characters (λ, α, β etc.) and using colours, we need to use a bunch of Windows-specific stuff to achieve this. If you are struggling with inputting strings, or are seeing strange, non-printable values in the console, simply set FANCY_UI to false in Console.Java.
The application uses RawConsoleInput courtesy of Christian d'Heureuse for the single-character-input, and Yin Shan's code for displaying of Greek symbols.