Sept. 18-24 – \$20k in total prizes!

# Lambda Calcu-wha? 0/3 points

You’ve discovered an ancient system of symbols, dating back to the Kids in the Woods. It appears to be a system for denoting an inventory system for counting and comparing various objects, such as sheep, fruit, and tools. In order to better understand the system, you’ve decided to build an interpreter for the symbols, so you can run them as programs!

In this problem, you’re going to build an interpreter for a simple functional programming language.

### For what programming language?

The programming language you will be interpreting is very simple: there are three main categories of syntax:

• Literals like 5 or true
• Lambda abstraction written lam <binding> <body>
• Function application written app <lambda> <value>

In case you’ve never heard the phrase “lambda abstraction” before, a lambda is simply an anonymous function. In Javascript, lam x y would be written as

function(x) { return y }


Similarly, function application (app f x) would be written f(x). We’ll use lam x y and app f x because it’s a bit simpler.

So, more formally, the grammar for this language can be written in EBNF as:

expr = "(", expr, ")"
| "lam", ident, expr
| "app", expr, expr
| literal
| ident ;

ident   = letter, { letter | digit } ;
literal = number | bool ;
number  = digit, { digit } ;
bool    = "true" | "false" ;

letter = "A" | "B" | "C" | "D" | "E" | "F" | "G"
| "H" | "I" | "J" | "K" | "L" | "M" | "N"
| "O" | "P" | "Q" | "R" | "S" | "T" | "U"
| "V" | "W" | "X" | "Y" | "Z" | "a" | "b"
| "c" | "d" | "e" | "f" | "g" | "h" | "i"
| "j" | "k" | "l" | "m" | "n" | "o" | "p"
| "q" | "r" | "s" | "t" | "u" | "v" | "w"
| "x" | "y" | "z" ;
digit = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ;


One thing that’s sometimes annoying about EBNF grammars is that they can be a bit cavalier about where whitespace goes. In our grammar, whitespace can go anywhere there is concatenation except inside the ident and number expressions and must separate the concatenations for the lambda abstractions and function applications.

As you might have been able to deduce from the grammar, there are two kinds of literals: non-negative integers, and booleans.

### Standard library

Our language has some additional values built in. They are

• add which adds two integers. Since functions can only take one argument in our language, add takes a number and returns a function that takes another number that sums both and returns the result. You use add to add 1 and 2 like app (app add 1) 2, which evaluates to 3.
• gt which compares two integers. Just like add, gt takes one number as an argument, returns a function that takes another number, then returns if the first argument was greater than the second argument. This strategy of using multiple functions to handle multiple arguments one at a time is called currying. You use gt to compare two numbers like app (app gt 1) 2, which evaluates to false.
• if which branches on a bool. Like add and gt, we’ll curry the arguments, but here we need to take three arguments. You can use if like app app app if false 2 3, which evaluates to 3.

As input your program should take lines of expressions in the above programming language, evaluate them fully, and output the result. The evaluation should be “strict”, meaning that your program’s behavior should be identical to if it fully evaluated the left and right hand sides of an app before doing the application. All of the input expressions are syntactically valid, eventually terminate, and the result is an integer.

### Hint

One thing you’ll want to do is keep track of what variables are defined and when. When you evaluate a lambda, you’ll want to keep a snapshot of the variables that were defined when you evaluated the lambda, so that when the lambda is called, it knows about all of those variables still. These mappings of variable definitions are usually called “environments” and evaluated lambdas that are bound to an environment are usually called “closures,” if you want to Google around to read more.

### Example

#### Input

2
app (lam x x) 2
app lam x (x) 2
app app app if (app app gt 3 1) 10 5


#### Output

2
2
2
4
10


These won’t be tested, but just for more ideas:

#### Input

app app (app (lam f lam y lam x (app (app f y) x)) (lam x lam y x)) 3 4


#### Output

3