Математическая логика и теория алгоритмов. Самохин А.В. - 216 стр.

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216 úÁÄÁÞÉ
3. òÁÓËÒÙÔØ ÓËÏÂËÉ.
4. ðÏ×ÔÏÒÑÀÝÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ×ÚÑÔØ ÐÏ ÏÄÎÏÍÕ ÒÁÚÕ.
5. ïÐÕÓÔÉÔØ ÔÏÖÄÅÓÔ×ÅÎÎÏ ÌÏÖÎÙÅ ÓÌÁÇÁÅÍÙÅ, ÔÏ ÅÓÔØ ÓÌÁÇÁÅÍÙÅ ×ÉÄÁ:
. . . · x
i
· x
i
· . . .
6. ðÏÐÏÌÎÉÔØ ÏÓÔÁ×ÛÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ÎÅÄÏÓÔÁÀÝÉÍÉ ÐÅÒÅÍÅÎÎÙÍÉ.
(ðÒÉÍÅÒ ÎÁ ÐÏÐÏÌÎÅÎÉÅ (ÐÅÒÅÍÅÎÎÙÅ x
1
, x
2
, x
3
):
. . . ∨ x
1
x
3
∨ . . . ≡ . . . x
1
(x
2
∨ x
2
)x
3
. . . ≡ . . . ∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ . . . )
7. ðÏ×ÔÏÒÑÀÝÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ×ÚÑÔØ ÐÏ ÏÄÎÏÍÕ ÒÁÚÕ.
ðÒÉÍÅÒ 8. îÁÊÔÉ óäîæ ÆÏÒÍÕÌÙ
(x
1
→ x
2
x
3
) → (x
1
∼ x
3
).
òÅÛÅÎÉÅ.
(x
1
→ x
2
x
3
)(x
1
∼ x
3
)
1
≡ x
1
→ x
2
x
3
∨ (x
1
x
3
∨ x
1
x
3
)
2
≡
≡ x
1
· (x
2
∨ x
3
) ∨ x
1
x
3
∨ x
1
x
3
3
≡
≡ x
1
x
2
∨ x
1
x
3
∨ x
1
x
3
∨ x
1
x
3
4
≡ x
1
x
2
∨ x
1
x
3
∨ x
1
x
3
6
≡
≡ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
7
≡
≡ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
.
óÏ×ÅÒÛÅÎÎÕÀ ËÏÎßÀÎËÔÉ×ÎÕÀ ÎÏÒÍÁÌØÎÕÀ ÆÏÒÍÕ (óëîæ) ÍÏÖÎÏ ÐÏ-
ÓÔÒÏÉÔØ ÐÏ ÓÌÅÄÕÀÝÅÊ ÓÈÅÍÅ:
f ≡ (f
∗
)
∗
≡ (óäîæ(f
∗
))
∗
ÐÒÉÎÃÉÐ
≡
Ä×ÏÊÓÔ×ÅÎÎÏÓÔÉ
óëîæ(f).
ðÒÉÍÅÒ 9. îÁÊÔÉ óëîæ ÆÏÒÍÕÌÙ
(x
1
→ x
2
x
3
) → (x
1
∼ x
3
).
òÅÛÅÎÉÅ.
(x
1
→ x
2
x
3
) → (x
1
∼ x
3
) ≡

(x
1
→ x
2
x
3
) ∨ (x
1
x
3
∨ x
1
x
3
)
∗
∗
≡
≡ (x
1
(x
2
∨ x
3
) ∨ x
1
x
3
∨ x
1
x
3
)
∗
)
∗
≡ ((x
1
∨ x
2
x
3
) · (x
1
∨ x
3
) · (x
1
∨ x
3
))
∗
≡
≡ ((x
1
∨ x
1
x
2
x
3
∨ x
1
x
3
∨ x
2
x
3
)(x
1
∨ x
3
))
∗
≡ (x
1
x
2
x
3
∨ x
1
x
3
)
∗
≡
≡ (x
1
x
2
x
3
∨ x
1
x
2
x
3
∨ x
1
x
2
x
3
)
∗
≡ (x
1
∨ x
2
∨ x
3
)(x
1
∨ x
2
∨ x
3
)(x
1
∨ x
2
∨ x
3
).
éÚ×ÅÓÔÎÏ, ÞÔÏ óäîæ É óëîæ ÏÐÒÅÄÅÌÅÎÙ ÆÏÒÍÕÌÏÊ ÏÄÎÏÚÎÁÞÎÏ É, ÚÎÁ-
ÞÉÔ, ÉÈ ÍÏÖÎÏ ÓÔÒÏÉÔØ ÐÏ ÔÁÂÌÉÃÅ ÉÓÔÉÎÎÏÓÔÉ ÆÏÒÍÕÌÙ.
óÈÅÍÁ ÐÏÓÔÒÏÅÎÉÑ óäîæ É óëîæ ÐÏ ÔÁÂÌÉÃÅ ÉÓÔÉÎÎÏÓÔÉ ÐÒÉ×ÅÄÅÎÁ ÎÉ-
ÖÅ ÄÌÑ ÆÏÒÍÕÌÙ (x
1
→ x
2
x
3
)(x
1
∼ x
3
).
216                                                                                 úÁÄÁÞÉ

  3. òÁÓËÒÙÔØ ÓËÏÂËÉ.
  4. ðÏ×ÔÏÒÑÀÝÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ×ÚÑÔØ ÐÏ ÏÄÎÏÍÕ ÒÁÚÕ.
  5. ïÐÕÓÔÉÔØ ÔÏÖÄÅÓÔ×ÅÎÎÏ ÌÏÖÎÙÅ ÓÌÁÇÁÅÍÙÅ, ÔÏ ÅÓÔØ ÓÌÁÇÁÅÍÙÅ ×ÉÄÁ:
     . . . · xi · xi · . . .
  6. ðÏÐÏÌÎÉÔØ ÏÓÔÁ×ÛÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ÎÅÄÏÓÔÁÀÝÉÍÉ ÐÅÒÅÍÅÎÎÙÍÉ.
     (ðÒÉÍÅÒ ÎÁ ÐÏÐÏÌÎÅÎÉÅ (ÐÅÒÅÍÅÎÎÙÅ x1, x2, x3):
     . . . ∨ x1x3 ∨ . . . ≡ . . . x1 (x2 ∨ x2)x3 . . . ≡ . . . ∨ x1x2x3 ∨ x1 x2 x3 ∨ . . . )
  7. ðÏ×ÔÏÒÑÀÝÉÅÓÑ ÓÌÁÇÁÅÍÙÅ ×ÚÑÔØ ÐÏ ÏÄÎÏÍÕ ÒÁÚÕ.
  ðÒÉÍÅÒ 8. îÁÊÔÉ óäîæ ÆÏÒÍÕÌÙ
                                (x1 → x2x3 ) → (x1 ∼ x3).
  òÅÛÅÎÉÅ.
                            1                                    2
 (x1 → x2x3 )(x1 ∼ x3) ≡ x1 → x2x3 ∨ (x1x3 ∨ x1 x3) ≡
                                                                 3
                            ≡ x1 · (x2 ∨ x3) ∨ x1 x3 ∨ x1 x3 ≡
                                                 4                          6
                 ≡ x1 x2 ∨ x1x3 ∨ x1x3 ∨ x1 x3 ≡ x1x2 ∨ x1x3 ∨ x1x3 ≡
                                                                                7
             ≡ x1x2 x3 ∨ x1x2x3 ∨ x1 x2x3 ∨ x1x2x3 ∨ x1x2 x3 ∨ x1 x2 x3 ≡
                                 ≡ x1x2 x3 ∨ x1x2 x3 ∨ x1x2x3 ∨ x1 x2x3 ∨ x1 x2 x3.
  óÏ×ÅÒÛÅÎÎÕÀ ËÏÎßÀÎËÔÉ×ÎÕÀ ÎÏÒÍÁÌØÎÕÀ ÆÏÒÍÕ (óëîæ) ÍÏÖÎÏ ÐÏ-
ÓÔÒÏÉÔØ ÐÏ ÓÌÅÄÕÀÝÅÊ ÓÈÅÍÅ:
                                                     ÐÒÉÎÃÉÐ
                  f ≡ (f ∗)∗ ≡ (óäîæ(f ∗))∗            ≡         óëîæ(f ).
                                                Ä×ÏÊÓÔ×ÅÎÎÏÓÔÉ

  ðÒÉÍÅÒ 9. îÁÊÔÉ óëîæ ÆÏÒÍÕÌÙ
                                (x1 → x2x3 ) → (x1 ∼ x3).
  òÅÛÅÎÉÅ.
                                                                      ∗ ∗
 (x1 → x2x3 ) → (x1 ∼ x3) ≡            (x1 → x2x3 ) ∨ (x1x3 ∨ x1 x3 )           ≡
      ≡ (x1(x2 ∨ x3 ) ∨ x1x3 ∨ x1x3 )∗)∗ ≡ ((x1 ∨ x2 x3) · (x1 ∨ x3) · (x1 ∨ x3 ))∗ ≡
       ≡ ((x1 ∨ x1x2 x3 ∨ x1x3 ∨ x2x3)(x1 ∨ x3 ))∗ ≡ (x1 x2 x3 ∨ x1x3 )∗ ≡
  ≡ (x1 x2 x3 ∨ x1x2x3 ∨ x1 x2 x3 )∗ ≡ (x1 ∨ x2 ∨ x3)(x1 ∨ x2 ∨ x3 )(x1 ∨ x2 ∨ x3 ).
  éÚ×ÅÓÔÎÏ, ÞÔÏ óäîæ É óëîæ ÏÐÒÅÄÅÌÅÎÙ ÆÏÒÍÕÌÏÊ ÏÄÎÏÚÎÁÞÎÏ É, ÚÎÁ-
ÞÉÔ, ÉÈ ÍÏÖÎÏ ÓÔÒÏÉÔØ ÐÏ ÔÁÂÌÉÃÅ ÉÓÔÉÎÎÏÓÔÉ ÆÏÒÍÕÌÙ.
  óÈÅÍÁ ÐÏÓÔÒÏÅÎÉÑ óäîæ É óëîæ ÐÏ ÔÁÂÌÉÃÅ ÉÓÔÉÎÎÏÓÔÉ ÐÒÉ×ÅÄÅÎÁ ÎÉ-
ÖÅ ÄÌÑ ÆÏÒÍÕÌÙ (x1 → x2x3 )(x1 ∼ x3).

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