# Формулы первого порядка, сохраняющиеся при минимальной

```&Ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&aring;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute;
&ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave;.
&Ecirc;&egrave;&ecirc;&icirc;&ograve;&uuml; &Ntilde;. &Iuml;.
&Egrave;&iacute;&ntilde;&ograve;&egrave;&ograve;&oacute;&ograve; &iuml;&eth;&icirc;&aacute;&euml;&aring;&igrave; &iuml;&aring;&eth;&aring;&auml;&agrave;&divide;&egrave; &egrave;&iacute;&ocirc;&icirc;&eth;&igrave;&agrave;&ouml;&egrave;&egrave; &ETH;&Agrave;&Iacute;,
&Igrave;&icirc;&ntilde;&ecirc;&icirc;&acirc;&ntilde;&ecirc;&egrave;&eacute; &ocirc;&egrave;&ccedil;&egrave;&ecirc;&icirc;-&ograve;&aring;&otilde;&iacute;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute; &egrave;&iacute;&ntilde;&ograve;&egrave;&ograve;&oacute;&ograve;,
&Acirc; &iacute;&agrave;&ntilde;&ograve;&icirc;&yuml;&ugrave;&aring;&eacute; &eth;&agrave;&aacute;&icirc;&ograve;&aring; &igrave;&ucirc; &auml;&agrave;&aring;&igrave; &ntilde;&egrave;&iacute;&ograve;&agrave;&ecirc;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &icirc;&iuml;&egrave;&ntilde;&agrave;&iacute;&egrave;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;,
&ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&otilde;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;.
&Agrave;&iacute;&iacute;&icirc;&ograve;&agrave;&ouml;&egrave;&yuml;
&Ecirc;&euml;&thorn;&divide;&aring;&acirc;&ucirc;&aring; &ntilde;&euml;&icirc;&acirc;&agrave;:
1
&ograve;&aring;&icirc;&eth;&egrave;&yuml; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute;, &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &icirc; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ccedil;&agrave;&ouml;&egrave;&egrave;, &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave;
&Acirc;&acirc;&aring;&auml;&aring;&iacute;&egrave;&aring;
&Ograve;&aring;&icirc;&eth;&aring;&ograve;&egrave;&ecirc;&icirc;-&igrave;&icirc;&auml;&aring;&euml;&uuml;&iacute;&agrave;&yuml; &ecirc;&icirc;&iacute;&ntilde;&ograve;&eth;&oacute;&ecirc;&ouml;&egrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave; &aacute;&ucirc;&euml;&agrave; &iuml;&eth;&egrave;&auml;&oacute;&igrave;&agrave;&iacute;&agrave; &acirc; 1960&aring; [18] &auml;&euml;&yuml; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&ucirc;&otilde; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute; &egrave;&ccedil; &aacute;&aring;&ntilde;&ecirc;&icirc;&iacute;&aring;&divide;&iacute;&ucirc;&otilde; &iuml;&eth;&egrave; &iuml;&icirc;&igrave;&icirc;&ugrave;&egrave; &ocirc;&agrave;&ecirc;&ograve;&icirc;&eth;&egrave;&ccedil;&agrave;&ouml;&egrave;&egrave;, &ograve;&icirc; &aring;&ntilde;&ograve;&uuml; &ntilde;&ecirc;&euml;&aring;&egrave;&acirc;&agrave;&iacute;&egrave;&yuml; &icirc;&auml;&iacute;&icirc;&ograve;&egrave;&iuml;&iacute;&ucirc;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; &acirc;
&icirc;&auml;&iacute;&oacute;. &Igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; &aring;&aring; &ntilde;&agrave;&igrave;&ucirc;&eacute; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&eacute; &divide;&agrave;&ntilde;&ograve;&iacute;&ucirc;&eacute; &ntilde;&euml;&oacute;&divide;&agrave;&eacute;. &Acirc; &iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&egrave;&otilde; &iacute;&aring;&auml;&agrave;&acirc;&iacute;&ucirc;&otilde; &eth;&agrave;&aacute;&icirc;&ograve;&agrave;&otilde; &oacute;&ntilde;&ograve;&icirc;&eacute;&divide;&egrave;&acirc;&icirc;&ntilde;&ograve;&uuml; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&eacute; &aacute;&ucirc;&euml;&agrave; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&agrave; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring;
&auml;&icirc;&ntilde;&ograve;&agrave;&ograve;&icirc;&divide;&iacute;&icirc;&atilde;&icirc; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml; &eth;&agrave;&ccedil;&eth;&aring;&oslash;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave; &auml;&euml;&yuml; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&ucirc;&otilde; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &euml;&icirc;&atilde;&egrave;&ecirc; [3], [6], [10].
&Ccedil;&agrave;&auml;&agrave;&divide;&egrave; &icirc;&aacute; &icirc;&iuml;&egrave;&ntilde;&agrave;&iacute;&egrave;&egrave; &acirc; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave; &acirc;&ntilde;&aring;&otilde; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &egrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&otilde; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;, &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&otilde;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave;, &ntilde;&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&aring; &Acirc;. &Oslash;&aring;&otilde;&ograve;&igrave;&agrave;&iacute;&icirc;&igrave;, &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &egrave;&iacute;&ograve;&aring;&eth;&aring;&ntilde;
&auml;&euml;&yuml; &ograve;&aring;&icirc;&eth;&egrave;&egrave; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute; &iuml;&eth;&icirc;&ntilde;&ograve;&icirc;&ograve;&icirc;&eacute; &iuml;&icirc;&ntilde;&ograve;&agrave;&iacute;&icirc;&acirc;&ecirc;&egrave; &egrave; &iacute;&aring;&icirc;&aelig;&egrave;&auml;&agrave;&iacute;&iacute;&icirc;&eacute; &ntilde;&euml;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn;. &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &iacute;&agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&aring; &yacute;&ograve;&icirc;&atilde;&icirc;
&icirc;&iuml;&egrave;&ntilde;&agrave;&iacute;&egrave;&yuml; &igrave;&icirc;&aelig;&iacute;&icirc; &aacute;&ucirc;&euml;&icirc; &aacute;&ucirc; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve;&uuml; &iacute;&icirc;&acirc;&ucirc;&aring; &eth;&agrave;&ccedil;&eth;&aring;&oslash;&egrave;&igrave;&ucirc;&aring; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &euml;&icirc;&atilde;&egrave;&ecirc;&egrave;. &Aring;&ntilde;&ograve;&aring;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&agrave;&yuml; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&ccedil;&agrave;,
&divide;&ograve;&icirc; &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&aring;&ntilde;&yuml; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&aring; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &ntilde; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &auml;&icirc; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave;&igrave;&aring;&thorn;&ograve; &acirc;&egrave;&auml;
3p → φ, &atilde;&auml;&aring; φ &iuml;&icirc;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&agrave;, &icirc;&ecirc;&agrave;&ccedil;&ucirc;&acirc;&agrave;&aring;&ograve;&ntilde;&yuml; &iacute;&aring;&acirc;&aring;&eth;&iacute;&icirc;&eacute; &aacute;&aring;&ccedil; &ograve;&eth;&aring;&aacute;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&ntilde;&ograve;&egrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;. &Ecirc;&icirc;&iacute;&ograve;&eth;&iuml;&eth;&egrave;-
&igrave;&aring;&eth;&icirc;&igrave; &yuml;&acirc;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml; &agrave;&iuml;&iuml;&eth;&icirc;&ecirc;&ntilde;&egrave;&igrave;&agrave;&iacute;&ograve;&ucirc; &Otilde;&icirc;&auml;&ecirc;&egrave;&iacute;&ntilde;&icirc;&iacute;&agrave; [7] &yacute;&ecirc;&ccedil;&egrave;&ntilde;&ograve;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;-&ecirc;&icirc;&iacute;&uacute;&thorn;&iacute;&ecirc;&ograve;&egrave;&acirc;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc;
&iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;
γm = 2(p1 ∨ . . . ∨ pm ) →
m
_
(3(pi ∧ 3pj ) ∧ 3(pj ∧ 3pi ))
i,j=1
&egrave;&ccedil; &eth;&agrave;&aacute;&icirc;&ograve;&ucirc; [9]. &Ograve;&agrave;&ecirc;&egrave;&igrave; &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&igrave;, &iuml;&icirc;&auml;&ograve;&acirc;&aring;&eth;&aelig;&auml;&aring;&iacute;&egrave;&aring; &egrave;&euml;&egrave; &icirc;&iuml;&eth;&icirc;&acirc;&aring;&eth;&aelig;&aring;&iacute;&egrave;&yuml; &yacute;&ograve;&icirc;&eacute; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&ccedil;&ucirc; &ntilde;&acirc;&yuml;&ccedil;&agrave;&iacute;&icirc; &ntilde; &iacute;&agrave;&otilde;&icirc;&aelig;&auml;&aring;&iacute;&egrave;&aring;&igrave; &iacute;&icirc;&acirc;&ucirc;&otilde; &acirc;&egrave;&auml;&icirc;&acirc; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&otilde; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;. &Egrave;&ccedil; &oacute;&iuml;&icirc;&igrave;&yuml;&iacute;&oacute;&ograve;&icirc;&eacute; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&ccedil;&ucirc; &ograve;&agrave;&ecirc;&aelig;&aring; &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;
&ecirc;&eth;&egrave;&ograve;&aring;&eth;&egrave;&eacute; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&igrave;&icirc;&ntilde;&ograve;&egrave; &auml;&euml;&yuml; &ecirc;&icirc;&iacute;&uacute;&thorn;&iacute;&ecirc;&ograve;&egrave;&acirc;&iacute;&ucirc;&otilde; &yacute;&ecirc;&ccedil;&egrave;&ntilde;&ograve;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &egrave;&ccedil; &eth;&agrave;&aacute;&icirc;&ograve;&ucirc; [11].
&Acirc; &iacute;&agrave;&ntilde;&ograve;&icirc;&yuml;&ugrave;&aring;&eacute; &eth;&agrave;&aacute;&icirc;&ograve;&aring; &igrave;&ucirc; &auml;&agrave;&aring;&igrave; &ntilde;&egrave;&iacute;&ograve;&agrave;&ecirc;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &icirc;&iuml;&egrave;&ntilde;&agrave;&iacute;&egrave;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&otilde;&ntilde;&yuml;
&iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;, &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &agrave;&auml;&agrave;&iuml;&ograve;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&icirc; &ecirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&aring;&igrave;&oacute; &yuml;&ccedil;&ucirc;&ecirc;&oacute;. &Iacute;&agrave;&oslash;
&eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve; &iuml;&eth;&aring;&auml;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve; &ntilde;&icirc;&aacute;&icirc;&eacute; &ograve;&egrave;&iuml;&egrave;&divide;&iacute;&oacute;&thorn; &ograve;&aring;&icirc;&eth;&aring;&igrave;&oacute; &icirc; &otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ccedil;&agrave;&ouml;&egrave;&egrave;, &ntilde;&acirc;&yuml;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ugrave;&oacute;&thorn; &ntilde;&egrave;&iacute;&ograve;&agrave;&ecirc;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&eacute;
&acirc;&egrave;&auml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &ntilde; &aring;&aring; &egrave;&iacute;&acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&icirc;&acirc; &ntilde;&iuml;&aring;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &acirc;&egrave;&auml;&agrave; (&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&igrave;&egrave; &iuml;&icirc; &ntilde;&oacute;&ograve;&egrave;
&yuml;&acirc;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave;). &Ograve;&agrave;&ecirc;&icirc;&atilde;&icirc; &eth;&icirc;&auml;&agrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc; &iuml;&icirc;&yuml;&acirc;&egrave;&euml;&egrave;&ntilde;&uuml; &acirc; 1950&aring;, &igrave;&icirc;&igrave;&aring;&iacute;&ograve; &ntilde;&icirc;&ccedil;&auml;&agrave;&iacute;&egrave;&yuml;
&ograve;&aring;&icirc;&eth;&egrave;&egrave; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute; &egrave; &oacute;&iacute;&egrave;&acirc;&aring;&eth;&ntilde;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &agrave;&euml;&atilde;&aring;&aacute;&eth;&ucirc; &acirc; &eth;&agrave;&aacute;&icirc;&ograve;&agrave;&otilde; &Euml;&icirc;&ntilde;&yuml; [12], &Ograve;&agrave;&eth;&ntilde;&ecirc;&icirc;&atilde;&icirc; [20], &Euml;&egrave;&iacute;&auml;&icirc;&iacute;&agrave; [14], &times;&aring;&iacute;&agrave;
[5], &Otilde;&icirc;&eth;&iacute;&agrave; [8], &Igrave;&agrave;&euml;&uuml;&ouml;&aring;&acirc;&agrave; [15] &egrave; &Ograve;&agrave;&eacute;&igrave;&agrave;&iacute;&icirc;&acirc;&agrave; [19]. &Acirc; &ecirc;&icirc;&iacute;&ouml;&aring; 1970&otilde; &acirc;&agrave;&iacute; &Aacute;&aring;&iacute;&ograve;&aring;&igrave; [2] &iuml;&aring;&eth;&aring;&iacute;&aring;&ntilde; &igrave;&iacute;&icirc;&atilde;&egrave;&aring; &egrave;&ccedil; &iacute;&egrave;&otilde;
&iacute;&agrave; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;. &Acirc; &ograve;&aring;&divide;&aring;&iacute;&egrave;&egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&egrave;&otilde; 20 &euml;&aring;&ograve; &acirc; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&aring; &aacute;&icirc;&euml;&uuml;&oslash;&icirc;&aring; &acirc;&iacute;&egrave;&igrave;&agrave;&iacute;&egrave;&aring; &oacute;&auml;&aring;&euml;&yuml;&aring;&ograve;&ntilde;&yuml;
&icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&yuml;&igrave; &yacute;&ograve;&egrave;&otilde; &ograve;&aring;&icirc;&eth;&aring;&igrave; &iacute;&agrave; &ntilde;&euml;&oacute;&divide;&agrave;&eacute; &ecirc;&icirc;&iacute;&aring;&divide;&iacute;&ucirc;&otilde; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute; [16], [1], [17]. &Aring;&ntilde;&ograve;&uuml; &oacute; &iacute;&egrave;&otilde; &egrave; &iuml;&icirc;&ograve;&aring;&iacute;&ouml;&egrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring;
&iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave;&aring; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&yuml; &acirc; &ecirc;&icirc;&igrave;&iuml;&uuml;&thorn;&ograve;&aring;&eth;&iacute;&ucirc;&otilde; &iacute;&agrave;&oacute;&ecirc;&agrave;&otilde;. &Iacute;&agrave;&iuml;&eth;&egrave;&igrave;&aring;&eth;, &acirc; &eth;&agrave;&aacute;&icirc;&ograve;&aring; [13] &icirc;&iacute;&egrave; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&thorn;&ograve;&ntilde;&yuml; &auml;&euml;&yuml;
&iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &egrave; &egrave;&ccedil;&oacute;&divide;&aring;&iacute;&egrave;&yuml; &ntilde;&euml;&icirc;&aelig;&iacute;&icirc;&ntilde;&ograve;&egrave; &agrave;&euml;&atilde;&icirc;&eth;&egrave;&ograve;&igrave;&icirc;&acirc;, &iuml;&eth;&icirc;&acirc;&aring;&eth;&yuml;&thorn;&ugrave;&egrave;&otilde;, &igrave;&icirc;&aelig;&aring;&ograve; &euml;&egrave; &icirc;&iacute;&ograve;&icirc;&euml;&icirc;&atilde;&egrave;&yuml;, &ccedil;&agrave;&auml;&agrave;&iacute;&iacute;&agrave;&yuml; &acirc; &aacute;&icirc;&euml;&aring;&aring;
&oslash;&egrave;&eth;&icirc;&ecirc;&icirc;&igrave; &yuml;&ccedil;&ucirc;&ecirc;&aring;, &aacute;&ucirc;&ograve;&uuml; &iuml;&aring;&eth;&aring;&iuml;&egrave;&ntilde;&agrave;&iacute;&agrave; &acirc; &aacute;&icirc;&euml;&aring;&aring; &oacute;&ccedil;&ecirc;&icirc;&igrave;.
2
&Ntilde;&icirc;&atilde;&euml;&agrave;&oslash;&aring;&iacute;&egrave;&aring; &icirc;&aacute; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&yuml;&otilde;
&Igrave;&ucirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&agrave;&atilde;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; &divide;&egrave;&ograve;&agrave;&ograve;&aring;&euml;&uuml; &ccedil;&iacute;&agrave;&ecirc;&icirc;&igrave; &ntilde; &aacute;&agrave;&ccedil;&icirc;&acirc;&ucirc;&igrave;&egrave; &iuml;&icirc;&iacute;&yuml;&ograve;&egrave;&yuml;&igrave;&egrave; &ograve;&aring;&icirc;&eth;&egrave;&egrave; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute;, &ograve;&agrave;&ecirc;&egrave;&igrave;&egrave; &ecirc;&agrave;&ecirc; &ocirc;&icirc;&eth;-
&igrave;&oacute;&euml;&agrave; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&ucirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;, &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&agrave;&ouml;&egrave;&yuml;, &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&ucirc;, &egrave;&ntilde;&ograve;&egrave;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
635
&acirc; &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&agrave;&ouml;&egrave;&egrave;, &egrave;&ntilde;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&aring; &iuml;&eth;&aring;&auml;&egrave;&ecirc;&agrave;&ograve;&icirc;&acirc; &egrave; &ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave;&igrave;&egrave; &icirc; &aring;&atilde;&icirc; &ecirc;&icirc;&eth;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave; &iuml;&icirc;&euml;&iacute;&icirc;&ograve;&aring;, &acirc; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&egrave; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &igrave;&ucirc; &ntilde;&euml;&aring;&auml;&oacute;&aring;&igrave; &ecirc;&iacute;&egrave;&atilde;&aring; [4]. &Acirc; &otilde;&icirc;&auml;&aring; &auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&agrave; &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &ograve;&agrave;&ecirc;&aelig;&aring; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&ograve;&uuml; &euml;&aring;&igrave;&igrave;&oacute; &icirc;
&ntilde;&acirc;&aring;&aelig;&egrave;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave;&otilde;, &ecirc;&icirc;&ograve;&icirc;&eth;&agrave;&yuml; &oacute;&ograve;&acirc;&aring;&eth;&aelig;&auml;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; &acirc; &egrave;&ntilde;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&egrave; &iuml;&eth;&aring;&auml;&egrave;&ecirc;&agrave;&ograve;&icirc;&acirc; &acirc;&ucirc;&acirc;&icirc;&auml;&egrave;&igrave;&agrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;
φ
&ntilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&icirc;&eacute;
c,
&ograve;&icirc; &ograve;&icirc;&atilde;&auml;&agrave; &acirc;&ucirc;&acirc;&icirc;&auml;&egrave;&igrave;&agrave; &egrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;, &iuml;&icirc;&euml;&oacute;&divide;&agrave;&thorn;&ugrave;&agrave;&yuml;&ntilde;&yuml; &egrave;&ccedil;
&iacute;&icirc;&acirc;&oacute;&thorn; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&oacute;&thorn;, &iacute;&aring; &acirc;&otilde;&icirc;&auml;&yuml;&ugrave;&oacute;&thorn; &acirc;
φ.
φ
&ccedil;&agrave;&igrave;&aring;&iacute;&icirc;&eacute; &acirc;&ntilde;&aring;&otilde; &acirc;&otilde;&icirc;&aelig;&auml;&aring;&iacute;&egrave;&eacute;
c
&iacute;&agrave;
&Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &igrave;&ucirc; &eth;&agrave;&aacute;&icirc;&ograve;&agrave;&aring;&igrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc; &ntilde; &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&ucirc;&igrave;&egrave; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&yuml;&igrave;&egrave;, &auml;&euml;&yuml;
xRy .
(W, R), &atilde;&auml;&aring; R &aacute;&egrave;&iacute;&agrave;&eth;&iacute;&icirc;&aring; &icirc;&ograve;&iacute;&icirc;&oslash;&aring;&iacute;&egrave;&aring; &iacute;&agrave; W . &Iuml;&oacute;&ntilde;&ograve;&uuml;
G
G
F
F
G
&auml;&agrave;&iacute;&ucirc; &auml;&acirc;&aring; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&aring; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc; G = (W , R ) &egrave; F = (W , R ). &Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : W
→ WF
1
&aacute;&oacute;&auml;&aring;&igrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&ograve;&uuml; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&aring;&eacute; , &aring;&ntilde;&euml;&egrave; &acirc;&ucirc;&iuml;&icirc;&euml;&iacute;&aring;&iacute;&ucirc; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;:
(1) f &ntilde;&thorn;&eth;&uacute;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;;
G
G
F
(2) &aring;&ntilde;&euml;&egrave; x, y ∈ W
&egrave; xR y , &ograve;&icirc; f (x)R f (y) (&igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;&ntilde;&ograve;&uuml;);
0 0
F
0 F 0
G
0
(3) &aring;&ntilde;&euml;&egrave; x , y ∈ W
&egrave; x R y , &ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; x, y ∈ W , &divide;&ograve;&icirc; f (x) = x , f (y) = y &egrave;
G
xR y (&ntilde;&euml;&agrave;&aacute;&ucirc;&eacute; &acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml; &iuml;&icirc;&auml;&uacute;&aring;&igrave;&agrave;).
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &yuml;&ccedil;&ucirc;&ecirc; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave; L &acirc; &ntilde;&egrave;&atilde;&iacute;&agrave;&ograve;&oacute;&eth;&aring; &ntilde; &icirc;&auml;&iacute;&egrave;&igrave; &auml;&acirc;&oacute;&igrave;&aring;&ntilde;&ograve;&iacute;&ucirc;&igrave; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&igrave; &ntilde;&egrave;&igrave;&acirc;&icirc;&euml;&icirc;&igrave;
R &egrave; &eth;&agrave;&acirc;&aring;&iacute;&ntilde;&ograve;&acirc;&icirc;&igrave;. &Igrave;&ucirc; &atilde;&icirc;&acirc;&icirc;&eth;&egrave;&igrave;, &divide;&ograve;&icirc; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &yacute;&ograve;&icirc;&atilde;&icirc; &yuml;&ccedil;&ucirc;&ecirc;&agrave; φ &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde;
&ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &iuml;&agrave;&eth;&ucirc; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&otilde; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth; G = (W G , RG ) &egrave; F = (W F , RF ), &ograve;&agrave;&ecirc;&egrave;&otilde;,
&divide;&ograve;&icirc; &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&aring;&ograve; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; f : G → F , &egrave;&ccedil; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; G |= φ, &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; F |= φ.
&ccedil;&agrave;&iuml;&egrave;&ntilde;&egrave; &agrave;&ograve;&icirc;&igrave;&icirc;&acirc; &igrave;&ucirc; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&igrave; &egrave;&iacute;&ocirc;&egrave;&ecirc;&ntilde;&iacute;&oacute;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute; &acirc;&egrave;&auml;&agrave;
&Aacute;&oacute;&auml;&aring;&igrave; &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&ograve;&uuml; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&icirc;&eacute; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&icirc;&eacute; &iuml;&agrave;&eth;&oacute;
&Iacute;&aring;&ntilde;&ecirc;&icirc;&euml;&uuml;&ecirc;&icirc; &aacute;&icirc;&euml;&aring;&aring; &atilde;&eth;&icirc;&igrave;&icirc;&ccedil;&auml;&ecirc;&icirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&aring;&aring; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &auml;&euml;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &ntilde; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave;&igrave;&egrave;: &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;
φ(x1 , . . . , xn ) &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;, &aring;&ntilde;&euml;&egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&otilde; &ntilde;&ograve;&eth;&oacute;&ecirc;G
F
F
F
F
G
G
G
G
→ W F &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml;
&ograve;&oacute;&eth; G = (W , R , x1 , . . . , xn ) &egrave; F = (W , R , x1 , . . . , xn ), &aring;&ntilde;&euml;&egrave; f : W
F
F
G
F
G
G
&ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;, &egrave; &aring;&ntilde;&euml;&egrave; f (xi ) = xi &auml;&euml;&yuml; 1 ≤ i ≤ n, &ograve;&icirc; G |= φ(x1 , . . . , xn ) &acirc;&euml;&aring;&divide;&aring;&ograve; F |= φ(x1 , . . . , xn ).
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&eacute; &ecirc;&euml;&agrave;&ntilde;&ntilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &yuml;&ccedil;&ucirc;&ecirc;&agrave; L, &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&yuml;&aring;&igrave;&ucirc;&eacute; &iuml;&icirc; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave;:
ψ ::= xRy | x = y | ψ1 ∨ ψ2 | ψ1 ∧ ψ2 | ∀vψ | ∃vψ | ∀x∀y(xRy → ψ).
&Aacute;&oacute;&auml;&aring;&igrave; &yacute;&ograve;&icirc;&ograve; &ecirc;&euml;&agrave;&ntilde;&ntilde; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&ograve;&uuml;
FP,
&agrave; &ecirc;&euml;&agrave;&ntilde;&ntilde;, &iuml;&icirc;&euml;&oacute;&divide;&agrave;&thorn;&ugrave;&egrave;&eacute;&ntilde;&yuml; &egrave;&ccedil;
&acirc;&otilde;&icirc;&auml;&yuml;&ugrave;&egrave;&aring; &acirc; &iacute;&aring;&atilde;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;, &aacute;&oacute;&auml;&aring;&igrave; &icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&ograve;&uuml; &divide;&aring;&eth;&aring;&ccedil;
FP
&iacute;&agrave;&acirc;&aring;&oslash;&egrave;&acirc;&agrave;&iacute;&egrave;&aring;&igrave; &icirc;&ograve;&eth;&egrave;&ouml;&agrave;&iacute;&egrave;&eacute; &iacute;&agrave; &acirc;&ntilde;&aring;
&not;F P .
&Ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &ntilde; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &ograve;&icirc; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; &acirc; &yacute;&ograve;&icirc;&igrave; &ecirc;&euml;&agrave;&ntilde;&ntilde;&aring; &auml;&icirc;&iuml;&oacute;&ntilde;&ograve;&egrave;&igrave;&ucirc; &ntilde;&euml;&icirc;&aelig;&iacute;&ucirc;&aring; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&eth;&ucirc; &acirc;&egrave;&auml;&agrave;
∀x1 ∀y1 . . . ∀xn ∀yn (x1 Ry1 ∧ . . . ∧ xn Ryn → ψ),
&ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;, &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&agrave;&yuml; (1), &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&ntilde;&yuml; &egrave;&ccedil; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
ψ
&iuml;&icirc;&ntilde;&euml;&aring;
(1)
n
&iuml;&eth;&egrave;&igrave;&aring;&iacute;&aring;&iacute;&egrave;&eacute; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc;
&iuml;&eth;&agrave;&acirc;&egrave;&euml;&agrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&eth;&oacute;&egrave;&eth;&icirc;&acirc;&agrave;&iacute;&egrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;.
&Icirc;&aacute;&icirc;&ccedil;&iacute;&agrave;&divide;&aring;&iacute;&egrave;&aring; P C
3
`
&icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve; &acirc;&ucirc;&acirc;&icirc;&auml;&egrave;&igrave;&icirc;&ntilde;&ograve;&uuml; &acirc; &ecirc;&euml;&agrave;&ntilde;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&igrave; &egrave;&ntilde;&divide;&egrave;&ntilde;&euml;&aring;&iacute;&egrave;&egrave; &iuml;&eth;&aring;&auml;&egrave;&ecirc;&agrave;&ograve;&icirc;&acirc;.
&Icirc;&ntilde;&iacute;&icirc;&acirc;&iacute;&icirc;&eacute; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve;
&Ograve;&Aring;&Icirc;&ETH;&Aring;&Igrave;&Agrave; 1. &Ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;
φ &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&aring;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde; &ograve;&icirc;&atilde;&auml;&agrave; &egrave; &ograve;&icirc;&euml;&uuml;&ecirc;&icirc;
FP.
&ograve;&icirc;&atilde;&auml;&agrave;, &ecirc;&icirc;&atilde;&auml;&agrave; &icirc;&iacute;&agrave; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&agrave; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&aring; &egrave;&ccedil; &ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave;
A = (W A , RA ) &egrave; B = (W B , RB ). &Ccedil;&agrave;&iuml;&egrave;&ntilde;&uuml; A FP B &icirc;&ccedil;&iacute;&agrave;&divide;&agrave;&aring;&ograve;, &divide;&ograve;&icirc; &auml;&euml;&yuml; &euml;&thorn;&aacute;&icirc;&eacute; &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&icirc;&eacute; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; φ &ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave; F P , &egrave;&ccedil; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; A |= φ, &ntilde;&euml;&aring;&auml;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; B |= φ. &Euml;&aring;&atilde;&ecirc;&icirc;
&acirc;&egrave;&auml;&aring;&ograve;&uuml;, &divide;&ograve;o &aring;&ntilde;&euml;&egrave; A FP B , φ ∈ &not;F P &egrave; B |= φ, &ograve;&icirc; A |= φ.
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &auml;&acirc;&aring; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&aring; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc;
B
B
B
A = (W A , RA , (cA
λ : λ ∈ Λ)), B = (W , R , (cλ : λ ∈ Λ)) &egrave; A FP B . &Ograve;&icirc;&atilde;&auml;&agrave;
0
0
A
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; B &oslash;&ecirc;&agrave;&euml;&ucirc; B &egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f : A → B , &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&yuml;&ugrave;&aring;&aring; cλ
A
0
A
(A, (a : a ∈ W )) FP (B , (f a : a ∈ W )).
&Euml;&Aring;&Igrave;&Igrave;&Agrave; 2. &Iuml;&oacute;&ntilde;&ograve;&uuml;
&iacute;&agrave;&eacute;&auml;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&icirc;&aring;
&acirc;
cB
λ,
&ograve;&agrave;&ecirc;&icirc;&aring; &divide;&ograve;&icirc;
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &yuml;&ccedil;&ucirc;&ecirc; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ugrave;&egrave;&eacute; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;, &ecirc;&eth;&icirc;&igrave;&aring; &egrave;&ntilde;&otilde;&icirc;&auml;&iacute;&ucirc;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;
cλ ,
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc;
a ∈ WA
&egrave;
b ∈ WB .
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ograve;&aring;&icirc;&eth;&egrave;&thorn; &yacute;&ograve;&icirc;&atilde;&icirc; &yuml;&ccedil;&ucirc;&ecirc;&agrave;
T = {ψ(a1 , . . . , an ) | ψ ∈ F P ; A |= ψ(a1 , . . . , an )} ∪ {φ(b1 , . . . , bm ) | B |= φ(b1 , . . . , bm )}.
&Iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc;
T
&iacute;&aring;&iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&acirc;&agrave;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc;
&ecirc;&icirc;&iacute;&uacute;&thorn;&iacute;&ecirc;&ouml;&egrave;&egrave;, &yacute;&ograve;&icirc; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
1
ψ ∈ F P , φ ∈ L,
&Ograve;&agrave;&ecirc;&egrave;&aring; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; &ograve;&agrave;&ecirc;&aelig;&aring; &egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&ucirc; &ecirc;&agrave;&ecirc; &ntilde;&euml;&agrave;&aacute;&ucirc;&aring; &atilde;&icirc;&igrave;&icirc;&igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&ucirc;.
636
FP
&ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve; &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;
&ograve;&agrave;&ecirc;&egrave;&aring;, &divide;&ograve;&icirc;
A |= ψ(a1 , . . . , an ),
B |= φ(b1 , . . . , bm ), &iacute;&icirc; P C ` ψ(a1 , . . . , an ) → &not;φ(b1 , . . . , bm ). &Acirc; &ntilde;&egrave;&euml;&oacute; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&agrave; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&yuml;2 , P C `
∀ā(ψ(a1 , . . . , an ) → &not;φ(b1 , . . . , bm )), &divide;&ograve;&icirc; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc; P C ` (∃āψ(a1 , . . . , an )) → &not;φ(b1 , . . . , bm ). &Ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; ∃āψ(ā) ∈ F P , A |= ∃āψ(ā). &Icirc;&ograve;&ntilde;&thorn;&auml;&agrave;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; A FP B , B |= ∃āψ(ā). &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;,
B |= (∃āψ(a1 , . . . , an )) → &not;φ(b1 , . . . , bm ), &icirc;&ograve;&ecirc;&oacute;&auml;&agrave; B |= &not;φ(b1 , . . . , bm ), &divide;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc;
B |= φ(b1 , . . . , bm ).
0
0
&Iuml;&oacute;&ntilde;&ograve;&uuml; B &igrave;&icirc;&auml;&aring;&euml;&uuml; &auml;&euml;&yuml; &ograve;&aring;&icirc;&eth;&egrave;&egrave; T . &Ograve;&icirc;&atilde;&auml;&agrave; &iuml;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&thorn; T , B &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; B .
&Icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f &ccedil;&agrave;&auml;&agrave;&aring;&ograve;&ntilde;&yuml; &ograve;&aring;&igrave; &ntilde;&acirc;&icirc;&eacute;&ntilde;&ograve;&acirc;&icirc;&igrave;, &divide;&ograve;&icirc; &icirc;&iacute;&icirc; &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&egrave;&ograve; &ecirc;&agrave;&aelig;&auml;&ucirc;&eacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve; a ∈ WA &acirc; &egrave;&iacute;&ograve;&aring;&eth;&iuml;&eth;&aring;&ograve;&agrave;0
A
0
A
&ouml;&egrave;&thorn; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&ucirc; a &acirc; &igrave;&icirc;&auml;&aring;&euml;&egrave; B , &egrave; &yacute;&ograve;&icirc; &atilde;&agrave;&eth;&agrave;&iacute;&ograve;&egrave;&eth;&oacute;&aring;&ograve;, &divide;&ograve;&icirc; (A, (a : a ∈ W )) FP (B , (f a : a ∈ W )), &egrave; &divide;&ograve;&icirc;
A
B
f &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&egrave;&ograve; cλ &acirc; cλ . 2
B
B
B
A = (W A , RA , (cA
λ : λ ∈ Λ)), B = (W , R , (cλ : λ ∈ Λ)) &egrave; A FP B . &Ograve;&icirc;&atilde;&auml;&agrave;
0
0
B
&iacute;&agrave;&eacute;&auml;&aring;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&icirc;&aring; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; A &oslash;&ecirc;&agrave;&euml;&ucirc; A &egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g : B → A , &iuml;&aring;&eth;&aring;&acirc;&icirc;&auml;&yuml;&ugrave;&aring;&aring; cλ
A
0
B
B
B
&acirc; cλ , &ograve;&agrave;&ecirc;&icirc;&aring; &divide;&ograve;&icirc; (A , (gb : b ∈ W )) FP (B, (b : b ∈ W )), &egrave;, &ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; b1 , b2 ∈ W , &aring;&ntilde;&euml;&egrave;
B |= b1 Rb2 , &ograve;&icirc; A0 |= g(b1 )Rg(b2 ).
&Euml;&Aring;&Igrave;&Igrave;&Agrave; 3. &Iuml;&oacute;&ntilde;&ograve;&uuml;
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &yuml;&ccedil;&ucirc;&ecirc; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;, &ntilde;&icirc;&auml;&aring;&eth;&aelig;&agrave;&ugrave;&egrave;&eacute; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;, &ecirc;&eth;&icirc;&igrave;&aring;
&yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&ucirc;
a ∈ WA
b ∈ WB .
&egrave;
cλ ,
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ograve;&aring;&icirc;&eth;&egrave;&thorn;
T = {φ(a1 , . . . , an ) | A |= φ(a1 , . . . , an )}∪
∪{ψ(b1 , . . . , bm ) | ψ ∈ &not;F P ; B |= ψ(b1 , . . . , bm )}∪
∪{b1 Rb2 | b1 , b2 ∈ WB ; b1 Rb2 }.
&not;F P &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve; &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;
ψ ∈ &not;F P , φ ∈ L, &egrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&ucirc; b1 , . . . , b2k ∈ W B ,
&ograve;&agrave;&ecirc;&egrave;&aring;, &divide;&ograve;&icirc; P C ` ψ(b1 , . . . , bm ) ∧ (b1 Rb2 ) ∧ . . . ∧ (b2k−1 Rb2k ) → &not;φ(a1 , . . . , an ). &Iuml;&icirc; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&oacute; &icirc;&aacute;&icirc;&aacute;&ugrave;&aring;&iacute;&egrave;&yuml;,
P C ` ∀b̄(ψ(b1 , . . . , bm ) ∧ (b1 Rb2 ) ∧ . . . ∧ (b2k−1 Rb2k ) → &not;φ(a1 , . . . , an )), &icirc;&ograve;&ecirc;&oacute;&auml;&agrave; P C ` ∃b̄(ψ(b1 , . . . , bm ) ∧
(b1 Rb2 ) ∧ . . . ∧ (b2k−1 Rb2k )) → &not;φ(a1 , . . . , an )). &Ccedil;&agrave;&igrave;&aring;&ograve;&egrave;&igrave;, &divide;&ograve;&icirc; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; ζ = ∃b̄(ψ(b1 , . . . , bm ) ∧ (b1 Rb2 ) ∧
. . . ∧ (b2k−1 Rb2k )) &euml;&aring;&aelig;&egrave;&ograve; &acirc; &ecirc;&euml;&agrave;&ntilde;&ntilde;&aring; &not;F P , &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &aring;&aring; &icirc;&ograve;&eth;&egrave;&ouml;&agrave;&iacute;&egrave;&aring; &egrave;&igrave;&aring;&aring;&ograve; &acirc;&egrave;&auml; (1). &Iacute;&icirc; &igrave;&ucirc; &ccedil;&iacute;&agrave;&aring;&igrave;, &divide;&ograve;&icirc; B |= ζ .
&Icirc;&ograve;&ntilde;&thorn;&auml;&agrave;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; A FP B , &ograve;&icirc; A |= ζ . &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, A |= ζ → &not;φ(a1 , . . . , an ), &icirc;&ograve;&ecirc;&oacute;&auml;&agrave; A |= &not;φ(a1 , . . . , an ),
&divide;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc; A |= φ(a1 , . . . , an ).
0
B
&Acirc;&icirc;&ccedil;&igrave;&uuml;&igrave;&aring;&igrave; &acirc; &ecirc;&agrave;&divide;&aring;&ntilde;&ograve;&acirc;&aring; A &igrave;&icirc;&auml;&aring;&euml;&uuml; &auml;&euml;&yuml; T , &egrave; &icirc;&aacute;&uacute;&yuml;&acirc;&egrave;&igrave; &iacute;&agrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&otilde; b ∈ W
g(b) &eth;&agrave;&acirc;&iacute;&ucirc;&igrave; &egrave;&iacute;&ograve;&aring;&eth;0
0
&iuml;&eth;&aring;&ograve;&agrave;&ouml;&egrave;&egrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&ucirc; b &acirc; &igrave;&icirc;&auml;&aring;&euml;&egrave; A . &Ograve;&icirc;&atilde;&auml;&agrave; &iuml;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&thorn; A &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; A, &egrave;,
0
B
B
B
B
&ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, (A , (gb : b ∈ W )) FP (B, (b : b ∈ W )), &egrave; &auml;&euml;&yuml; &euml;&thorn;&aacute;&ucirc;&otilde; b1 , b2 ∈ W , b1 R b2 , &acirc;&aring;&eth;&iacute;&icirc;, &divide;&ograve;&icirc;
0
A |= g(b1 )Rg(b2 ). 2
&Iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc;
T
&iacute;&aring;&iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&acirc;&agrave;. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;. &Ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc;
&ecirc;&icirc;&iacute;&uacute;&thorn;&iacute;&ecirc;&ouml;&egrave;&egrave;, &yacute;&ograve;&icirc; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
A = (W A , RA ), B = (W B , RB ), &egrave; A FP B . &Ograve;&icirc;&atilde;&auml;&agrave;
B ≺ B ∗ &egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; f : A∗ → B ∗ .
&Euml;&Aring;&Igrave;&Igrave;&Agrave; 4. &Iuml;&oacute;&ntilde;&ograve;&uuml;
&eth;&aring;&iacute;&egrave;&yuml;
A≺A
∗
,
&iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;-
A0 = A, B0 = B , &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&igrave; &Euml;&aring;&igrave;&igrave;&oacute; 2, &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;B1 &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc; B0 &egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f0 : A0 → B1 , &ograve;&agrave;&ecirc;&icirc;&aring;, &divide;&ograve;&icirc; (A0 , (a : a ∈ W A )) FP (B1 , (f0 (a) :
a ∈ W A )). . &Ograve;&aring;&iuml;&aring;&eth;&uuml; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&egrave;&igrave; &Euml;&aring;&igrave;&igrave;&oacute; 3 &ecirc; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave;&igrave; (A, (a : a ∈ W A )) &egrave; (B1 , (f a : a ∈ W A )),
&egrave; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&icirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&aring; A1 &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&ucirc; A0 &egrave; &icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; g1 : B1 → A1 , &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc;.. &Iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;
&eth;&aring;&iacute;&egrave;&aring;
&eth;&yuml;&thorn;&ugrave;&aring;&aring; &ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&aring;&iacute;&egrave;&thorn; &Euml;&aring;&igrave;&igrave;&ucirc; 3. &Iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&aring;&igrave; &oslash;&agrave;&atilde;&aring; &igrave;&ucirc; &aacute;&oacute;&auml;&aring;&igrave; &iuml;&eth;&egrave;&igrave;&aring;&iacute;&yuml;&ograve;&uuml; &Euml;&aring;&igrave;&igrave;&oacute; 2 &ecirc; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave;&igrave;
(A, (a : a ∈ W A ), (g1 (b) : b ∈ W B1 )) &egrave; (B, (f0 (a) : a ∈ W A ), (b : b ∈ W B1 )), &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&oacute; B2 &egrave;
A
B
A
0
&icirc;&ograve;&icirc;&aacute;&eth;&agrave;&aelig;&aring;&iacute;&egrave;&aring; f1 : A1 → B2 &ograve;&agrave;&ecirc;&egrave;&aring; &divide;&ograve;&icirc; (A, (a : a ∈ W ), (g1 (b) : b ∈ W 1 ), (a : a ∈ W 1 )) FP (B , (f0 (a) :
A
B1
A1
a ∈ W ), (b : b ∈ W ), (f1 (a) : a ∈ W )). &Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&oacute;&yuml; &ograve;&agrave;&ecirc; &auml;&icirc; &aacute;&aring;&ntilde;&ecirc;&icirc;&iacute;&aring;&divide;&iacute;&icirc;&ntilde;&ograve;&egrave;, &iuml;&eth;&egrave;&igrave;&aring;&iacute;&yuml;&yuml; &iuml;&icirc; &icirc;&divide;&aring;&eth;&aring;&auml;&egrave;
&Euml;&aring;&igrave;&igrave;&ucirc; 1 &egrave; 2, &igrave;&ucirc; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&igrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&oacute;&thorn; &aacute;&agrave;&oslash;&iacute;&thorn; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&eacute;
A0 ≺ A1 ≺ A2 ≺ . . .
f0
g1
f1
g2
B0 ≺ B1 ≺ B2 ≺ . . .
&egrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&egrave; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute;
2
fn
&egrave;
gn .
&Ccedil;&auml;&aring;&ntilde;&uuml; &iacute;&aring;&yuml;&acirc;&iacute;&icirc; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&oacute;&aring;&ograve;&ntilde;&yuml; &euml;&aring;&igrave;&igrave;&agrave; &icirc; &ntilde;&acirc;&aring;&aelig;&egrave;&otilde; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&agrave;&otilde;. &Ntilde;&ograve;&eth;&icirc;&atilde;&icirc; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;, &igrave;&ucirc; &ccedil;&agrave;&igrave;&aring;&iacute;&yuml;&aring;&igrave; &ecirc;&icirc;&iacute;&ntilde;&ograve;&agrave;&iacute;&ograve;&ucirc; ai &iacute;&agrave;
&iacute;&icirc;&acirc;&ucirc;&aring; &iuml;&aring;&eth;&aring;&igrave;&aring;&iacute;&iacute;&ucirc;&aring; xi , &egrave; &oacute;&aelig;&aring; &iacute;&agrave; &iacute;&egrave;&otilde; &iacute;&agrave;&acirc;&aring;&oslash;&egrave;&acirc;&agrave;&aring;&igrave; &ecirc;&acirc;&agrave;&iacute;&ograve;&icirc;&eth; &acirc;&ntilde;&aring;&icirc;&aacute;&ugrave;&iacute;&icirc;&ntilde;&ograve;&egrave;.
637
&Aacute;&agrave;&oslash;&iacute;&yuml; &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&aring;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;&igrave;
(A0 , (a : a ∈ W A )) FP (B0 , (f0 (a) : a ∈ W A )),
(A1 , (a : a ∈ W A ), (g1 (b) : b ∈ W B1 )) FP (B1 , (f0 (a) : a ∈ W A ), (b : b ∈ W B1 )),
(A, (a : a ∈ W A ), (g1 (b) : b ∈ W B1 ), (a : a ∈ W A1 )) FP
FP (B 0 , (f0 (a) : a ∈ W A ), (b : b ∈ W B1 ), (f1 (a) : a ∈ W A1 )),
...
fn &oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&yuml;&aring;&ograve; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&yuml;&igrave; fn ⊆ fn+1 .
Aω = ∪{An | n ≥ 0}, Bω = ∪{Bn | n ≥ 0}, fω = ∪{fn | n ≥ 0}. &Ograve;&icirc;&atilde;&auml;&agrave; fω &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml;
&ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; &egrave;&ccedil; Aω &acirc; Bω . &Auml;&aring;&eacute;&ntilde;&ograve;&acirc;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;, fω &ntilde;&thorn;&eth;&uacute;&aring;&ecirc;&ograve;&egrave;&acirc;&iacute;&icirc;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &aring;&ntilde;&euml;&egrave; b ∈ Bn , &ograve;&icirc; gn (b) ∈ An ,
&egrave; fn+1 (gn (b)) = b; fω &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&icirc;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; &acirc;&ntilde;&aring; fn &igrave;&icirc;&iacute;&icirc;&ograve;&icirc;&iacute;&iacute;&ucirc;, &egrave; &aring;&ntilde;&euml;&egrave; b1 , b2 ∈ Bn , &ograve;&icirc; &auml;&euml;&yuml; i ∈ {1, 2}
fn+1 gn (bi ) = bi , &egrave; Aω |= gn (b1 )Rgn (b2 ). &Iuml;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &icirc;&aacute; &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&otilde; &ouml;&aring;&iuml;&yuml;&otilde; (&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; 3.1.9 &acirc; [4]), Aω
&egrave; Bω &yacute;&euml;&aring;&igrave;&aring;&iacute;&ograve;&agrave;&eth;&iacute;&ucirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&yuml; A &egrave; B . 2
&Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&ntilde;&ograve;&uuml; &ocirc;&oacute;&iacute;&ecirc;&ouml;&egrave;&eacute;
&Iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;
&Auml;&icirc;&ecirc;&agrave;&ccedil;&agrave;&ograve;&aring;&euml;&uuml;&ntilde;&ograve;&acirc;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&ucirc;
&ecirc;&euml;&agrave;&ntilde;&ntilde;&agrave;
FP
1.
&Egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&aring;&eacute; &iuml;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&thorn; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &auml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; &acirc;&ntilde;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;
(&acirc; &ograve;&icirc;&igrave; &divide;&egrave;&ntilde;&euml;&aring;, &egrave; &ntilde; &iuml;&agrave;&eth;&agrave;&igrave;&aring;&ograve;&eth;&agrave;&igrave;&egrave;) &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ograve;&ntilde;&yuml; &iuml;&eth;&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;. &Acirc;&ntilde;&aring; &iuml;&eth;&agrave;-
&acirc;&egrave;&euml;&agrave; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;, &ecirc;&eth;&icirc;&igrave;&aring; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc;, &eth;&agrave;&ccedil;&aacute;&egrave;&eth;&agrave;&thorn;&ograve;&ntilde;&yuml; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&icirc; &ograve;&aring;&icirc;&eth;&aring;&igrave;&aring; &icirc; &iuml;&icirc;&ccedil;&egrave;&ograve;&egrave;&acirc;&iacute;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;&otilde; (&Ograve;&aring;&icirc;&eth;&aring;&igrave;&agrave; 3.2.4 &egrave;&ccedil; [4]). &Auml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave; &ecirc;&icirc;&eth;&eth;&aring;&ecirc;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&atilde;&icirc; &iuml;&eth;&agrave;&acirc;&egrave;&euml;&agrave;. &ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&oacute;
ψ = ∀x∀y(xRy → ψ 0 (x, y, v1 , . . . , vn )), &oslash;&ecirc;&agrave;&euml;&ucirc; G = (W G , RG , v1G , . . . , vnG ) &egrave; F = (W F , RF , v1F , . . . , vnF ),
G
&egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&oacute;&thorn; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&thorn; f : W
→ W F , &ograve;&agrave;&ecirc;&oacute;&thorn;, &divide;&ograve;&icirc; f (viG ) = viF &auml;&euml;&yuml; 1 ≤ i ≤ n. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;F
F
F
G
G
&egrave;
&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; G |= ψ(v1 , . . . , vn ), &iacute;&icirc; F 6|= φ(v1 , . . . , vn ). &Iuml;&icirc;&ntilde;&euml;&aring;&auml;&iacute;&aring;&aring; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &ograve;&icirc;&divide;&ecirc;&egrave; x
F
F F F
0 F
F
F
F
y , &ograve;&agrave;&ecirc;&egrave;&aring;, &divide;&ograve;&icirc; x R y &egrave; F 6|= ψ (x , y , v1 , . . . , vn ). &Ograve;&icirc;&atilde;&auml;&agrave;, &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; f &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;G G
&ouml;&egrave;&yuml;, &ograve;&icirc; &iacute;&agrave;&eacute;&auml;&oacute;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&egrave;&aring; &ograve;&icirc;&divide;&ecirc;&egrave; x , y
∈ W G , &divide;&ograve;&icirc; xG RG y G , &egrave; f (xG ) = xF , f (y G ) = y F . &Iacute;&icirc; &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc;
G
G
0 G G G
G |= ψ(v1 , . . . , vn ), &ograve;&icirc; G |= ψ (x , y , v1 , . . . , vnG ), &agrave; &icirc;&ograve;&ntilde;&thorn;&auml;&agrave;, &iuml;&icirc; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&thorn; &egrave;&iacute;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &auml;&euml;&yuml; ψ 0 ,
F |= ψ 0 (xF , y F , v1F , . . . , vnF ). &Iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&aring;.
&Ograve;&aring;&iuml;&aring;&eth;&uuml; &iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave;, &divide;&ograve;&icirc; φ &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;&yuml; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave;, &ntilde;&icirc;&otilde;&eth;&agrave;&iacute;&yuml;&thorn;&ugrave;&agrave;&yuml;&ntilde;&yuml; &iuml;&eth;&egrave; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml;&otilde;. &Iuml;&oacute;&ntilde;&ograve;&uuml;
F P C(φ) = {ψ | ψ ∈ F P, ψ &ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;, φ |= ψ}. &Iuml;&icirc;&ecirc;&agrave;&aelig;&aring;&igrave;, &divide;&ograve;&icirc; F P C(φ) |= φ. &Iuml;&eth;&aring;&auml;&iuml;&icirc;&euml;&icirc;&aelig;&egrave;&igrave; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&aring;.
B
B
&Yacute;&ograve;&icirc; &ccedil;&iacute;&agrave;&divide;&egrave;&ograve;, &divide;&ograve;&icirc; &iacute;&agrave;&oslash;&euml;&agrave;&ntilde;&uuml; &ograve;&agrave;&ecirc;&agrave;&yuml; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&agrave;&yuml; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth;&agrave; B = (W , R ), &divide;&ograve;&icirc; B |= F P C(φ) &egrave; B |= &not;φ.
&ETH;&agrave;&ntilde;&ntilde;&igrave;&icirc;&ograve;&eth;&egrave;&igrave; &ograve;&aring;&icirc;&eth;&egrave;&thorn; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;
T = {φ} ∪ {ψ | ψ ∈ &not;F P, ψ
T
&ccedil;&agrave;&igrave;&ecirc;&iacute;&oacute;&ograve;&agrave;, B
|= ψ}.
P C ` φ → &not;ψ1 ∨ . . . ∨ &not;ψn , &auml;&euml;&yuml; &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &ocirc;&icirc;&eth;&igrave;&oacute;&euml; ψi ,
B |= ψi , &icirc;&ograve;&ecirc;&oacute;&auml;&agrave; &not;ψ1 ∨ . . . ∨ &not;ψn ∈ F P C(φ), &divide;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve;
&iacute;&aring;&iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&acirc;&agrave;: &acirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&iacute;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring;,
&ograve;&agrave;&ecirc;&egrave;&otilde; &divide;&ograve;&icirc; &auml;&euml;&yuml; &ecirc;&agrave;&aelig;&auml;&icirc;&atilde;&icirc;
i ψi ∈ &not;F P
&egrave;
B |= F P C(φ).
A &igrave;&icirc;&auml;&aring;&euml;&uuml; &auml;&euml;&yuml; T . &Ograve;&icirc;&atilde;&auml;&agrave; &iuml;&icirc; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&aring;&iacute;&egrave;&thorn; A FP B . &Icirc;&ograve;&ntilde;&thorn;&auml;&agrave; &iuml;&icirc; &Euml;&aring;&igrave;&igrave;e 4 &ntilde;&oacute;&ugrave;&aring;&ntilde;&ograve;&acirc;&oacute;&thorn;&ograve; &yacute;&euml;&aring;&igrave;&aring;&iacute;∗
∗
∗
∗
&ograve;&agrave;&eth;&iacute;&ucirc;&aring; &eth;&agrave;&ntilde;&oslash;&egrave;&eth;&aring;&iacute;&egrave;&yuml; A ≺ A &egrave; B ≺ B &egrave; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&agrave;&yuml; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&yuml; f : A → B . &Agrave; &ograve;&agrave;&ecirc; &ecirc;&agrave;&ecirc; A |= φ,
∗
∗
&ograve;&icirc; A |= φ, B |= φ &egrave; B |= φ. &Yacute;&ograve;&icirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&eth;&aring;&divide;&egrave;&ograve; &ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc; B |= &not;φ. 2
&ograve;&icirc;&igrave;&oacute;, &divide;&ograve;&icirc;
&Iuml;&oacute;&ntilde;&ograve;&uuml;
4
&Ccedil;&agrave;&ecirc;&euml;&thorn;&divide;&aring;&iacute;&egrave;&aring;
&Acirc; &yacute;&ograve;&icirc;&eacute; &eth;&agrave;&aacute;&icirc;&ograve;&aring; &igrave;&ucirc; &icirc;&ograve;&ograve;&icirc;&euml;&ecirc;&iacute;&oacute;&euml;&egrave;&ntilde;&uuml; &icirc;&ograve; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&yuml; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&otilde; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&eacute; &egrave;&ccedil; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&icirc;&eacute; &euml;&icirc;&atilde;&egrave;&ecirc;&egrave;,
&iuml;&icirc;&euml;&oacute;&divide;&egrave;&euml;&egrave; &ecirc;&euml;&agrave;&ntilde;&ntilde; &igrave;&icirc;&eth;&ocirc;&egrave;&ccedil;&igrave;&icirc;&acirc; &eth;&aring;&euml;&yuml;&ouml;&egrave;&icirc;&iacute;&iacute;&ucirc;&otilde; &ntilde;&ograve;&eth;&oacute;&ecirc;&ograve;&oacute;&eth; &egrave; &icirc;&iuml;&egrave;&ntilde;&agrave;&euml;&egrave; &ntilde;&egrave;&iacute;&ograve;&agrave;&ecirc;&ntilde;&egrave;&divide;&aring;&ntilde;&ecirc;&egrave; &ntilde; &ograve;&icirc;&divide;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; &auml;&icirc; &yacute;&ecirc;&acirc;&egrave;&acirc;&agrave;&euml;&aring;&iacute;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave;&iacute;&acirc;&agrave;&eth;&egrave;&agrave;&iacute;&ograve;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &iuml;&aring;&eth;&acirc;&icirc;&atilde;&icirc; &iuml;&icirc;&eth;&yuml;&auml;&ecirc;&agrave;. &Igrave;&ucirc; &iacute;&agrave;&auml;&aring;&aring;&igrave;&ntilde;&yuml;, &divide;&ograve;&icirc; &yacute;&ograve;&icirc; &iacute;&agrave;&aacute;&euml;&thorn;&auml;&aring;&iacute;&egrave;&aring; &iuml;&icirc;&igrave;&icirc;&aelig;&aring;&ograve;
&icirc;&iuml;&egrave;&ntilde;&agrave;&ograve;&uuml; &acirc;&ntilde;&aring; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc;, &auml;&icirc;&iuml;&oacute;&ntilde;&ecirc;&agrave;&thorn;&ugrave;&egrave;&aring; &igrave;&egrave;&iacute;&egrave;&igrave;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &ocirc;&egrave;&euml;&uuml;&ograve;&eth;&agrave;&ouml;&egrave;&egrave;, &egrave; &iuml;&icirc;&ntilde;&ograve;&eth;&icirc;&egrave;&ograve;&uuml; &iacute;&icirc;&acirc;&ucirc;&aring; &eth;&agrave;&ccedil;&eth;&aring;&oslash;&egrave;&igrave;&ucirc;&aring; &igrave;&icirc;&auml;&agrave;&euml;&uuml;&iacute;&ucirc;&aring; &euml;&icirc;&atilde;&egrave;&ecirc;&egrave;.
&Ntilde;&iuml;&egrave;&ntilde;&icirc;&ecirc; &euml;&egrave;&ograve;&aring;&eth;&agrave;&ograve;&oacute;&eth;&ucirc;
1. N. Alechina and Y. Gurevich. Syntax vs. semantics on nite structures. In Structures in logic and computer
science, pages 1433. Springer, 1997.
2. J. van Benthem. Modal correspondence theory. PhD thesis, University of Amsterdam, 1977.
3. N. Bezhanishvili and B. ten Cate. Transfer results for hybrid logic. Part I: The case without satisfaction
operators. J. of Logic and Computation, 16(2):177197, 2006.
638
4. C. Chang and J. Keisler. Model theory. Elsevier, 1990.
5. C. Chang. On unions of chains of models. Proceedings of the American Mathematical Society, 10(1):120127,
1959.
6. D. Gabbay, I. Shapirovsky, and V. Shehtman. Products of modal logics and tensor products of modal algebras.
Journal of Applied Logic, 2014. To appear.
7. I. Hodkinson. Hybrid formulas and elementarily generated modal logics. Notre Dame Journal of Formal
Logic, 47(4):443478, 2006.
8. A. Horn. On sentences which are true of direct unions of algebras. The Journal of Symbolic Logic, 16(01):14
21, 1951.
9. S. Kikot. A dichotomy for some elementarily generated modal logics. Studia Logica, to appear, 2015.
10. S. Kikot, I. Shapirovsky, and E. Zolin. Filtration safe operations on frames. In Advances in Modal Logic,
volume 10, pages 326330, Milton Keynes, UK, 2014. College Publication.
11. S. Kikot and E. Zolin. Modal denability of rst-order formulas with free variables and query answering.
Journal of Applied Logic, 11(2):190216, 2013.
12. J. Los. On the extending of models (i). Fundamenta Mathematicae, 42(1):3854, 1955.
13. C. Lutz, R. Piro, and F. Wolter. Description logic TBoxes: Model-theoretic characterizations and rewritability.
In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Articial Intelligence, Barcelona,
Catalonia, Spain, July 16-22, 2011, pages 983988, 2011.
14. R. Lyndon. Properties preserved under algebraic constructions. Bulletin of the American Mathematical
Society, 65(5):287299, 1959.
15. &Agrave;. &Igrave;&agrave;&euml;&uuml;&ouml;&aring;&acirc;. &Icirc; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&iacute;&ucirc;&otilde; &icirc;&iuml;&aring;&eth;&agrave;&ouml;&egrave;&yuml;&otilde; &egrave; &iuml;&eth;&aring;&auml;&egrave;&ecirc;&agrave;&ograve;&agrave;&otilde;. &Auml;&icirc;&ecirc;&euml;&agrave;&auml;&ucirc; &Agrave;&Iacute; &Ntilde;&Ntilde;&Ntilde;&ETH;, &ograve;&icirc;&igrave; 116, &ntilde;&ograve;&eth;. 2427, 1957.
16. E. Rosen. Modal logic over nite structures. Journal of Logic, Language and Information, 6(4):427439, 1997.
17. B. Rossman. Homomorphism preservation theorems. Journal of the ACM (JACM), 55(3):15, 2008.
18. K. Segerberg. Decidability of four modal logics. Theoria, 34:2125, 1968.
19. &Agrave;. &Ograve;&agrave;&eacute;&igrave;&agrave;&iacute;&icirc;&acirc;. &Otilde;&agrave;&eth;&agrave;&ecirc;&ograve;&aring;&eth;&egrave;&ntilde;&ograve;&egrave;&ecirc;&agrave; &agrave;&ecirc;&ntilde;&egrave;&icirc;&igrave;&agrave;&ograve;&egrave;&ccedil;&egrave;&eth;&oacute;&aring;&igrave;&ucirc;&otilde; &ecirc;&euml;&agrave;&ntilde;&ntilde;&icirc;&acirc; &igrave;&icirc;&auml;&aring;&euml;&aring;&eacute;. I. &Egrave;&ccedil;&acirc;. &Agrave;&Iacute; &Ntilde;&Ntilde;&Ntilde;&ETH;. &Ntilde;&aring;&eth;. &igrave;&agrave;&ograve;&aring;&igrave;.,
25(4):601620, 1961.
20. A. Tarski. Contributions to the theory of models, I, II. 1954.
639
```