“可計(jì)算性(Computability)”是可計(jì)算性理論的核心概念，具有深刻的數(shù)學(xué)內(nèi)涵和哲學(xué)底蘊(yùn)，圖靈、丘奇、哥德?tīng)柕惹拜叺墓ぷ鳛榇烁拍畲蛳铝藞?jiān)實(shí)的基礎(chǔ)，應(yīng)該說(shuō)對(duì)此概念的理解已經(jīng)不成問(wèn)題了，然而從“NP是可計(jì)算的”流行觀念看，此概念并未得到人們充分而正確的解讀，這或許是造成千禧年難題“P versus NP”的最根本原因。

我們從回答“什么是可計(jì)算性？”說(shuō)起，來(lái)解讀學(xué)術(shù)界流行觀點(diǎn)的回答，指出“可計(jì)算性”蘊(yùn)含著“機(jī)器”與“問(wèn)題”二個(gè)不同的層次：

一，學(xué)術(shù)界回答“什么是可計(jì)算性？”的典型答案

可計(jì)算性(calculability)是指一個(gè)實(shí)際問(wèn)題是否可以使用計(jì)算機(jī)來(lái)解決。

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

One goal of computability theory is to determine which problems, or classes of problems, can be solved in each model of computation.

3，Michael Sipser的書(shū)“Introduction to the Theory of Computation”

此書(shū)是計(jì)算理論領(lǐng)域的經(jīng)典著作，常被大學(xué)選為教材。書(shū)中干脆就不談“可計(jì)算性”，直接用“可計(jì)算函數(shù)”代替(p. 234)：

COMPUTABLE FUNCTIONS：A Turing machine computes a function by starting with the input to the function on the tape and halting with the output of the function on the tape.

A mathematical problem is computable if it can be solved in principle by a computing device.

二，解讀“可計(jì)算性(Computability)”與“可計(jì)算的(computable)”

可見(jiàn)，上述回答實(shí)際上是在答“什么是可計(jì)算問(wèn)題(computable problem)？”，而不是直接答“什么是可計(jì)算性(Computability)？”，那么，我們不禁要問(wèn)：“computability”與“computable problem”是一回事嗎？

對(duì)于“computable problem”的標(biāo)準(zhǔn)答案是由丘奇－圖靈論題給出的(https://en.wikipedia.org/wiki/Church–Turing_thesis)：

J. B. Rosser (1939) addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps". Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing a result".

In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1939 Ph.D. thesis Systems of Logic Based on Ordinals, supervised by Church, are virtually the same:

"? We shall use the expression 'computable function' to mean a function calculable by a machine, and let 'effectively calculable' refer to the intuitive idea without particular identification with any one of these definitions."

The thesis can be stated as follows:

Every effectively calculable function is a computable function.[8]

Turing stated it this way:

"It was stated ... that 'a function is effectively calculable if its values can be found by some purely mechanical process.' We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability? with effective calculability." (? is the footnote above, ibid.)

“如果一個(gè)函數(shù)的值能被某個(gè)純粹的機(jī)械過(guò)程求得，那么此函數(shù)就是能行可計(jì)算的”，可見(jiàn)，“可計(jì)算函數(shù)”是須要前提的，此前提就是存在著一臺(tái)可以求解函數(shù)值的機(jī)器，換句話說(shuō)，“可計(jì)算性”涉及到“問(wèn)題”與“機(jī)器”二個(gè)層次，“機(jī)器的能力”為本，“問(wèn)題的性質(zhì)”為末。于是，用“可計(jì)算問(wèn)題”代替“可計(jì)算性”，實(shí)際上是忽略了“可計(jì)算性”蘊(yùn)含的二個(gè)層次的關(guān)系，從而造成認(rèn)知混淆，。。。

所以，對(duì)于“可計(jì)算性”的理解應(yīng)該回到對(duì)“機(jī)器的能力”的認(rèn)知上，即回到對(duì)“圖靈機(jī)”的認(rèn)知上來(lái)。追本溯源，“圖靈機(jī)”是在圖靈1936年那篇重要論文《論可計(jì)算數(shù)及其在判定問(wèn)題上的應(yīng)用》( On Computable Numbers, with an Application to the Entscheidungsproblem)中提出來(lái)的，讓我們一起回到這篇具有歷史性意義，但又令人望而生畏，至今使人無(wú)法深入的奠基性的論文。

三，《論可計(jì)算數(shù)及其在判定問(wèn)題上的應(yīng)用》中關(guān)于“圖靈機(jī)”的論述

圖靈在論文中開(kāi)門(mén)見(jiàn)山指出，“按照我的定義，一個(gè)數(shù)是可計(jì)算的，如果它的十進(jìn)制的表達(dá)能被機(jī)器寫(xiě)下來(lái)。”就是說(shuō)，圖靈強(qiáng)調(diào)預(yù)期的結(jié)果一定要被機(jī)器寫(xiě)下來(lái)，才能認(rèn)為一個(gè)數(shù)是可計(jì)算的。

接下來(lái)，他開(kāi)始設(shè)計(jì)這樣的機(jī)器，將之稱(chēng)為“computing machine”。于“Computing machine”，圖靈又區(qū)分了“circular machine”和“circle-free machine”，“circular machine”是因某些因素如“死循環(huán)”而無(wú)法寫(xiě)下計(jì)算結(jié)果的機(jī)器；而“circle-free machine”沒(méi)有這樣阻礙，能寫(xiě)下預(yù)期結(jié)果，換句話說(shuō)， “circle-free”表達(dá)了圖靈稱(chēng)之的“可計(jì)算性(computability)”。

然而，一個(gè)問(wèn)題是否“computable”須要判斷，這就是圖靈這篇奠基性的論文的主題，回答“可計(jì)算性的判斷”，進(jìn)而回答著名的希爾伯特第十問(wèn)題，。。。

讓我們回到問(wèn)題：“什么是可計(jì)算性？”，相對(duì)于流行答案“可計(jì)算性是指一個(gè)實(shí)際問(wèn)題是否可以使用計(jì)算機(jī)來(lái)解決”，應(yīng)該說(shuō)：“可計(jì)算性是指算法(圖靈機(jī))具有解決一個(gè)實(shí)際問(wèn)題的能力”，“可計(jì)算問(wèn)題是指存在著具有可計(jì)算性的算法的問(wèn)題”。。。

附：

1，圖靈論文“On Computable Numbers, with an Application to the Entscheidungsproblem”原文：https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

2，《論可計(jì)算數(shù)及其在判定問(wèn)題上的應(yīng)用》的前言，第一，二章部分譯文：

“可計(jì)算數(shù)”簡(jiǎn)單說(shuō)是，其十進(jìn)制的表達(dá)用有限的手段可計(jì)算的實(shí)數(shù)。雖然本文的主題表面上講可計(jì)算數(shù)，然而幾乎可以同樣容易定義和研究變量為整數(shù)或?qū)崝?shù)或可計(jì)算變量的可計(jì)算函數(shù)，可計(jì)算謂詞等。在每種情況下，基本的問(wèn)題是一樣的，我選擇可計(jì)算數(shù)來(lái)解釋，是因?yàn)檫@樣可以涉及最少的技術(shù)細(xì)節(jié)。不久我希望給出可計(jì)算數(shù)與可計(jì)算函數(shù)等之間的關(guān)系，這將包括用可計(jì)算數(shù)表達(dá)的實(shí)數(shù)變量的函數(shù)理論。按照我的定義，一個(gè)數(shù)是可計(jì)算的，如果它的十進(jìn)制的表達(dá)能被機(jī)器寫(xiě)下來(lái)。

在§9,10，我給出一些論點(diǎn)，想指出可計(jì)算數(shù)包括所有的能自然看作可計(jì)算的數(shù)。比如，它們包括代數(shù)數(shù)的實(shí)數(shù)部分，Bessel函數(shù)的大小的實(shí)數(shù)部分，PI, e等等。然而可計(jì)算數(shù)并不包括所有可定義的數(shù)。

盡管可計(jì)算數(shù)類(lèi)如此之大，在許多方面類(lèi)似于實(shí)數(shù)類(lèi)，它仍然是可枚舉。在§8章，我調(diào)查某些證明似乎證明相反的論點(diǎn)。通過(guò)正確應(yīng)用一個(gè)論點(diǎn)，可達(dá)成一個(gè)與哥德?tīng)柕慕Y(jié)論表面上相似的結(jié)論。這些結(jié)果可有價(jià)值的應(yīng)用，尤其是，指出(§11)Hilbertian Entscheidungsproblem沒(méi)有解。

The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine.

In §§ 9, 10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers X, e, etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable.

Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable. In §8 I examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of G?del [1] . These results {231} ?have valuable applications. In particular, it is shown (§11) that the Hilbertian Entscheidungsproblem can have no solution.

1，計(jì)算機(jī)器(Computing machine)

我們已經(jīng)說(shuō)過(guò)，可計(jì)算數(shù)是那些可用有限的手段計(jì)算而得的表達(dá)為十進(jìn)制的數(shù)，這需要較明確的定義。在第九章之前，不對(duì)定義作真正合理化的說(shuō)明，目前我僅說(shuō)理由是人類(lèi)的記憶需要限制。

可將一個(gè)在計(jì)算實(shí)數(shù)的人與某機(jī)器比較，此機(jī)器具有有限數(shù)目的狀態(tài)：q1;q2;…qk(m-格局，m-configurations)。給機(jī)器提供一條紙帶(類(lèi)似紙張)，機(jī)器在上面運(yùn)行，紙帶被分為段(方格，squares)，每個(gè)方格上放置一個(gè)符號(hào)。在任何時(shí)刻，只有一個(gè)方格，比如第r個(gè)方格，放置符號(hào)了S(r)，“在機(jī)器里”，我們可以稱(chēng)此方格為“掃描方格”，“掃描方格”里的符號(hào)稱(chēng)為“掃描符號(hào)”，“掃描符號(hào)”是機(jī)器“直接感知”的唯一符號(hào)。然而，通過(guò)變化格局，機(jī)器可以有效記憶已經(jīng)“見(jiàn)過(guò)(掃描)”的符號(hào)。機(jī)器在任何時(shí)刻可能的行為由m－格局qn，掃描符號(hào)S(r)決定，這對(duì)(qn，S(r))稱(chēng)為“格局”：于是，格局決定機(jī)器可能的行為。在有些格局，掃描方格是空的(即沒(méi)有符號(hào))，機(jī)器就在掃描方格上寫(xiě)下一個(gè)新的符號(hào)；在別的格局，它擦掉掃描符號(hào)。機(jī)器也可能改變正在掃描的方格，但是僅僅向左右移動(dòng)一個(gè)方格。此外，對(duì)任何一個(gè)這樣的操作，m－格局可能變化。某些寫(xiě)下的符號(hào)將形成數(shù)字序列( sequence of figures)，它是正在計(jì)算的表達(dá)為十進(jìn)制的實(shí)數(shù)，別 的符號(hào)只是“輔助記憶”的粗糙紀(jì)錄，只有那些粗糙紀(jì)錄可能被擦掉。

我的觀點(diǎn)是，這些操作包括了所有用在計(jì)算一個(gè)數(shù)時(shí)所需的操作，為此論點(diǎn)辯護(hù)要在讀者較熟悉機(jī)器理論后才會(huì)較容易。在下一節(jié)，我將著手發(fā)展此理論，假設(shè)讀者已經(jīng)知道“機(jī)器”，“紙帶”，“掃描”等。

1. Computing machines.

We have said that the computable numbers are those whose decimals are calculable by finite means. This requires rather more explicit definition. No real attempt will be made to justify the definitions given until we reach §9. For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.

We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q1, q2, ..., qR which will be called “m-configurations”. The machine is supplied with a “tape”, (the analogue of paper) running through it, and divided into sections (called “squares”) each capable of bearing a “symbol”. At any moment there is just one square, say the r-th, bearing the symbol S(r) which is “in the machine”. We may call this square the “scanned square”. The symbol on the scanned square may be called the “scanned symbol”. The “scanned symbol” is the only one of which the machine is, so to speak, “directly aware”. However, by altering its m-configuration the machine can effectively remember some of the symbols which it has “seen” (scanned) previously. The possible behaviour of the machine at any moment is determined by the m-configuration qn and the scanned symbol S(r). This pair qn, S(r) will be called the “configuration”: thus the configuration determines the possible behaviour of the machine. In some of the configurations in which the scanned square is blank (i.e. bears no symbol) the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or 1eft. In addition to any of these operations the m-configuration may be changed. Some of the symbols written down {232} will form the sequence of figures which is the decimal of the real number which is being computed. The others are just rough notes to “assist the memory”. It will only be these rough notes which will be liable to erasure.

It is my contention that these operations include all those which are used in the computation of a number. The defence of this contention will be easier when the theory of the machines is familiar to the reader. In the next section I therefore proceed with the development of the theory and assume that it is understood what is meant by “machine”, “tape”, “scanned”, etc.

2，定義

自動(dòng)機(jī)(Automatic machines)

如果每個(gè)階段機(jī)器的運(yùn)動(dòng)(在第一章的意義上)完全由格局確定，我們稱(chēng)此機(jī)器為?自動(dòng)機(jī)?(a-machine)。

為了某種目的，我們可能使用一些機(jī)器(choice machines or c-machines)，它的運(yùn)動(dòng)部分由格局決定。當(dāng)這樣的機(jī)器到達(dá)一個(gè)這樣模棱兩可的格局時(shí)，只有當(dāng)外界的操作做某個(gè)任意的選擇，它才能繼續(xù)運(yùn)行。如果我們用機(jī)器處理公理系統(tǒng)時(shí)會(huì)出現(xiàn)這樣的情況，在本文中，我只處理自動(dòng)機(jī)，因此往往忽略前綴a-。

計(jì)算機(jī)器(Computing machines)

如果一臺(tái)機(jī)器打印兩類(lèi)符號(hào)，第一類(lèi)(稱(chēng)為數(shù)字)全是0和1，其它被稱(chēng)為第二類(lèi)符號(hào)，則機(jī)器將被稱(chēng)為“計(jì)算機(jī)器”。如果給機(jī)器裝置一條空白紙帶，讓它運(yùn)動(dòng)起來(lái)，從正確的初始m－格局出發(fā)，機(jī)器打印的第一類(lèi)符號(hào)的子序列稱(chēng)作“機(jī)器計(jì)算的序列”；在表達(dá)為二進(jìn)制的十進(jìn)制實(shí)數(shù)前放上小數(shù)點(diǎn)，稱(chēng)作“機(jī)器計(jì)算的數(shù)”。

在機(jī)器運(yùn)動(dòng)的任何階段，掃描方格的數(shù)，在紙帶上所有符號(hào)的完整序列，及m－格局，被說(shuō)成是描述那時(shí)刻的“完全格局”。機(jī)器和紙帶在一系列完整格局之間的變化被稱(chēng)作“機(jī)器的運(yùn)動(dòng)”。

循環(huán)和非循環(huán)機(jī)器(Circular and circle-free machines)

如果計(jì)算機(jī)不再寫(xiě)下有限數(shù)目的第一類(lèi)符號(hào)，被稱(chēng)作“循環(huán)(Circular)”；否則，被稱(chēng)作“無(wú)循環(huán)(circle-free)”。

如果一臺(tái)機(jī)器達(dá)到一個(gè)格局，從此不再運(yùn)動(dòng)，或者即使繼續(xù)運(yùn)動(dòng)，只能打印第二類(lèi)符號(hào)，而不能打印任何第一類(lèi)符號(hào)，此機(jī)器則是“circular”。“circle-free”的意義將在第8章解釋。

可計(jì)算序列和可計(jì)算數(shù)(Computable sequences and numbers)

一個(gè)序列被說(shuō)成“可計(jì)算的”，如果能夠通過(guò)一臺(tái)“circle-free machine”計(jì)算而得。一個(gè)數(shù)是可計(jì)算的，如果它與由“circle-free machine”計(jì)算的數(shù)只差一個(gè)整數(shù)。

我們應(yīng)該避免混淆，說(shuō)可計(jì)算序列比說(shuō)可計(jì)算數(shù)更經(jīng)常。

2. Definitions.

Automatic machines.

If at each stage the motion of a machine (in the sense of §1) is completely determined by the configuration, we shall call the machine an “automatic machine” (or a-machine). For some purposes we might use machines (choice machines or c-machines) whose motion is only partially determined by the configuration (hence the use of the word “possible” in §1). When such a machine reaches one of these ambiguous configurations, it cannot go on until some arbitrary choice has been made by an external operator. This would be the case if we were using machines to deal with axiomatic systems. In this paper I deal only with automatic machines, and will therefore often omit the prefix a-.

Computing machines.

If an a-machine prints two kinds of symbols, of which the first kind (called figures) consists entirely of 0 and 1 (the others being called symbols of the second kind), then the machine will be called a computing machine. If the machine is supplied with a blank tape and set in motion, starting from the correct initial m-configuration, the subsequence of the symbols printed by it which are of the first kind will be called the sequence computed by the machine. The real number whose expression as a binary decimal is obtained by prefacing this sequence by a decimal point is called the number computed by the machine.

At any stage of the motion of the machine, the number of the scanned square, the complete sequence of all symbols on the tape, and the m-configuration will be said to describe the complete configuration at that stage. The changes of the machine and tape between successive complete configurations will be called the moves of the machine.{233}

Circular and circle-free machines.

If a computing machine never writes down more than a finite number of symbols of the first kind it will be called circular. Otherwise it is said to be circle-free.

A machine will be circular if it reaches a configuration from which there is no possible move, or if it goes on moving, and possibly printing symbols of the second kind, but cannot print any more symbols of the first kind. The significance of the term “circular” will be explained in §8.

Computable sequences and numbers.

A sequence is said to be computable if it can be computed by a circle-free machine. A number is computable if it differs by an integer from the number computed by a circle- free machine.

We shall avoid confusion by speaking more often of computable sequences than of computable numbers.

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