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1、1、 Introduction 简介In this paper I shall discuss the prospects for a mathematical science of computation . In a mathematical science ,it is possible to deduce from the basic assumptions,the important properties of the entities treated by the science. Thus，from Newtons law of gravitation and his laws
2、of motion,one can duduce that the planetary orbits obey kerplers laws.我将在这篇文章中谈谈数学化科学计算的前景。在数学化的科学中，从一些基本的结论中可以推断：被数学科学处理地重要的实体前景。从而，从牛顿万有引力定律和他的运动定律，有人推出了行星轨道满足开普勒定理。What are the entities with which the science of computation deals? 计算科学处理的实体是什么？What kinds of facts about these entities would we lik
3、e to derive?关于这些实体我们想要的推导什么种类的事实？What are the basic assumptions from which we should start?从我们开始什么事基本的假设？What important results have already been obtained? 已经获得什么重要的结论？How can the mathematical science help in the solution of practical problems?在实际问题上，数学化的科学怎么帮忙的？I would like to propose some partial
4、answers to these questions. These partial answers suggest some problems for future work. First I shall give some very sketchy general answers to the questions. First ,I shall give some very sketchy general answers to the questions. Then I shall present some recent results on three specific questions
5、. Finally, I shall try to draw some conclusions about practical applications and problems for future work.关于这些问题我想给出一些部分答案。这些部分答案对未来的工作问题有些建议。我先给出一些概要关于这些问题的答案。然后我将给出三个特殊问题一些当前的结果。最后我将试着得出一些关于实际应用和来来研究问题的结论。2. What Are The Entities With Which Computer Science Deals?什么是“计算机科学处理与实体？These are problems,
6、 procedures, data spaces, programs representing procedures in particular programming languages, and computers.这些问题，程序，数据空间，代表在特定的编程语言程序的方案，和计算机。A problem is defined by the criterion which determines whether a proposed solution is accepted. One can understand a problem completely without having any m
7、ethod of solution. Procedures are usually built up from elementary procedures. What these elementary procedures may be, and how more complex procedures are constructed from them, is one of the first topics in computer science. This subject is not hard to understand since there is a precise notion of
8、 a computable function to guide us, and computability relative to a given collection of initial functions is easy to define.一个问题定义的标准，决定建议的解决方案能否被接受。人们能够理解的问题，完全没有任何解决方法。程序通常建立了基本程序。 这些基本程序，以及如何从他们建造更复杂的程序，在计算机科学中的首要议题之一。这个题目是不难理解的，因为有一个可精确计算函数的概念引导我们，很容易定义可计算相对的初始功能集合。Procedures operate on members
9、of certain data spaces and produce members of other data spaces, using in general still other data spaces as intermediates. A number of operations are known for constructing new data spaces from simpler ones, but there is as yet no general theory of representable data spaces comparable to the theory
10、 of computable functions.程序操作上的某些数据空间成员和其他数据空间的成员，还有其他作为中间体使用的数据空间。 从简单的空间已知的操作，为其构建新的数据，但目前还没有表示数据位的一般理论与可计算函数理论。Programs are symbolic expressions representing procedures. The same procedure may be represented by different programs in different programming languages. We shall discuss the problem of
11、 defining a programming language semantically by stating what procedures the programs represent. As for the syntax of programming languages, the rules which allow us to determine whether an expression belongs to the language have been formalized, but the parts of the syntax which relate closely to t
12、he semantics have not been so well studied. The problem of translating procedures from one programming language to another has been much studied, and we shall try to give a definition of the correctness of the translation.程序是代表程序的符号表达式。 同样的过程可能由不同的编程语言表示不同的程序。 我们将讨论说明什么程序方案代表一种编程语言的语义定义的问题。至于编程语言，语法
13、规则，这使我们能够确定一个表达式是否属于正式语言，但密切相关的语义，语法的部分没有得到很好的研究。从一种编程语言的翻译程序的问题已经很多研究，我们会尽力给翻译的正确性的定义。Computers are finite automata. From our point of view, a computer is defined by the effect of executing a program with given input on the state of its memory and on its outputs. Computer science must study the var
14、ious ways elements of data spaces are represented in the memory of the computer and how procedures are represented by computer programs. From this point of view, most of the current work on automata theory is beside the point.计算机是有限自动机。 从我们的角度来看，是指由计算机执行其内存的状态，并在其输入输出方案与效果。 计算机科学必须研究各种方式的数据空间的元素，在计算
15、机的内存空间表示数据和如何由计算机程序表示程序。 从这个角度来看，自动机理论的大部分目前的工作是毫无意义的。3 What Kinds of Facts About Problems, Procedures, data Spaces, Programs, And Computers Would We Like to Derive?我们愿意派生出什么种类的关于问题、程序、数据空间、方案以及计算机的事实？Primarily, we would like to be able to prove that given procedures solve given problems. However, p
16、roving this may involve proving a whole host of other kinds of statement such as: 实际上，我们更愿意去证明那些给定程序的特定问题。然而，证明这类问题很可能涉及要证明整个主机其它种类的声明，例如：1. Two procedures are equivalent, i.e. compute the same function.2. A number of computable functions satisfy a certain relationship, such as an algebraic identity or a formula of the functional calculus.3. A certain procedure terminates for certain initial data, or for all initial data.4. A certain translation procedure correctly translates pr
