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                                                                       k
                                                        π |=k ϕ ⇐⇒ π |=0 ϕ
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                      i  k π |=ik+1 ψê             ⇐⇒      Xψ          |=ik   π
                                     k
                      ∃j, i j kπ |=j ψ              ⇐⇒      Fψ          |=ik   π
                                           òïí
                                           qñð%îì   ⇐⇒      Gψ          |=ik   π
                   π |=ik ψ1   êë
                               ›¦ê
                                π |=ik ψ2           ⇐⇒      ψ1 ∨ ψ 2    |=ik   π
                                  ê
                   π |=ik ψ1 π |=ik ψ2              ⇐⇒      ψ1 ∧ ψ 2    |=ik   π
                   p ∈ L(π(i))
                     /                              ⇐⇒      ¬p          |=ik   π
                   p ∈ L(π(i))                      ⇐⇒      p           |=ik   π
            ϕ
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                               π |=ik ϕ                    π = q 0 , q1 , . . . , q k
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             π   |=ik   ¬p           ⇐⇒   p ∈ L(π(i))
                                            /
             π   |=ik   ψ1 ∧ ψ 2     ⇐⇒   π |=ik ψ1 π |=ik ψ2
                                                    ê
             π   |=ik   ψ1 ∨ ψ 2     ⇐⇒   π |=ik ψ1 ›¦ê
                                                    êë    π |=ik ψ2
                                            k              j
             π   |=ik   Gψ           ⇐⇒     j=min(1,j) π |=k ϕ
                                            k              j
             π   |=ik   Fψ           ⇐⇒     j=min(1,j) π |=k ϕ
                                             π |=lk ψ,          ›ë ™÷
                                                               ê ø      i =k
             π |=ik Xψ               ⇐⇒
                                             π |=ik+1 ψ,     ÷ a%¤ê
                                                               ûúù
          π |=k ϕ ⇐⇒ π |=0 ϕ
                         k




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                                        00          01




                           11           10




         0
[S]k := q01 ∧
     0      0         0      0         0      0
 (¬q00 ∨ ¬q01 ) ∧ (¬q00 ∨ ¬q10 ) ∧ (¬q00 ∨ ¬q11 )∧
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 (q01 → q00 ) ∧ (q00 → (q11 ∨ q10 )) ∧ (q11 → q10 ) ∧ (q10 → (q11 ∨ q10 ))∧
 ...
   k−1     k        k−1     k       k      k−1     k    k−1    k     k
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                                        00           01




                           11           10




         0    0
[S]k := b0 ∧ b1 ∧

      ...
        0    0    1    1       0    0     1    1    1    1
      (b0 ∧ b1 → b0 ∧ b0 ) ∧ (b0 ∧ b0 → (b1 ∧ b1 ∨ b1 ∧ b0 )) ∧ . . .
      ...




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                     [p]ik := p(qi )
                     [¬p]ik := ¬p(qi )
                     [ψ1 ∧ ψ2 ]ik := [ψ1 ]ik ∧ [ψ2 ]ik
                     [ψ1 ∨ ψ2 ]ik := [ψ1 ]ik ∨ [ψ2 ]ik
                     [Gψ]ik := [ψ]ik ∧ [Gψ]ik+1
                     [Fψ]ik := [ψ]ik ∨ [Fψ]ik+1
                     [Xψ]ik := [ψ]ik+1
   DSD†vœW¦™iœU
   wp…u‘ w ˆge!p‡
                            [ϕ]k+1 =
                               k        qñð%îì
                                        òïí



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                                 (k, l )   
                      i
                 l [p]k := p(qi )
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                             i             i         i
                 l [ψ1 ∧ ψ2 ]k := l [ψ1 ]k ∧ l [ψ2 ]k
                             i := [ψ ]i ∨ [ψ ]i
                 l [ψ1 ∨ ψ2 ]k      l 1 k     l 2 k
                                              succ(i )
                 l [Gψ]ik := l [ψ]ik ∧ l [Gψ]k
                        i         i          succ(i )
                 l [Fψ]k := l [ψ]k ∨ l [Fψ]k
                         i         succ(i )
                 l [Xψ]k := l [ψ]k
   P › xt
                                    i + 1, i  k
                   succ(i) =
                                    l, i = k




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20131006 h10 lecture3_matiyasevich
 

20071104 verification konev_lecture12

  • 1. qqŒ Ž ‹ŠŠ` “d ‡ Ve ‡†“d ‡ V… ˜˜˜‰i¦a©ˆqi¦vhR „ ƒ‚ €r~lus yiWqqzxwlus yq vt r} |{yy vt r •†pn¤lDD¤D˜¦f¤†¦S¦˜¦• oj m kj f – – i h ™ d g f e d ™ — – gP€e ” UgP“d ’ ‘Ude SW¦©ƒWShiESi¦4 ‰§i¦W†„uƒ¦WiS‚iDWy¦ES¦v¤a'sqiSh¦f©¦aYWSSQI ˆ€ U ‡… x w e€€P U€ pdx e wP uP te r pRdPg ed c b `X VU TRP £5GG 8 7¨ $!C !3 8 7¨ 5$!$33 '4¦'©HFEE'D#BA@96%4'$ 2 1$0 )!£¡ £¨$¡ ! £ ¥ ¡ £¨£ ¥£¡ §¤©©¤('%# §©©§¦¤¢ 
  • 2. q™Ž Ž 1010 − 1020 © ¤ £§¦£ ˜W¨hqˆWˆ¦ ©µ µ ·£ ±¶´µ £´² ¨ ¤° ¢£¯ ® f˜¢yDv%®Fy‚ay¯Dv‚³v±™™¦ay'˜  ­ ¬6ª « « ©˜‚hq§ˆWyš¥WDa ¦‚™ ¡Ÿ  ¤ ¨£ ¦£¦ ¤ £ ¢¢ rS‚UiSP¦ˆgD‚…h›tvfDži‚gB9œSPv›ae'špqidShg¦fc P € u p ‘ Pd r ˆ€ V ‘ w uP t r R P ed ‘‘Pdx ed ™™fy¦f§c 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C ” £ £ ¥ $ 0 ! £ ¡ ! –ˆ@–B'©4†§'§§“¤'¤#’‘
  • 3. q™Ž Ž º¹ %¦¸ 1010 − 1020 © ¤ £§¦£ ˜W¨hqˆWˆ¦ ©µ µ ·£ ±¶´µ £´² ¨ ¤° ¢£¯ ® f˜¢yDv%®Fy‚ay¯Dv‚³v±™™¦ay'˜  ­ ¬6ª « « ©˜‚hq§ˆWyš¥WDa ¦‚™ ¡Ÿ  ¤ ¨£ ¦£¦ ¤ £ ¢¢ rS‚UiSP¦ˆgD‚…h›tvfDži‚gB9œSPv›ae'špqidShg¦fc P € u p ‘ Pd r ˆ€ V ‘ w uP t r R P ed ‘‘Pdx ed ™™fy¦f§c 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C ” £ £ ¥ $ 0 ! £ ¡ ! –ˆ@–B'©4†§'§§“¤'¤#’‘
  • 4. q™Ž Ž ÀÀ qˆÀ 10100 ¥W˜D¿vS„¢¼FDv‚¤aa˜‚‚huD£ a˜yDvB9z‚ayDv´ ¤   ¾ © £¤ µ½  ¤±´¼ ©µ ¢µ ·£ ® ±¶´µ¯ £ » %¦¸ º¹ 1010 − 1020 © ¤ £§¦£ ˜W¨hqˆWˆ¦ ©µ µ ·£ ±¶´µ £´² ¨ ¤° ¢£¯ ® f˜¢yDv%®Fy‚ay¯Dv‚³v±™™¦ay'˜  ­ ¬6ª « « ©˜‚hq§ˆWyš¥WDa ¦‚™ ¡Ÿ  ¤ ¨£ ¦£¦ ¤ £ ¢¢ rP €P gp… tPd r ˆ€ V ‘ w uP t r R P ed S‚UiS¦uˆD‚h‘›vfDži‚gB9œSPv›ae'špqidShg¦fc ‘‘Pdx ed ™™fy¦f§c 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C ” £ £ ¥ $ 0 ! £ ¡ ! –ˆ@–B'©4†§'§§“¤'¤#’‘
  • 5. q™Õ Ž ¥¤¦ ´µ§¤ µ Σ ¥¤   ¿ ¼ © £¤¤µ½  ¤±´¼  ¶ ¿ Ô ‚™qµhavh˜ Qhv§s‚˜ih¾S„¢FihWfa˜W‚™ui£ fia  #Ó ¨ Ò±   Р´ ¶± ¥ Î ¯ µ Î ¶±§ ¥´µ ® ´²´§¤£ ™ i‚y¶˜ÑfWfµy¯ ‚y¶Ï‚¤ v±ˆ9–‚vsWf%˜‚W‚fhiBÍ b VP t v›4Ì d rUdd…€ eR “…Ug V Ë tPd SP'†iDiDa¦qÆW‚BB¤vfc r¦hDhSw¦vSuv¤ae'ršUBu³hf%a¦‚˜B‚D‚iSh¦fc eg…‘ eg‘ “ P t …Pp tpu‡e Ê“…UdPged ‰ eg…‘weg‘ U “ uP te r p€ ¤v†h‚¦hSœ¢Bv›™'’WDp É ”£ £ ¥ 0!£¡ !3 £5GG Ũ£¨£ !3 £ 7¨ • 3 '§§È©©¤#Ç'4¦''Æ'Ä'‰HÃÂÁ
  • 6. q Ù Ž ¬p ¬p ¬p ¬p ¬p sk sl +1 sl s1 s0 Fp R dP rUdd…€e šS'iii†a¦©” ¬p ∨ ¬p ∨ ¬p ∨ ¬p ∨ ¬p sk sl +1 sl s1 s0 Gp R dP rUdd…€e šS'iii†a¦©” 7!£ !3 £ × 2 —! ¡ 5 '‰§'ØÏ6›§4©Ö
  • 7. q å Ž k “… U uPg…Êeu ã ˆg P€ U u‘ †WUBvhD¦¦ä‚œfšvâ ±¢¶  Ò áµ iy˜D9q¤à ®£¢¶  DSˆ˜iÒ e … w e g w U ß Þ p x d P g­ e d  hgDh‘S¦vS‘E‰QDÝiSh¦fi¦e k Ê e P P r … U ‚g¦e›t™aÜqfdSfœP'SP‚h‘‚S‘Fg i€Bu¦t †f™œ†aqSS¦©c 1 ˆ U “…… “…pR‘U e k “…pg ed U‘R U Û Dfh¦fSˆ¢fp Ú 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C ” £ £ ¥ $ 0 ! £ ¡ ! –ˆ@–B'©4†§'§§“¤'¤#’‘
  • 8. q™õ Ž k π |=k ϕ ⇐⇒ π |=0 ϕ U W ô e€  Ux e up€ p PU€P uP tPd  DW†y¦aWD³SiSv›vfD¦e U Vu D¦t ó i k π |=ik+1 ψê ⇐⇒ Xψ |=ik π k ∃j, i j kπ |=j ψ ⇐⇒ Fψ |=ik π òïí qñð%îì ⇐⇒ Gψ |=ik π π |=ik ψ1 êë ›¦ê π |=ik ψ2 ⇐⇒ ψ1 ∨ ψ 2 |=ik π ê π |=ik ψ1 π |=ik ψ2 ⇐⇒ ψ1 ∧ ψ 2 |=ik π p ∈ L(π(i)) / ⇐⇒ ¬p |=ik π p ∈ L(π(i)) ⇐⇒ p |=ik π ϕ U r VU€pTUd…e U r ˆ€‘P… ‘ P rde Û we€“ up rde 6BWiD˜i†S¢6§iS™WF'i¦’¦fW™©D¦f€ π |=ik ϕ π = q 0 , q1 , . . . , q k g ˆBˆ…Ärid¦e’D¦t u Û Vu rU uP tPd Bv›vfD „ ‰ “…‘… †hh'c $ 0 1 é £ ˜ 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C $ 0 5 ¨ $ –§“¤è#ĈÈ#'''D§ç¦6B³'£ æ
  • 9. ö#ĈÈ#'''D§ç¦6B³'£ æ G 7 ¨ ˜ — C ” ¨ ¨ £ • ¨ $ !C $ 0 5 ¨ $ ##1 0 (k, l ) Bv›vfD „ ‰ rU uP tPd π = q 0 , q1 , . . . , q k †hh'c “…‘… π |=ik ϕ ˆBˆÄi¦’D¦t u… rd e Û Vu ϕ g U6rBVWiD˜i†S¢6§iS™WF'i¦’¦fW™©D¦f€ U€pTUd…e U r ˆ€‘P… ‘ P rde Û we€“ up rde π |=ik p ⇐⇒ p ∈ L(π(i)) π |=ik ¬p ⇐⇒ p ∈ L(π(i)) / π |=ik ψ1 ∧ ψ 2 ⇐⇒ π |=ik ψ1 π |=ik ψ2 ê π |=ik ψ1 ∨ ψ 2 ⇐⇒ π |=ik ψ1 ›¦ê êë π |=ik ψ2 k j π |=ik Gψ ⇐⇒ j=min(1,j) π |=k ϕ k j π |=ik Fψ ⇐⇒ j=min(1,j) π |=k ϕ π |=lk ψ, ›ë ™÷ ê ø i =k π |=ik Xψ ⇐⇒ π |=ik+1 ψ, ÷ a%¤ê ûúù π |=k ϕ ⇐⇒ π |=0 ϕ k q™ Ž
  • 10. q™¨ Ž V p WU¦Ti‚€†U¦‡'r¦ˆRšUu³p©r†Ui€‚ef‚iDS'S˜W’Sha†i¦'¦ˆR § e u  ˆg rUd…e r‘e U rP…dU€U‡ re → p u… rd e Û pgP u… ¤yÄi¦¢™Sy‚‡ v†h‚¦hg § eg…‘we i =1 T (qi , qi +1 ) [S]k = I (q0 ) ∧ k k → p u… rd e Û pgP u… ¦ˆÄi¦¢hS¦y‚‡ ˆ€U u t g e teÜPdP ˆ rP…‘U‘ U…… iB¦F¦›™aqfSfH6SWhSS¢†f4c ˆ€U‡… i¦WD xu w e€ P€ eR ÜV uP te r g u… r d e Û ¥ …R U …€ p rP U uU uP tPd ˆ ¦W†Sf¦qsi†v›™'ÃYyÄD¦Ã¥ º ¦vDaD©S™‘ fv¤vWi¦e ’¤ ¢   ÿ ý ¡ £G¡ ü £¡þH›¤Æ ƒ#¡‘
  • 11. #BDÄ'Ä3 7 'Ĥ''G ''§†©¦§¦Ä’‘ ¡ ¥8£!£ £5G £¨£ ¡$5G ¥£! 00 01 11 10 0 [S]k := q01 ∧ 0 0 0 0 0 0 (¬q00 ∨ ¬q01 ) ∧ (¬q00 ∨ ¬q10 ) ∧ (¬q00 ∨ ¬q11 )∧ 0 0 0 0 0 0 (¬q01 ∨ ¬q10 ) ∧ (¬q01 ∨ ¬q11 ) ∧ (¬q10 ∨ ¬q11 )∧ 0 1 0 1 1 0 1 0 1 1 (q01 → q00 ) ∧ (q00 → (q11 ∨ q10 )) ∧ (q11 → q10 ) ∧ (q10 → (q11 ∨ q10 ))∧ ... k−1 k k−1 k k k−1 k k−1 k k (q01 → q00 ) ∧ (q00 → (q11 ∨ q10 )) ∧ (q11 → q10 ) ∧ (q10 → (q11 ∨ q10 )) k k k k k k (¬q00 ∨ ¬q01 ) ∧ (¬q00 ∨ ¬q10 ) ∧ (¬q00 ∨ ¬q11 )∧ k k k k k k (¬q01 ∨ ¬q10 ) ∧ (¬q01 ∨ ¬q11 ) ∧ (¬q10 ∨ ¬q11 )∧ q™© Ž
  • 12. qaŒ Ž := q00 ≡ q00 ∧ q01 ≡ q01 ∧ . . . k l k l k l ¥ 10 11 01 00 (k, l ) $ 0 –§1 £¨£ ¡$5G ¥£! ''§†©¦§¦Ä’‘
  • 13. qvŒŒ Ž l=0 k l := k ¥ k ¥ 10 11 01 00 $ 0 1 C 0 $0 £ ¨ £ ¡ $ 5G ¥£ ! –§§Æ5 a#D¤ž''§†©¦§¦Ä’‘
  • 14. #BDÄ'Ä3 E'4¦''G ''§†'¦§¤Hn¤43 #‰ ¡ ¥8£!£ 7 £5G £¨£ ¡$5G ¥£!3 £¨50$ 00 01 11 10 0 0 [S]k := b0 ∧ b1 ∧ ... 0 0 1 1 0 0 1 1 1 1 (b0 ∧ b1 → b0 ∧ b0 ) ∧ (b0 ∧ b0 → (b1 ∧ b1 ∨ b1 ∧ b0 )) ∧ . . . ... qa„ŽŒ Ž
  • 15. –#¤ö7 i9– ) ''§†©¦§¦Ä’‘ $ 0 1 飘 — ! £¨£ ¡$5G ¥£! [p]ik := p(qi ) [¬p]ik := ¬p(qi ) [ψ1 ∧ ψ2 ]ik := [ψ1 ]ik ∧ [ψ2 ]ik [ψ1 ∨ ψ2 ]ik := [ψ1 ]ik ∨ [ψ2 ]ik [Gψ]ik := [ψ]ik ∧ [Gψ]ik+1 [Fψ]ik := [ψ]ik ∨ [Fψ]ik+1 [Xψ]ik := [ψ]ik+1 DSD†vœW¦™iœU wp…u‘ w ˆge!p‡ [ϕ]k+1 = k qñð%îì òïí qa„ÕŒ Ž
  • 16. Ç7 i9– ) ''§†©¦§¦Ä’‘ G — ! £¨£ ¡$5G ¥£! ##1 0 (k, l ) i l [p]k := p(qi ) l [¬p]ik := ¬p(qi ) i i i l [ψ1 ∧ ψ2 ]k := l [ψ1 ]k ∧ l [ψ2 ]k i := [ψ ]i ∨ [ψ ]i l [ψ1 ∨ ψ2 ]k l 1 k l 2 k succ(i ) l [Gψ]ik := l [ψ]ik ∧ l [Gψ]k i i succ(i ) l [Fψ]k := l [ψ]k ∨ l [Fψ]k i succ(i ) l [Xψ]k := l [ψ]k P › xt i + 1, i k succ(i) = l, i = k q ŒÙ Ž
  • 17. q Œå Ž k π |=0 ϕ k ⇐⇒ ‰ ‰… g e teÜPdP … rP…‘U‘ !PdP ¦xh¡¦¤a™qWSf’ÄS‚h‚S‰$WS‚ [S, ϕ]k ˆ€U u t “…… …P…g…‘P ß… iB¦9Df™’hhqDh™ÑDv‘ p rU€ue ˆ ©iB‚WWg p u… rd e ¦yÄi¦Ã# l=0 k ∧ l [ϕ]0 )) k (l k ∧ [ϕ]0 ∧ k [S, ϕ]k := [S]k ∧ (¬ ¥ k ¥ £5G£ ¡ £G Ħ%‰4H%F 
  • 18. qa„õŒ Ž ‰‰ ™™‰ ‰‰ ™™‰ ‰‰ ™™‰ c ‹ed ˜C` e Y P Y ŠaX P bT a`EB 8X fq¦bg ‰q t e b` qhP bT a`B 8X Y PEY fi¤Ud ‰c t e b` ehP bT a`B 8X Y PEY ig bb` cq‹ b˜` Š ˜` ‹ Y P Y #P bT a`EB 8X sc¤c ‰ dcg b` Š aX ihP bT a`B 8X Y PEY Š aX iqX bG›d X ˜X c Y PEY rP bT a`B 8X fq˜X ‰ qeX bG¦i X ˜X d Y PEY pP bT a`B 8X ` iŠ baX ‹ g Y PEY hP bT a`B 8X fe¦Š ‰ ` ‹ ` Y P Y #P bT a`EB 8X d ¦` ‰ dX a˜X k c Y P Y X P bT a`EB 8X º ¹ B WV T R ô « « PHFE B ô B¦¸ ó D UUSº §¬6ª ó QIG$DC A “ uP te v¤aä @ 6 7 7 6 4 2 1 ) '% $ 0 5 G 5 $ 8398$530(¤6¦'6%'5 æ
  • 19. qa„Œ Ž Š`eŠ ˜˜baX Šccd ibUbd dbaX d qX G¦i ŠgX‹ ˜G¦˜` ib‹ ‹gg ˜bb` Šdid DCbd iUc cqq bGX Šigd ˜bCc gUc q bg ŠŠie ˜˜bGX ebg ie bUc Šq`X ˜ba˜X d g q GX Šdq` Db˜g ib` c e Še`` ˜b˜˜i k c ` BD“ B ’  ‰ ‘3¸ @ ˆ 7 7 64 ƒ  y x w u $0 5G 5 $ „p‡†a…„‚€h¡v¤6¦'6%'5 æ
  • 20. qa¨Œ Ž ™ fe¦q ‰ t t i`i b˜bg t d aX g GX V H D ™ Y ae« gf›GX ‰dq t t i`X ba˜X qbDaX `d X¦g ` aX V ™ Y ae« HD ñ‹›g ‰d t t Š ˜i `ic$X ` ˜g X ˜X V H D ™ Y ae« sc¦˜bg ‰Še `X a¦g † Ãt t t X ¦g Š aX V ™ Y ae« HD fi˜¦GX ‰Xg c ` cgŠ b˜q t t ebg q V HD ™ Y ae« c e Š ˜` ‹Xg a¦bi t t c g e V ™ Y ae« HD X ‰f ` Š aX X‹i ¦˜b` t q ‹ V HD ™ Y ae« `ñ¦aX ‰` e GX t ic Ug t `˜` i V ™ Y ae« HD X‰f g d ` de GX gŠ b˜e ` aX c V HD ™ Y ae« ñ‹¦g ‰ ` aX Ša¤c X Xi ¦b` Š˜biq d ` d V ™ Y ae« HD X‰f g i b‹ c‹ ˜g q‹ bic d aX g V ™ Y ae« HD X‰f fe¦Š ‰ X ˜X ‰ e Š˜‹ c HD ` V ™ Y ae« ñ`¦`‰ g ™ `e ˜b` ™ 2 i 1 ‹ e k GX X V Y ae«HD d $F R E V H ô W D UT P ó $I`FB HF $I`B ª˜ ª˜ “ uP te Bv›™Ã¤ @ 7 7 7 6 – ”% $ 0 5 G 5 $ ‡9‡†a4 —•¤6¦'6%'5 æ
  • 21. qa©Œ Ž ¬p ¬p ¬p ¬p ¬p ¬p sk+1 sk sl +1 sl s1 s0 = sl sl sl l=0 := k l k k ú Bë n mú lj o¢bkë ¥ k ¥ l := q00 ≡ q00 ∧ q01 ≡ q01 ∧ . . . k l k l r qp †ki i 5h ¥ ¬p ¬p ¬p ¬p ¬p sk sl +1 sl s1 s0 Fp 0 $!£ !3! 5¨ $0G Å 1$é 5 ''¡6B#†0 ¤''#3 #4E§B3 g
  • 22. qa(Ž Ž u S |= ¬ϕ U u‘P e… Bv™œ‚W“t k π |=0 ϕ k ⇐⇒ ‰¦xh…¡¦¤a™qWSf’ÄS‚h‚S‰$WS‚ ‰ g e teÜPdP … rP…‘U‘ !PdP [S, ϕ]k ˆ€U u t “…… …P…g…‘P ß… iB¦9Df™’hhqDh™ÑDv‘ p rU€ue ˆ ©iB‚WWg p u… rd e ¦yÄi¦Ã# l=0 k ∧ l [ϕ]0 )) k (l ∧ k ∧ [ϕ]k [S, ϕ]k := [S]k ∧ (¬ ¥ 0 ¥ k s $ 5 3©'¨ 4¡‘
  • 23. qvîŽ ŽŒ ôp€puuU rRp ¤ p€P ” ˆ…e‡pd‰ r SWD¦DW6Si¢|WSnHDS˜WW4S‘ ó S |= ¬ϕ eW‚'‚D†Va¦f’œ†hvvWfaIv¦¤œÄD†‚%yvf¢@¦i©D¦f†U … Ê“ … € e U g …‘“ uP…p!pR e t Ü ˆ € p t…P€ !U Þ U Tp r d e Û€ “ … g e! ue U ˆ te…P r P ˆ€g U …RP Û Ûz PP ue‡ PUx …d “ …‘ Dfph¦™a“B‚fS‘S4aWh'éi‚††hS™{™‚˜³‚y†™4t †hâ U…‘e rU yU…‘e †h˜'†³I†h˜¤t dDh'DpS4t¡W€šSRiS˜¦¡©D†¦‚™‚˜‰†WDD‚W f¡G³¤ …P r U p U €PTe P ˆ€e… PPue‡ “…U…ue e€ ye 2|ϕ| k et “…eu “…Pd…e r‘‘p e€e…p…‘e ›FDSaWg †hfDS'S™WWd f¦Wf‚h˜QÉ len(σ1 ) + len(σ2 ) 2|ϕ| U σ = σ1 (σ2 )∗ ϕ p tU UUTp…PddP…€ %WWg †¦D‚hfiDSWaU g p rU€ue ˆg p€e e…Êp rU€ue ˆ s©iB‚f§W’W¦œ‚BD©†DB‚f‚g p u… rd e Û ¥ U u‘ ¦yÄi¦¢¥ º xBv¦â 5G w 5G ¥ ! 5£ $ 6¦„6¦nç6¦‰ v
  • 24. qaŽŽ Ž ‰Ü ˆ€€P rPdP PU€pgUpg‘Ud ÄiDS'SfSf’‚iD™†DhSSii ôPe€U…‘p w e‡e‘ …P Vugp…‘ tPd “…… w ˆ t yp f˜f††hf˜ ó ¦˜˜™Ãh¦DqiSh¤vWi³†f™ 4QGn” A A A º º A A A ¬r r ¬r r ¬r r ¬r r ¬q q ¬q q ¬p p (¬p ∨ ¬r ) ∧ (p ∨ q) ∧ (r ∨ ¬q) rUd…e r‘‘p iDS'S™W›‚ € € ~ h}
  • 25. qaÕŽ Ž ‹ ‰ˆ •Œ‡ q ‹ ‰ˆ ŠQ‡ q= § ¦  ¦¢zvµÑ£z™°¦¢aQv‘£‹ b(Ž’ h¤W¤˜WQys§‹  ‘  Ž  ¦ ¶ ‚ £ f ¬ ∧( ∨ q) ‹Ž Q‘ b¬r ∧ (r ∨ ¬q)  ¢yqµÑ£u™°¦¢fq§S’£v‚˜fˆ•f‡  ¦ ¶ £ ¤ ¤   b§ ‰ ¦ Œ r= ( ∨ ¬r ) ∧ ∧ (r ∨ ¬q) ‹ ˆ ‹Ž Q‘ b (¬ ∨ ¬r ) ∧ ( ‹ ‰ˆ •Œ‡ ∨ q) ∧ (r ∨ ¬q) ‹ ‰ˆ b•Œ‡ ‹ ‰ˆ ŠQ‡ (¬p ∨ ¬r ) ∧ (p ∨ q) ∧ (r ∨ ¬q) °Wyq¾ † §¦ p= » (2n − 1) eRp€ te Ê rU…P r qiW4a%6††h'ip Ú n ‰geu! S¦v… Ê Ü ˆ € €P rPdP V u ɉ‰ †ÄiDS'SfSf Däu™™‰ ……… £$ • — £ ¥—8 ¡ „ 5G ¨ w kk8n6G 'Ïü †Âž†¤nƒ æ
  • 26. q ŽÙ Ž ò ˜— †š$– ò ˜— †š$– r= U u‘P eR“ ue… Ê e€€U…‘ Bv™’ˆW‚W›¦Wi††h‚U r òïí vñð›îì ) (r ∨ ò ˜— )™G– ∧ (r ∨ ¬ò ˜— †š$– ò ˜— †š$– q= U u‘P eR“ ue… Ê e€€ U…‘ Bv™Eˆ‚B‚‚›¦fiDhSU q ∧ (r ∨ ¬q) ∧ q ∧ (r ∨ ¬q) ò ˜— †š$– ∨ ¬r ) ∧ q ∧ (r ∨ ¬q) ò ˜— ™G–( ∨ q) ∧ (r ∨ ¬q) òïí qñð%îì ∨ ¬r ) ∧ ( òïí vñð›îì(¬ òïí vñð›Ýì p= “…‘… Dh™™ • (¬p ∨ ¬r ) ∧ (p ∨ q) ∧ (r ∨ ¬q) ……… ¨ • C $ ¨ ƒk”†§›'Ï“
  • 27. q Žå Ž 5 œgeu! U¦v… eR“ ue ˆ‚B‚W… º º q, r ¬q, ¬r ¬p p (¬p ∨ ¬r ) ∧ (p ∨ q) ∧ (r ∨ ¬q) Vu †äÉ 5 •$ ħ4'¨ ›
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  • 29. qaŽ Ž ‰ V‘“ … U tex U d  …P ye r R ‘ U e w ˆ € …€ P r P d R € U ‰ ‰ zhD‚4™îiišh¬$'œSS¦f i†aS'SfSi'¤e ° k k+1 k k+1 [ψ]0 ≡ [ψ]0 ; [ψ]1 ≡ [ψ]1 . . . ÊPuP t re rp‘ p e S¦v›Â¦'iaÃW€ n‡ k+1 k+1 k+1 k+1 k+1 [Gψ]0 := [ψ]0 ∧ [ψ]1 ∧ · · · ∧ [ψ]k ∧ [ψ]k+1 . p k k k k [Gψ]0 := [ψ]0 ∧ [ψ]1 ∧ · · · ∧ [ψ]k , b p uR U T !P‡ ˆ u… rd e Û PU€P ug p…‘ tPd D¤q¦¢$˜þyÄi¦Ã‚iS¦ˆD‚h›vf§c 0G ” 7¨ 5¨£ £!0¨ ''–3 ’6B'‰'Ä'’¯