Formal Definition
The terms and derivations in this system can also be more formally defined:
Terms: The set T of terms is defined recursively by the following rules.
- S, K, and I are terms.
- If τ1 and τ2 are terms, then (τ1τ2) is a term.
- Nothing is a term if not required to be so by the first two rules.
Derivations: A derivation is a finite sequence of terms defined recursively by the following rules (where all Greek letters represent valid terms or expressions with fully balanced parentheses):
- If Δ is a derivation ending in an expression of the form α(Iβ)ι, then Δ followed by the term αβι is a derivation.
- If Δ is a derivation ending in an expression of the form α((Kβ)γ)ι, then Δ followed by the term αβι is a derivation.
- If Δ is a derivation ending in an expression of the form α(((Sβ)γ)δ)ι, then Δ followed by the term α((βδ)(γδ))ι is a derivation.
Assuming a sequence is a valid derivation to begin with, it can be extended using these rules.
Read more about this topic: SKI Combinator Calculus
Famous quotes containing the words formal and/or definition:
“This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
—Leontine Young (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)