Theory of symbolic expressions, I

Abstract

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 representa tiou of these finita* . objects. Thus, for instance, a natural number n can be represented by a c *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 Lt. 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 a z the COES