Editing Scheme (programming language) (section)

===Lambda calculus===
{{See also|Lambda calculus}}
[[Alonzo Church]]'s mathematical notation, the lambda calculus, has inspired Lisp's use of "lambda" as a keyword for introducing a procedure, as well as influencing the development of [[functional programming]] techniques involving the use of [[higher-order function]]s in Lisp. But early Lisps were not suitable expressions of the lambda calculus because of their treatment of [[Free variables and bound variables|free variables]].<ref name="revisited"/>

A formal lambda system has axioms and a complete calculation rule. It is helpful for the analysis using mathematical logic and tools. In this system, calculation can be seen as a directional deduction. The syntax of lambda calculus follows the recursive expressions from x, y, z, ...,parentheses, spaces, the period and the symbol λ.<ref>{{Cite journal |last=van Tonder |first=André |date=1 January 2004 |title=A Lambda Calculus for Quantum Computation |journal=SIAM Journal on Computing |volume=33 |issue=5 |pages=1109–1135 |arxiv=quant-ph/0307150 |doi=10.1137/S0097539703432165 |s2cid=613571}}</ref> The function of lambda calculation includes: First, serve as a starting point of powerful mathematical logic. Second, it can reduce the requirement of programmers to consider the implementation details, because it can be used to imitate machine evaluation. Finally, the lambda calculation created a substantial meta-theory.<ref>{{Cite journal |last1=Niehren |first1=J. |last2=Schwinghammer |first2=J. |last3=Smolka |first3=G. |date=November 2006 |title=A concurrent lambda calculus with futures |url=https://hal.inria.fr/inria-00090434/file/0.pdf |journal=Theoretical Computer Science |volume=364 |issue=3 |pages=338–356 |doi=10.1016/j.tcs.2006.08.016}}</ref>

The introduction of lexical scope resolved the problem by making an equivalence between some forms of lambda notation and their practical expression in a working programming language. Sussman and Steele showed that the new language could be used to elegantly derive all the imperative and declarative semantics of other programming languages including ALGOL and [[Fortran]], and the dynamic scope of other Lisps, by using lambda expressions not as simple procedure instantiations but as "control structures and environment modifiers".<ref name="lambda_paper_2">{{Cite journal |last1=Gerald Jay Sussman |last2=Guy Lewis Steele Jr. |name-list-style=amp |date=March 1976 |title=Lambda: The Ultimate Imperative |url=http://library.readscheme.org/page1.html |url-status=dead |format=postscript or PDF |journal=AI Memos |publisher=[[MIT Computer Science and Artificial Intelligence Laboratory|MIT AI Lab]] |volume=AIM-353 |archive-url=https://web.archive.org/web/20160510140804/http://library.readscheme.org/page1.html |archive-date=2016-05-10 |access-date=2012-08-09}}</ref> They introduced [[continuation-passing style]] along with their first description of Scheme in the first of the Lambda Papers, and in subsequent papers, they proceeded to demonstrate the raw power of this practical use of lambda calculus.