rybla / Escaigne

A programming language for defining integer sequences.

Escaigne

A programming language for defining integer sequences.

Language

Each Escaigne program defines exactly one integer sequence, using the following grammar:

«program» ::=
  Begin Head.      «header»     End Head.
  Begin Prelude.   «statements» End Prelude.
  Begin Generator. «expression» End Generator.

«header» ::=
  | Title:       «string».
  | Author:      «string».
  | Description: «string».
  | «name»:      «string»

«statement» ::=
  | Value «name» := «expression».
  | Function «name» «parameters» := «expression».
  | Symbolic Value «name».
  | Symbolic Function «name» «integer».
  | Symbolic Operation «name».
  | Table «name».

«expression» ::=
  | «value»
  | ( «parameters» => «expression» )
  | ( «expression» «arguments» )
  | ( value «name» := «expression» ; «expression» )
  | ( function «name» «parameters» := «expression» ; «expression» )
  | ( «name» [ «expression» ] )

«parameters» ::= «name» «name» ... «name»

«arguments» ::= «name» «name» ... «name»

«value» ::=
  | «integer»
  | «boolean»
  | «string»
  | «name»

The Head contains a set of headers which provide metadata about the file and program.

The Prelude contains a set of definitions that may be used elsewhere in the Prelude and also in the Generator.

The Generator contains a single expression that is expected to be a function that takes a single parameter, n, and evaluates to the nth term in the sequence it represents.

Note that the grammar is purposely kept very sparse. For example, simple operations are not included. That is because all strings, other than ., ;, (, ), and keywords, can be used in «name». Of course, many useful operations and functions will be defined in standard libraries.

Statements

Value n := e binds n to e.

Function f x1 ... xn := e binds f to a function with parameters x1, ..., xn and evaluates to e.

Symbolic Value n and Symbolic Function f # bind their names to the appropriate type of value, which is marked as symbolic. A symbolic value cannot be reduced, and persists in the output. Symbolic values are automatically assumed to extend equality, but no other properties. Symbolic values are useful for forcing a certain representations - making explicit kinds of abstractions about the output sequence.

Table t declares a new table that can store values. Notably, these values persist during the evaluation of the generator over many entries of the sequence. This construct is useful for taking advantage of a dynamic-programming approach to computing some sequences (e.g. the Fibbonacci sequence).

Application

Escaigne is used to provide a simple representation of integer sequences for online reference. This is useful because it allows for efficient generation of sequences on the fly, given a specification of how many entries to count out to.