Theory of Symbolic Expressions, I
Theory of Symbolic Expressions, I
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符号表达理论，I
DOI：
10.1016/0304-3975(83)90137-8
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发表时间：
1983
影响因子：
1.1
通讯作者：
M. Sato
中科院分区：
文献类型：
--
作者：
M. Sato
本系列论文的目的是引入一个新的符号表达式域S（s~ rps，简称：），并在S的框架内研究有限数学。虽然有限数学包含了传统的数学理论，如数论或有限集合论，但我们的重点将主要放在计算理论上，包括它的元理论。有限的妈妈？它处理有限对象，如自然数，符号串或证明树。我们的域S是足够灵活的，以允许这些finita* 的自然表示。对象因此，例如，自然数n可以在S中的符号表达式s中用ac* n表示。类似地，证明树T可以由sexp f表示。然而，我们将不把s看作自然数的表示，而是看作自然数n本身。同样，我们将把I看作一棵证明树。很可能是这样的。在这种情况下，我们的ciewpoint决定它是（被认为是）自然数还是证明树。在我们的理论中，每一个有限对象都是一个scxp。(Note在集合论中也采用类似的staridpoint，其中每个对象都是一个集合。这样，我们的域S就成为有限对象的泛域。每一个有限对象都是sexp的原理看起来似乎很简单，但它在我们的研究中会产生重要的影响。想象一个无限的无叶二叉树，如图1所示，其中每个节点都画了一个小圆圈。最上面的节点称为根。任意选择有限数量的节点，并将其标记为黑色，如图2所示。我们称之为性感的结果。没有标记节点的sexp被denoecd为0。图I，作为一个sexp，是0。3e仅标记节点为根的sexp用1表示。如果一个sexp的根被标记，则该sexp被称为原子。根没有标记的性粒子称为分子。对于任意的sexp z，它的左子树称为2的cur，它的右子树称为z的dr。修复实例，汽车和0和1的cdr为0。对于任意的sexp x和y，存在一个分子，其car是s，cdr是JZ，我们称之为COES
The purpose of this series of papers is to introduce a new domain S of symbolic expressions (s~ rps, for short:, and to study finite mathematics within the framework of S. Although finite mathematics contains traditional mathematical theory such as number theory or finite set theory, our emphasis will tc; mainly on the theory of computation including its metatheory. Finite ma? ematics deals with finitary objects such as natural numbers, strings of symbols or proof trees. Our domain S is flexible enough to permit natural representatiou of these finita*. objects. Thus, for instance, a natural number n can be represented by ac* rtp, in symbolic expression s in S. Similarly a proof tree T may be represented by a sexp f. We will, however, consider s not as a representation of a natural number but as the natural number n itself. Similarly we will consider I as a proof tree. It may well be the case that s L-t. In such a case, it is our ciewpoint that determines whether it is (considered to be) a natural number or a proof tree. in our theory, every finitary object is a scxp.(Note that a similar staridpoint is taken in set theory, where every object is a set.) In this way, our domain S becomes a universal domain of finitary objects. The principle that every finitary object is a sexp may seem quite innocent, but it will have important consequences throughout our study.An intuitive definition of a sexp can be given as follows. Imagine an infinite leaf-free binary tree like Fig. 1, where a small circle is drawn at each node. Th~ z topmost node is called the root. Choose a finite number of nodes arbitrarily and mark them black as in Fig. 2. We call the resulting figure a sexy. The sexp with no marked nodes is denoecd by 0. Fig. I, considered as a sexp, is 0. The sexp who, 3e only marked node is the root is denoted by 1. A sexp is called an atom if its root is marked. A sexp whose root is unmarked is called a molecule. For any sexp z, its left subtree is called the cur of 2 and its right subtrce is calied the dr of z. FIX instance, the car and the cdr of 0 and 1 is 0. For any sexp x and y, there uniquzlJ* exists a molecule of whose car is s and whose cdr is JZ We call such az the COES