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File: Programming Pdf 174857 | Lecturelambda Handout
02157 functional program ming 02157 functional programming michaelr hansen abrief introduction to lambda calculus lambda calculus background syntax reductions michael r hansen lambda terms as programs church numerals fixpoint combinators ...

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                                                                                                                                                       02157
                                                                                                                                                     Functional
                                                                                                                                                      Program-
                                                                                                                                                        ming
             02157 Functional Programming                                                                                                           MichaelR.Hansen
             Abrief introduction to Lambda calculus                                                                                                 Lambda
                                                                                                                                                    calculus
                                                                                                                                                     Background
                                                                                                                                                     Syntax
                                                                                                                                                     reductions
             Michael R. Hansen                                                                                                                       Lambda
                                                                                                                                                     terms as
                                                                                                                                                     programs
                                                                                                                                                     Church
                                                                                                                                                     numerals
                                                                                                                                                     Fixpoint
                                                                                                                                                     combinators
 1       DTUInformatics,Technical University of Denmark                                           Abrief introduction to Lambda calculus       MRH25/10/2012
    Purpose
                                                                                                                                                       02157
                                                                                                                                                     Functional
                                                                                                                                                      Program-
                                                                                                                                                        ming
                                                                                                                                                    MichaelR.Hansen
             Thetheoreticalunderpinningof functional languagesis λ-calculus.                                                                        Lambda
                                                                                                                                                    calculus
                                                                                                                                                     Background
             Thepurposeisto hint on this underpinningand to introduce                                                                                Syntax
                                                                                                                                                     reductions
             concepts of functional languages.                                                                                                       Lambda
                                                                                                                                                     terms as
                                                                                                                                                     programs
                 • Informal introduction to λ-calculus                                                                                               Church
                                                                                                                                                     numerals
                          • computations of lambda-calculus and functional languages                                                                 Fixpoint
                                                                                                                                                     combinators
             Todayyouwill be introduced to basic concepts of λ-calculus and you
             will get a feeling for the theoretical power of these concepts by the
             construction of an interpreter for a λ-calculus based language .
 2       DTUInformatics,Technical University of Denmark                                           Abrief introduction to Lambda calculus       MRH25/10/2012
    Lambdacalculus: background
                                                                                                                                                       02157
                                                                                                                                                     Functional
                                                                                                                                                      Program-
                                                                                                                                                        ming
                                                                                                                                                    MichaelR.Hansen
                 • Invented in the 1930’s by the logician Alonzo Church in logical                                                                  Lambda
                     studies and in investigations of function definition and                                                                        calculus
                                                                                                                                                     Background
                     application, and recursion.                                                                                                     Syntax
                                                                                                                                                     reductions
                 • Comprisefull computability.                                                                                                       Lambda
                                                                                                                                                     terms as
                                                                                                                                                     programs
                 • First uncomputability results were discovered using λ-calculus.                                                                   Church
                                                                                                                                                     numerals
             Somequestions                                                                                                                           Fixpoint
                                                                                                                                                     combinators
                 • Doesthemathematicalexpressionx −y denotea function,say
                     f, of x or a function, say g, of y, or :::?
                 • Doesthenotationh(z) mean a function h or h applied to z
 3       DTUInformatics,Technical University of Denmark                                           Abrief introduction to Lambda calculus       MRH25/10/2012
    Lambdacalculus: informal ideas
                                                                                                                                                       02157
                                                                                                                                                     Functional
                                                                                                                                                      Program-
                                                                                                                                                        ming
                 • λx:e denotes the anonymousfunction of x which e is.                                                                              MichaelR.Hansen
             Examplesoffunction definitions:                                                                                                         Lambda
                                                                                                                                                    calculus
                                                                                                                                                     Background
                 • Let f be λx:x − y                                                                                                                 Syntax
                     Theexpressionx −y consideredas a function f : Z → Z of x                                                                        reductions
                                                                                                                                                     Lambda
                 • g = λy:x −y                                                                                                                       terms as
                                                                                                                                                     programs
                     Theexpressionx −y consideredas a function g : Z → Z of y                                                                        Church
                                                                                                                                                     numerals
                                                                                                                                                     Fixpoint
                 • h = λx:λy:x −y                                                                                                                    combinators
                     Theexpressionx −y consideredas a higher-orderfunction
                     h : Z → (Z → Z)
             Examplesoffunction applications:
                 • f(1) = 1−y andg(1) = x −1
                 • h(2) = λy:2 −y and h(2)(5) = 2−5 = −3
 4       DTUInformatics,Technical University of Denmark                                           Abrief introduction to Lambda calculus       MRH25/10/2012
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...Functional program ming programming michaelr hansen abrief introduction to lambda calculus background syntax reductions michael r terms as programs church numerals fixpoint combinators dtuinformatics technical university of denmark mrh purpose thetheoreticalunderpinningof languagesis thepurposeisto hint on this underpinningand introduce concepts languages informal computations and todayyouwill be introduced basic you will get a feeling for the theoretical power these by construction an interpreter based language lambdacalculus invented in s logician alonzo logical studies investigations function denition application recursion comprisefull computability first uncomputability results were discovered using somequestions doesthemathematicalexpressionx y denotea say f x or g doesthenotationh z mean h applied ideas e denotes anonymousfunction which is...

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