Introduction to Lambda Calculusλ演算，λ(Lambda（大写Λ，小写

文件名称: Introduction to Lambda Calculus

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详细说明：λ演算，λ(Lambda（大写Λ，小写λ）读音：lan b（m） da（兰亩达）['læ;mdə])演算是一套用于研究函数定义、函数应用和递归的形式系统。它由 Alonzo Church 和 Stephen Cole Kleene 在 20 世纪三十年代引入，Church 运用 lambda 演算在 1936 年给出判定性问题 (Entscheidungsproblem) 的一个否定的答案。这种演算可以用来清晰地定义什么是一个可计算函数。关于两个 lambda 演算表达式是否等价的命题无法通过一个通用的算法来解决，这是不可判定性能够证明的头一个问题，甚至还在停机问题之先。Map of the Universe We are here C omputer Mathematics Cience 3/131 What|s日 Function? Formula. Interpretation f( =2x f(ar) yields 2. c 9(m,y)=5 g is a constant function ·h:X→Y h maps every X to an y +>3 anonymous constant map °(aob)(x)=a(b(x) composition of a and b 4/131 TwoA∩ SWers Intensional Extensional Xsy GCD(×,y) f(× false true X=0 (×,y)←(y°X,) return y 5/131 TwoA∩ SWers Intensional Extensional Xsy GCD(×,y) FIX false true X=0 (×,y)←(y°X,) return y 6/131 My Answer According to Wiktionary performance" execution, 3 The word function" comes from a latin word that means The Latin word"calculus " means a pebble or stone used for counting Rhetorical question: Does modern set theoretical mathematics interpret these concepts appropriately This presentation tries to honour the traditional interpretations 7/131 Pros and cons of Constructive Approach Pros C ons Compatible with empirical Lack of certain symmetries science and skepticism such as pvp Theory may also work in Our culturally conditioned practice without extra work Euclidean world view Involves less ambiguity becomes challenged Subset of classical toolset, Proofs may become lengthy 8/131 Pros and cons of Constructive Approach Pros Cons Compatible with empirical Lack of certain symmetries science and skepticism such as pV-p Theory may also work in Our culturally conditioned practice without extra work Euclidean world view Involves less ambiguity becomes challenged Subset of classical toolset, Proofs may become lengthy 9/131 Terminology Functions are abstract dependencies between objects algorithms are representations of functions in a, formal system A function may have multiple algorithms Strings can represent functions as well as their arguments Computation is the art of algorithmic string manipulation A model of computation is a formal system that defines effectively computable functions in terms of algorithms 10/131

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