The self-reduction in lambda calculus

Self-Representation in Girard's System U

“…A similar idea is the reflective tower, which uses an infinite tower of self-interpreters to add reflective capabilities to a language. There are many examples of self-interpreters in the literature, including ones for λ-calculus [4,7,8,18,22,23,28,31], Haskell [25], JavaScript [13], Lisp [21], Python [36], Ruby [37], Scheme [2], Standard ML [29], and many others [19,32,38]. In each of these the representations are untyped, in the sense that (1) it is possible to build representations of ill-typed terms, and (2) all representations are either untyped or else have the same type.…”

Typed self-interpretation by pattern matching

“…He used the term self-reducer for any λ-term enact that satisfies Equation (2). Mogensen [25], Berarducci and Böhm [5] (who preferred the term reductor), and Song, Xu, and Qian [35] presented λ-terms enact that satisfy Equation (2). The name self-enactor seems apt because Equation (2) implies that enact must do work to turn a quotation into action.…”

“…Self-interpreters are popular and available for Standard ML [34], Haskell [28], Scheme [1], JavaScript [12], Python [33], Ruby [41], λ-calculus [2,4,5,21,25,26,32,35], and many other languages [23,38,42]. A self-interpreter enables programmers to easily modify, extend, and grow a language [31], do other forms of meta-programming [8], and even derive an algorithm for normalisation by evaluation [6].…”