“问题”是什么？

“问题”是什么

刘建文略译（ http://www.semi-translate.com/blog ）

KEY: 问题 科学哲学 方法论

“问题”是什么？

“问题”是什么？“问题”不是一个面包，不是一辆汽车，但它确实的存在，有它自己的性质或存在的客观性。我今天把它揪出来研究一下。 先看看《现代汉语词典》的定义：

问题

1. ①要求回答或解释的题目：这次考试一共有五个～│我想答复一下这一类的～。
2. ②须要研究讨论并加以解决的矛盾、疑难：思想～│这种药治感冒很解决～。
3. ③关键；重要之点：重要的～在善于学习。
4. ④事故或麻烦：那部车床又出～了。

从这个定义可以看出，“问题”和科学、方法、理论等概念一样的“普通”，是一个很高抽象级别范畴概念，已经深入到我们话语系统、思维结构之中了。这也说明了“问题”与其它的高抽象级别范畴概念一样具有极大的思考价值。我们再看看维基百科的定义：

A problem is an obstacle障碍 which makes it difficult to achieve a desired goal, objective or purpose. It refers to a situation, condition, or issue that is yet unresolved. In a broad sense, a problem exists when an individual becomes aware of a significant difference between what actually is and what is desired.

“问题”是使实现预定的目标、目的或宗旨进入困境的障碍。它涉及了一个没有解决的状况、条件或事件。在更广泛的意思说，当个人开始意识到“渴望的”与“实际的”有足够的差别时，问题产生了。

从这个定义我们了解到，“问题”的确不是像西瓜那样是一个具体存在的直觉上的“东西”，它和“计算”一样是一个抽象的概念；问题与[一个过程性的概念]——“实现一个目标”有关系，阻碍关系。问题阻碍了目标的实现。“问题”与“计算”有不同之处，问题是一个状况，一个相对静止的抽象概念；而“计算”和“实现一个目标”都是一个过程性的带时间性的抽象概念。看看一些常见的问题：

• “如何能追到那女孩？”
• “我肚子饿了，有什么东西我吃？”
• 我的生日是哪一天？
• 炒青菜应该放多少盐？
• 1515*34-443=?
• 怎么样设计交通信号灯才能使用交通流量最大？

一些定义

定义一

问题很难有一个确定的，无异议的定义，但是，一般来说都问题包含有以下三个基本成分：

• 上下文： 和问题相关的场景，指一组已经是明确已知的，关于问题的条件的描述。
• 目标： 指关于构成问题的结论的明确的描述。
• 障碍： 指问题的正确解决方法不是显而易见的，必须通过一定的思维活动，才能找到答案。

一般而言，问题是由于某些导致不能达到目的或者实现目标的认识障碍。它是指不期待的现状没有被解决或者事态出现意外。

定义二（摘自《数学方法论与解题研究》）

一般说来，问题是给定的信息和目标之间有某些障碍需要加以克服的情景。所有问题都会有三个基本成分：

• 给定（Givens），即一组给予的信息；
• 目标（Goals），问题要求的或结尾的状态，即关于构成问题的结论描述；
• 障碍（Obstacles），思维者无法立即找到正确答案，必须通过一定的方式来改变给定状态，逐步达到目标状态。

定义三《数学的发现--对解题的理解、研究的讲授 第一卷》

“有问题”指的是：有意识地寻求某一适当的行动，以便达到一个被清楚地意识到但不能立即达到的目的。解决问题指的是寻找这种活动。

“求解”问题的目的是要求一个确定的对象--问题的未知量，要求的是满足这种问题的条件的未知量，这个条件把未知量与问题的已知量联系起来。未知量可能属于任一可能想像的种类。一个表述清楚的问题必须规定其未知量的种类（集合），也必须规定未知量应该满足的条件。在由问题规定的对象集合（未知量必定属于该集合）中，有满足条件的那些对象组成的子集，并且任一个属于这个子集的对象被称为一个解。

我们把未知量、条件和已知量称为“求解”问题的主要部分。

我们探索的目标，可以是任何类型的未知量，或者是发现任何种类问题的真理：我们的问题可能是理论的或实际的，重要的或无足轻重的。为了解决我们的问题，我们必须制定一个深思熟虑的、有条有理的行动计划（逻辑推理、数学运算或者是具体工作），以从我们已有的东西得出我们缺少的东西--从前提得出结论、从已知量得出未知量等。

计算机处理的是什么样的“问题”？

先一个例子，摘自<<Algorithms and Data Structures: The Science of Computing>>

1.1.1 Problems

Some people (including one of the authors) chill bottles or cans of soft drinks or fruit juice by putting them in a freezer for a short while before drinking them. This is a nice way to get an extra-cold drink, but it risks disaster: a drink left too long in the freezer begins to freeze, at which point it starts to expand, ultimately bursting its container and spilling whatever liquid isn't already frozen all over the freezer. People who chill drinks in freezers may thus be interested in knowing the longest time that they can safely leave a drink in the freezer, in other words, the time that gives them the coldest drink with no mess to clean up afterwards. But since neither drinks nor freezers come with the longest safe chilling times stamped on them by the manufacturer, people face the problem of finding those times for themselves. This problem makes an excellent example of the kinds of problems and problem solving that exist in computer science. In particular, it shares two key features with all other problems of interest to computer science.

以上“冰冻饮料”的问题展示了计算机科学关心的问题及其解决方法的一个极好的例子。特别地，这些问题有两个共同的特征：

First, the problem is general enough to appear over and over in slightly different forms, or instances. In particular, different freezers may chill drinks at different speeds, and larger drinks will generally have longer safe chilling times than smaller drinks. Furthermore, there will be some margin of error on chilling times, within which more or less chilling really doesn't matter-for example, chilling a drink for a second more or a second less than planned is unlikely to change it from unacceptably warm to messily frozen. But the exact margin of error varies from one instance of the problem to the next (depending, for example, on how fast the freezer freezes things and how willing the person chilling the drink is to risk freezing it). Different instances of the longest safe chilling time problem are therefore distinguished by how powerful the freezer is, the size of the drink, and what margin of error the drinker will accept. Things that distinguish one problem instance from another are called parameters or inputs to the problem.

第一，这些问题都足够的一般，只需稍作修改（更改其中的某个条件）就可以演化出很多的有类似形式的问题（实例）。比如，“冰冻饮料”的问题里的冰箱（功力）和饮料（体积）都可以不同的，这样也有不同的答案--最长冰冻时间。此外，这个冰冻时间可以有一定的误差，有些问题适量误差是无关紧要的，因问题而定。“冰冻饮料”的问题的实例的不同由冰箱功力、饮料体积和可接受的误差值决定。这些区分问题实例的条件称为参数或问题输入。

Also note that different instances of a problem generally have different answers. For example, the longest safe chilling time for a two-liter bottle in a kitchenette freezer is different from the longest safe chilling time for a half-liter in an commercial deep freeze. It is therefore important to distinguish between an answer to a single instance of a problem and a process that can solve any instance of the problem. It is far more useful to know a process with which to solve a problem whenever it arises than to know the answer to only one instance-as an old proverb puts it, "Give a man a fish and you feed him dinner, but teach him to fish and you feed him for life."

（形式相似的）不同问题实例会有不同的答案，因此，找到一个特定问题实例 的答案和找到解答所有问题实例的过程（也就是解题算法）有很大的差别的。后者具有更大的价值，就像那句古训：授之以鱼不如授之以渔。

The second important feature of any computer science problem is that you can tell whether a potential answer is right or not. For example, if someone tells you that a particular drink can be chilled in a particular freezer for up to 17 minutes, you can easily find out if this is right. Chill the drink for 17 minutes and see if it comes out not quite frozen; then chill a similar container of the same drink for 17 minutes plus the margin of error and see if it starts to freeze. Put another way, a time must meet certain requirements in order to solve a given instance of the problem, and it is possible to say exactly what those requirements are: the drink in question, chilled for that time in the freezer in question, shouldn't quite freeze, whereas the drink in question, chilled for that time plus the margin of error in the freezer in question, would start to freeze. That you need to know what constitutes a correct answer seems like a trivial point, but it bears an important moral nonetheless: before trying to find a process to solve a problem, make sure you understand exactly what answers you will accept.

第二，这些问题的答案必须确定的，不能模棱两可。例如，某人告诉你某台冰箱冰冻一个特定大小的饮料的最长冰冻时间是17分钟。你可以做数次直接测试这个答案是否正确。在这里，这个答案--时间--必须满足一定的前提条件，比如，饮料有多大，饮料不能太冻和冰箱的功力。因此，你必须清楚知道一个正确答案是什么。这看似简单而微不足道，但实际上在问题解决中起着重的角色。当你试图为解决一个问题寻找解决过程（算法）时，必须先明确答案是什么。

Not every problem has these two features. Problems that lack one or the other are generally outside the scope of computer science. For example, consider the problem, "In what year did people first walk on the moon?" This problem lacks the first feature of being likely to appear in many different instances. It is so specific that it only has one instance, and so it's easier to just remember that the answer is "1969" than to find a process for finding that answer. As another example, consider the problem, "Should I pay parking fines that I think are unfair?" This problem lacks the second feature of being able to say exactly what makes an answer right. Different people will have different "right" answers to any instance of this problem, depending on their individual notions of fairness, the relative values they place on obeying the law versus challenging unfair actions, etc.