Proving TheoremsBy Using Abstraction Interactively
Proving TheoremsBy Using Abstraction Interactively
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通过交互使用抽象来证明定理
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发表时间：
1994
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通讯作者：
Fausto Giunchiglia
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作者：
R. Sebastiani;Adolfo Villa orita;Fausto Giunchiglia
属于这样一个集合的元素f，对于任何基础定理'，存在于2中抽象定理f（'）的证明2;此外，存在（至少）一种将”（至少一种可能的）"映射回2到1的方法，这产生了基础证明的轮廓。可以随后再进行。重要的是要注意，在交互式抽象定理证明中，不需要只与人类交互。因此，为了实现自动定理证明在有限的领域，一个互动的抽象定理证明，如ABSFOL，可以与外部控制程序的基础上euristic策略[Kno 91，BGW 91]。此外，这种系统可以提供一个内部ML类元语言，用于编写实现控制策略的程序，称为战术[GT 94]。自动化定理证明可以通过编写复杂的策略来实现。3地图着色问题作为交互式抽象定理证明的一个例子，在本文中，我们将考虑一个非常著名的问题，即只用四种不同的颜色对平面地图的各个区域进行着色。在图1中，我们找到了一个取自[McC 90]的非常简单的示例。根据McCarthy的说法，我们可以用如下的定理证明来形式化这个问题。我们需要一组变量（如阿尔巴尼亚、安道尔、奥地利、：或r1、r2、：），每个变量代表给定地图的一个区域，一个二元谓词n（）\next”，代表边界约束（例如n（奥地利、意大利）），以及一组四个独立的常量y; B; g; r（黄色、蓝色、绿色、红色）。然后，我们定义了一个公理“颜色”，写为着色约束的合取，如n（y，B）（"黄色状态可以与蓝色状态相邻”）。使用ABSFOL，问题可以如下初始化：4 NONAME：：NAMECONTEXT gmap; GMAP：：DESIGNE INDCONST y B g r; GMAP：：DESIGNE INDVAR r1 r2 r3 r4 r5 r6; GMAP：：DESIGNE PREDCONST n 2; GMAP：：AXIOM colors：n（y，B）和n（y，g）和n（y，r）和n（B，y）和n（B，g）和n（B，r）和n（g，y）和n（g，B）和n（g，r）和n（r，y）和n（r，B）和n（r，g）;目标是由所有且只有地图的边界约束的conjerion给出的：存在r1. R6.（n（r1，r2）和n（r1，r3）和.和n（r5，r6））4在ABSFOL中，电传字体用于写入ABSFOL的输入和输出。大写Teletype用于关键字。\<string>：：“是ABSFOL提示符：\：：“之前的字符串是当前上下文的名称，这是我们正在工作的理论。NAMECONTEXT命名当前上下文。DESCRIBE将新符号添加到当前上下文。AXIOM将公理添加到当前上下文中。输入和输出都进行了轻微的编辑，使其更具可读性。4地面问题抽象问题抽象解决方案地面解决方案ii）抽象iv）映射回v）提炼iii）抽象证明3 6 1 2 3 6 1 5 4 5 4蓝色黄色黄色
ion f belonging to such a set, for any ground theorem ', exists in 2 a proof 2 of the abstracted theorem f('); moreover exists (at least) one way to \map back" (at least one of the possible) 2 to 1 which gives rise to an outline of the ground proof. can be subsequently re ned. It is important to observe that in interactive abstract theorem proving there is no need to interact only with human beings. Thus, in order to achieve automatic theorem proving in limited domains, an interactive abstract theorem prover, like ABSFOL, could be interfaced with external control programs based on euristic strategies [Kno91, BGW91]. Moreover, such system can be provided with an internal ML-like metalanguage for writing programs which implements control strategies, called tactics [GT94]. Automated theorem proving can then be implemented by writing complex tactics. 3 The map colouring problem As an example of interactive abstract theorem proving, in this paper we will consider the very well-known problem of colouring the various regions of a planar map using only four di erent colours. In Figure 1 we nd a very simple example taken from [McC90]. According to McCarthy, we can formalize the problem in terms of theorem proving as follows. We need a set of variables (like Albania, Andorra, Austria, : : : or r1,r2, : : : ) each representing a region of the given map, a binary predicate n() \next", representing the frontier constraints (e.g. n(Austria,Italy)), and a set of the four individual constants y; b; g; r (yellow, blue, green, red). Then we de ne an axiom \colours", written as a conjunction of colouring constraints like n(y,b) (\a yellow-coloured state can border a blue-coloured one"). Using ABSFOL, the problem can be initialized as follows: 4 NONAME:: NAMECONTEXT gmap; GMAP:: DECLARE INDCONST y b g r; GMAP:: DECLARE INDVAR r1 r2 r3 r4 r5 r6; GMAP:: DECLARE PREDCONST n 2; GMAP:: AXIOM colours : n(y,b) and n(y,g) and n(y,r) and n(b,y) and n(b,g) and n(b,r) and n(g,y) and n(g,b) and n(g,r) and n(r,y) and n(r,b) and n(r,g); The goal is given by the conjuction of all and only the frontier constraints of the map: exists r1 ... r6. ( n(r1,r2) and n(r1,r3) and ... and n(r5,r6) ) 4In ABSFOL, Teletype font is used to write the input and output of ABSFOL. UPPERCASE TELETYPE is used for key-words. \<string>::" is the ABSFOL prompt: the string before \:: " is the name of the current context, that is the theory we are working in. NAMECONTEXT names the current context. DECLARE adds new symbols to the current context. AXIOM adds an axiom to the current context. Both the input and the output have been slightly edited to make them more readable. 4 Ground Problem Abstract Problem Abstract Solution Ground Ground Solution ii) abstract iv) map back v) refine iii) abstract prove 3 6 1 2 3 6 1 5 4 5 4 blue blue blue yellow yellow
日本临床麻醉学会杂志 4-3 (1984)。
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