在

《 Simply Logical
Intelligent Reasoning by Example 》

之第一部份起頭處， Peter Flach 開宗明義的講︰

Logic and Logic Programming

Logic Programming is the name of a programming paradigm which was developed in the 70s. Rather than viewing a computer program as a step-by-step description of an algorithm, the program is conceived as a logical theory, and a procedure call is viewed as a theorem of which the truth needs to be established. Thus, executing a program means searching for a proof. In traditional (imperative) programming languages, the program is a procedural specification of how a problem needs to be solved. In contrast, a logic program concentrates on a declarative specification of what the problem is. Readers familiar with imperative programming will find that Logic Programming requires quite a different way of thinking. Indeed, their knowledge of the imperative paradigm will be partly incompatible with the logic paradigm.

This is certainly true with regard to the concept of a program variable. In imperative languages, a variable is a name for a memory location which can store data of certain types. While the contents of the location may vary over time, the variable always points to the
same location. In fact, the term ‘variable’ is a bit of a misnomer here, since it refers to a value that is well-defined at every moment. In contrast, a variable in a logic program is a variable in the mathematical sense, i.e. a placeholder that can take on any value. In this respect, Logic Programming is therefore much closer to mathematical intuition than imperative programming.

Imperative programming and Logic Programming also differ with respect to the machine model they assume. A machine model is an abstraction of the computer on which programs are executed. The imperative paradigm assumes a dynamic, state-based machine model, where the state of the computer is given by the contents of its memory. The effect of a program statement is a transition from one state to another. Logic Programming does not assume such a dynamic machine model. Computer plus program represent a certain amount
of knowledge about the world, which is used to answer queries.

，指出『邏輯編程』和『 Von Neumann 程式語言』──

摘自《 CPU 機器語言的『解譯器』》︰

事實上 Von Neumann 的計算機架構，對於電腦程式語言的發展，有著極為深遠的影響，產生了現在叫做 Von Neumann 程式語言，與Von Neumann 的計算機架構，同形 isomorphism 同構︰

program variables ↔ computer storage cells
程式變數對映計算機的儲存單元

control statements ↔ computer test-and-jump instructions
控制陳述對映計算機的『測試．跳至』指令

assignment statements ↔ fetching, storing instructions
賦值陳述對映計算機的取得、儲存指令

expressions ↔ memory reference and arithmetic instructions.
表達式對映記憶體參照和算術指令

── 之『觀點』有極大的不同。通常這造成了學過一般程式語言的人『理解』上的困難。也可以說『邏輯編程』之『推導歸結』得到『證明』的想法遠離『圖靈機』的『狀態轉移』到達『接受』狀態。反倒是比較接近『 λ運算』與『 Thue 字串改寫系統』抽象建構。讀者可以試著參考讀讀那些文章，看看能否將『思考』方式的

Logic Programming requires quite a different way of thinking. Indeed, their knowledge of the imperative paradigm will be partly incompatible with the logic paradigm.

『不相容性』鎔鑄成整體思維之『結晶』，嫻熟用於解決各式各樣的『問題』！

也許『學習』到了一定的時候，或早或遲人們需要想想『學習方法』，經由對自己的了解，找到『事半功倍』的有效作法。雖說人人都是獨特的，然而在『學習』上確實有很多類似的『難處』。比方說一般寫程式的思維，有所謂『從上往下』 Top-Down 以及『由底向頂』 Button-Up 思路取向不同。一者類似『歐式幾何』，從公理公設出發，逐步推演定理定律，邏輯清楚明白。然而一旦要自己去證明 □□ 定理時，有時總覺的無處下手。這是因為由『公理』通往『定理』的『推導歸結』之道路遙遠，常常看不出 ○□ 兩個『概念』間竟然有如此的『聯繫』。另一彷彿『學會做菜』，由觀察親朋做菜開始，知道什麼菜要怎麼洗？怎麼切？用什麼方法調理？久而久之，學會了一道二道三道很多很多的菜的作法。也可以講，這樣的『學法』知道了許多『如何作』 Know-How ，很少思考『為什麼』 Know-Why 這麼作，也很少形成一般『這一類』 Know-What 的菜，多半這樣作的『通則』。我們將要如何回答別人，自己『想都沒想過』，這菜『卻得這麼作』的『問題』？或許異地嫁娶之人，更能體會，這種『不同』就是『不同』的意思。

─── 《勇闖新世界︰《 PYDATALOG 》導引《一》》

突如其來如，早年『邏輯編程』遭遇一事，闖入腦海震盪心胡！

憶起了曾經改寫之難表達之苦，而今已免『豆鵝狐人』之堵塞耶？

忽爾思及『因式分解』問題？？

『窮舉法』是一種通用方法。它以『問題』為『解』之『對錯』的『判準』，窮盡『所有可能』的『解』作『斷言』之求解法。或可同時參閱《 M♪o 之學習筆記本《寅》井井︰【䷝】試釋是事》文中所說之『蠻力法』以及例子︰

☿ 行︰雖說是蠻力法，實則乃用『窮舉』，數ㄕㄨˇ數ㄕㄨˋ數不盡，耗時難為功，怎曉機心迅捷後，此法遂真可行耶！？實習所用機，登入採『學號』與『針碼』【※ Pin Code 四位數字碼】。『學號』之制 ── 班碼-位碼 ──，班不過十，位少於百，故而極其數不足千。針碼有四位，總其量，只有萬。試而盡之，『千萬』已『窮舉』。問題當在『咸澤碼訊』有多快？『登錄』之法有多嚴？破解程式幾人會？設使一應具足，那個『駭黑』之事，怕是恐難免！！……

此處特別指出計算機『速度提昇』使得『窮舉法』的實用性大增，對此法的了解也就更顯得重要了。

假使將『所有可能解』看成『解空間』，將『問題』當成『約束』條件，此時『問題』的『求解』，就轉換成『尋找』『解空間』中滿足『約束』條件的『解』。一般將之稱為『蠻力搜尋法』︰

Brute-force search

In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem’s statement.

A brute-force algorithm to find the divisors of a natural number n would enumerate all integers from 1 to the square root of n, and check whether each of them divides n without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard, and, for each arrangement, check whether each (queen) piece can attack any other.

While a brute-force search is simple to implement, and will always find a solution if it exists, its cost is proportional to the number of candidate solutions – which in many practical problems tends to grow very quickly as the size of the problem increases. Therefore, brute-force search is typically used when the problem size is limited, or when there are problem-specific heuristics that can be used to reduce the set of candidate solutions to a manageable size. The method is also used when the simplicity of implementation is more important than speed.

This is the case, for example, in critical applications where any errors in the algorithm would have very serious consequences; or when using a computer to prove a mathematical theorem. Brute-force search is also useful as a baseline method when benchmarking other algorithms or metaheuristics. Indeed, brute-force search can be viewed as the simplest metaheuristic. Brute force search should not be confused with backtracking, where large sets of solutions can be discarded without being explicitly enumerated (as in the textbook computer solution to the eight queens problem above). The brute-force method for finding an item in a table — namely, check all entries of the latter, sequentially — is called linear search.

上一篇因數的例子︰

── 摘自《勇闖新世界︰《 PYDATALOG 》【專題】之約束編程‧三》

故地重遊，想那時為何未論及『質數』哩！！