Translator

Lambda calculus:
\f.(\x.xx)(\x.f(xx))

Klambda:

Explanation

Klambda is a compact language for representing untyped higher-order functions. It only differs in two ways from the standard lambda calculus language:

• Function application is represented using the + character in prefix notation. This eliminates ambiguity and the need for parentheses (which are not allowed in klambda) and makes klambda easy to programmatically parse.
• Variables are not named. Lambda abstractions are introduced with just "." instead of "\x.", and the letter used when a variable occurs in an expression is determined by the occurrence's distance (structurally) from the lambda to which it is bound: "a" refers to the the closest enclosing lambda abstraction, "b" to the second-closest enclosing lambda abstraction, and so on. For example, "..a" represents \x.\y.y and "..b" represents \x.\y.x. In ".+a.+ab", the "b" refers to the same variable as the first "a", but the second "a" refers to another variable.

Issues of "alpha-equivalence" (variable renaming) and "capture-avoiding" (preventing variable name collisions during substitution) don't exist in klambda; identical structures always have identical representations regardless of where they appear, and you can safely copy/move subexpressions around without messing things up.

I think these properties (conciseness and context-insensitivity/representation uniqueness) make klambda appealing, even though it's not very human-readable.