Welcome to the pLam wiki! (Work in progress...)

This home page serves as a reference to all other content related to pLam language and λ-calculus. The content described in these wiki pages deals not just with the documentation of pLam's syntax and semantics, but also serves as a mild introduction to λ-calculus and computability, along with most important definitions and results, pointing to links for further reading.

• λ-calculus
  • Language
  • β-reduction
  • α-equivalence
• Computability
  • Partial recursive functions
  • Church numerals
  • λ-definability
• pLam
  • Syntax and semantics
  • Libraries
  • Examples

---

(go to detailed wiki)

---

(go to detailed wiki)

---

(go to detailed wiki)

λ-variable is required to be lowercase and a single letter. For example, x is a good λ-variable for pLam and X, var,... are not. There are also environment variables (names for defined λ-expressions) which are every string that is not parsed as λ-variable, λ-abstraction or λ-application.

λ-abstraction is written the same as in the language of pure (untyped) λ-calculus, except that pLam treats a symbol \ as λ and it is required to write a space after .. For example, λx.λy.x would be written \x. \y. x in pLam. One can also write λ-abstraction in the curried form: \xy. x or \x y. x.

λ-application is written like 2 λ-expressions separated by a space, for example (\x. x) (\xy.x) or (\x. x) MyExpression or myexp1 myexp2. Brackets ( and ) are used as usual and are not required to be written; the association is to the left, so M N P is parsed as (M N) P and one needs to specify with brackets if the intended expression should be M (N P).

A block of code in pLam is a line, and possible lines (commands) are the following:

• syntax: <string> = <λ-expression>
• semantics: let the <string> be a name for <λ-expression>.
• examples: T = \x y. x, myexpression = T (T (\x. x) T) T
• restriction: <string> needs to be of length>1 or starting with uppercase letter

• syntax: <λ-expression> or :d <λ-expression>
• semantics: reduce the <λ-expression> to β-normal form. If :d is put in front of expression, all the reduction steps will be shown (manually or automatic, depends on what one chooses when asked)
• example: \x y. x, :d T (T (\x. x) T) T
• restriction: none

• syntax: :import <string>
• semantics: put all the expressions defined in the file import/<string>.plam into the list of environment variables.
• example: :import std
• restriction: <string>.plam has to be inside import/ directory within the pLam project directory

• syntax: --<string>
• semantics: a comment line
• example: -- this is a comment
• restriction: none

• syntax: :run <string>
• semantics: runs a .plam file with relative path <string>.plam
• example: :run <relative-path-to-plam>/examples/2.5.2
• restrictions: ~ for home does not work

• syntax: :print <string>
• semantics: prints <string> to a new line. It mostly makes sense to use it in .plam programs to be executed, not in interactive mode where a comment should do the job better.
• example: :print this is a message
• restrictions: none