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
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