探索λ演算基础：函数、转换与定义

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Lambda演算是一种基础且强大的数学逻辑系统，由Alonzo Church在20世纪初提出，用于研究计算的本质和函数的概念。此学习资料是英文版，详细介绍了Lambda演算的核心概念、规则以及其在理论计算机科学中的应用。以下是对文档各部分的详细解读： 1. **Lambda-Conversion**: 这部分主要探讨了Lambda演算中的转换规则，如原始符号和公式的基本构成（2.1），以及如何通过一系列转换操作将一个Lambda表达式变换到正常形式，这是Lambda演算的关键步骤，有助于简化表达式并理解其有效性（2.2）。这包括对Well-Formed Formulas（WFF）和正常形式的深入阐述。 2. **Lambda-Deﬁnability**: Lambda演算的另一个核心概念是lambda-定义性，即证明函数是否可以用Lambda表达式表示。这部分讨论了如何用Lambda表达式定义整数函数（3.1），有序对和三元组以及预设函数的构造（3.2），以及命题函数的定义（3.3）。递归定义也是重要的方法之一（3.4），展示了Lambda演算在处理复杂结构的能力。 3. **Combinations & Gödel Numbers**: 当涉及到更复杂的数学结构时，组合论和Gödel编码被引入进来。组合是构建更高级数学对象的基础（4.1），而如何将Lambda表达式的特性映射到特定的公式集合（4.2）是关键。一个关于转换的组合等价概念也被探讨，以及Gödel号码的使用，它将表达式编码为自然数，便于形式化推理（4.4 和 4.5）。 4. **λ-K-Conversion and λ-δ-Conversion**: 最后部分，文档深入讲解了λ-K转换和λ-δ转换（5.1），这两种转换方法在Lambda演算的证明理论中起着至关重要的作用。它们在简化表达式、证明性质和构造模型等方面有着显著的作用。通过这些内容，学习者能够掌握Lambda演算的核心思想，了解其在函数式编程、逻辑学和计算理论中的基础地位，以及如何运用转换规则来分析和设计计算过程。这对于理解现代计算机科学中的抽象数据类型、函数式语言以及理论基础至关重要。

1.3 Functions o f Several Variables

Functions in λ-Calculus can take more than one argument, however the meaning is interpreted slightly

diﬀerent from our previous notions of functions. In λ-Calculus, a function of n + 1 arguments is treated as

a function of one argument whose values are functions of (up to) n arguments.

For example, let f be a function of two arguments. Then ((f a)b) is how we would interpret providing

two arguments, a and b to f .

Notation 1.3.1. We adopt the following shorthand: ((fa)b) = (fab) = fab.

Notice the following: let f denote a function of 3 variables. Then fabc denotes the value of f for the

arguments a, b, and c. Given our interpretation of functions, note that f a is a function whose value is a

function of two variables, f ab is a function whose value is a function of one variable, and fabc denotes an

actual value (potentially another function of any number of arguments). In this respect, we may provide as

many arguments as we wish to any function!

Deﬁnition 1.3.2 (K, the constancy function). K is deﬁned such that Kxy is x for whatever x and y may

be.

Note that KII is I and KHI is H. More interestingly, note that KI is H and that KK is, simply,

Next, we deﬁne the following functions. . .

Deﬁnition 1.3.3 (T ). T xf is (fx)

Deﬁnition 1.3.4 (J ). Jfxyz is fx(fzy)

Deﬁnition 1.3.5 (B, the product function). Bfgx is f(gx)

Deﬁnition 1.3.6 (C, the converse function). Cf xy is (f yx)

Deﬁnition 1.3.7 (D). Dx is (xx)

Deﬁnition 1.3.8 (W ). W fx is (fxx)

Deﬁnition 1.3.9 (S). Snf x is f (nfx)

NOTE: This is distinctly diﬀerent from the function S deﬁned in Barendregt’s text. There, Sfgx is fx(gx).

1.4 Abstraction

Notation 1.4.1. Let M denote an expression containing a variable x that is free (that is, the meaning of

M depends upon the value of x). Then (λxM ) denotes a function whose value for some argument α is

denoted by substituting all occurrences of α in M .

Notice that if M does not contain the symbol x, then the value of (λxM) is constant and equal to M .

Also, although x is free in M , it is bound in (λxM ) and that λx is an incomplete expression! (λx has no

meaning on its own.

In the above, note the distinction between an expression and a function. Take the following two expres-

sions, M = (x + x) and N = (y + y). The equation M = N then expresses a relation between the numbers

denoted by x and y, and its truth depends on determining x and y.

Now lo ok at (λxM) = (λyN):

(λx(x + x)) = (λy(y + y)) (1.4.1)

This equation denotes something entirely diﬀerent; it says that the two functions (λxM) and (λyN) are

equivalent (which is true) and makes no mention of the values of x or y.

When we added the λx to the expression M , we created a function. This operation is called abstraction

and the symbol λx is called the abstract operator.

Notation 1.4.2. The expression (λx(λyM )) is abbreviated (λxy.M ) and denotes a function whose value

for the argument x is denoted by (λyM ).

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