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ΠΛΗ31 ΜΑΘΗΜΑ 2.2 (ΕΚΤΥΠΩΣΗ)

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Α.Θεωρία
1) Νόμοι Κατηγορηματικής Λογικής
1.1) Νόμοι Κατηγορηματικής Λογικής
1.2) Ταυτολογία-Αντίφαση
1.3) Μετονομασία μεταβλητών ποσοδεικτών
2) Κανονικές Μορφές
2.1) Κανονική Διαζευκτική Μορφή
2.2) Κανονική Συζευκτική Μορφή
2.3) Clausal Form
2.4) Πρόταση Horn
3) Μετατροπή wff σε ΣΚΜ
3.1) Μεγάλη Άσκηση Σ.Κ.Μ.
3.2) Γρήγορη Σ.Κ.Μ.
Β.Μεθοδολογία
1) Τυπική Απάντηση ΣΚΜ
2) Προτάσεις Horn (και μικρές παραλλαγές)
Ασκήσεις

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ΠΛΗ31 ΜΑΘΗΜΑ 2.2 (ΕΚΤΥΠΩΣΗ)

  1. 1. 31 2: 2.2: – . . ! . 1. 1. 2. - 3. 2. 1. ! " 2. # " 3. Clausal Form 4. $ Horn 3. wff #. . 1. % & #. . . 2. ' #. . . (. ) 1. $ % # 1. % # 2. $ % Horn ( ) !"" " "# " . $"% & 1. "& ' ( : !"" # $ % &'( )*$+ &+ , ( &+ ( & +-. - / ( ( )% -& ' . - 01 23 456789 01 - :;2 .( < =& - ( > / - -( ? (- ( @ / . -( - $ A B( C% . - & ( C-. - , ( ( C-. - Β∨Α≡Β∧Α Β∧Α≡Β∨Α ~~)(~ ~~)(~ )()()( )()()( Γ∨Α∧Β∨Α≡Γ∧Β∨Α Γ∧Α∨Β∧Α≡Γ∨Β∧Α Α∨Β≡Β∨Α Α∧Β≡Β∧Α Γ∨Β∨Α≡Γ∨Β∨Α Γ∧Β∧Α≡Γ∧Β∧Α )()( )()( Α≡Α)(~~ Β∨Α≡ΒΑ ~ Α≡ΒΑ ~~ B Α∃≡Α∀ Α∀≡Α∃ ~~ ~~ xx xx Β∀∧Α∀≡Β∧Α∀ Β∃∨Α∃≡Β∨Α∃ xxx xxx }{ }{ . $"% & 1. 1. ( ' & - & ) ' "& ) *( &" ( ). ( ) & " " ) ( " ' + " ( + + ) ( " + ) , + !+ " !+ + ( #" &* ' " " # + ) . &, ( " "+&) : <!"" ( ' & "& + * + " + ) ) ' "& + ' & ) "& + * + " + ) ) "& + -"(* . *" : ' ( " + )" "& ) *! " : (" ,% * +' ) ) (" ,% ) + ) *"& ) (" ,% ) + ) *"& )
  2. 2. A. $"% & 1. 2. " )& " .' /# ( " * &% " ) ( " " .' " + ) *"& , "& " ) ( " '" " &)" ) +"*& " " .' . + *" # ( " + ) ) : + "& "& % : +" %: >!"" A. $"% & 2. )"% + *" !" " wff !+ + "& " +"& )" & + + + % + )"% ?!"" * : • + )"% "& (+ + " ) + &" + "& " +"& ( " ) *! ) " ' )# ) + ) ' (wff). • $ ( " 3 + )"% : (,"( ( ) ,"( ( ) clausal form (CF) • * ! " + " + ( " + * + " !+ ' )" )(,"( . A. $"% & 2. )"% 1. (,"( @!"" * : / !+ "& )" # " (# ), "& : + ( " "& : " !" "& (literal) * ' * + ) ) + ) *"& : .#. + )" : #$% &'# ($) %& &'# ($# #$% *'* ($) %& ($# *'* + $ , - - - + - "& )" (,"( ( ) A. $"% & 2. )"% 2. ,"( A!"" * : / !+ "& )" ! " (! ), "& : + ( " "& : " !" "& ( ' (literal) * ' * + ) ) + ) *"& : .#. + )" : #$% &'# ($) %& &'# ($# #$% *'* ($) %& ($# *'* + $ , - - - + - "& )" ,"( ( )
  3. 3. . $"% & 2. )"% 3. Clausal Form D!"" ) : / !+ "& )" clausal form "& : ./ .0 .1 2 / 0 3 + ( 0 ( ' ! (+ )" ) ( ' ! )( +" ) ) "& + )" (* ' * #% & )" ) * " + 1)" : n=0: ./ .0 .1 2 ((+ )" #% & )( + ) ) " + ) "& m=0: 2 / 0 3 ()( +" ) + ( )#! ( "2 + (+ )" ) " + ) "& + ' * ' * n=m=0: 2 + ( )( . '&," " 4 "& + ' , * ' * $ : C.F. "& " * +" &+ %) ! +.#.: ./ .0 2 / 0 ./ .0 / 0 ./ .0 / 0 ./ .0 / 0 . $"% & 2. )"% 4. ) Horn E!"" * : • M& $ Horn "& !+ )" clausal form " + . $ : • + )" Horn + &, ( ) ' ) ) Prolog ) " " ) ( + )( +" ) . • ) % . ! * " + 1)" : • " &+ %) 1: .1 " " + ()& (+ )"% : • : ./ .0 .1 2 5 • + (+1 " )('' ) : « )#! ( 6 78 )#!" » • 3) *! (+ (+ )" 6 7 +" )( + ) . • & + ) '"& • " &+ %) 2: .1 " #% & (+ )" : • : 2 5 • + (+1 " )('' ) : «3)#!" » • 3) *! )#!" "( (+ )"% ( )#!" & ). • & + ) '"& *! ( ) #"& + "' ! " ' '1)) Prolog. . $"% & 3. " + wff )" !"" * : • / ' )# ) !+ (wff) + "& " +"& )" (,"( )% " ' & ( + ( + "'"& + "2 . : 1. 2 '" - % ( "+ % 1 2. )" ) )" 3. 2 '" - % 4+ 2 1 ) *" 1 4. + ) " .' 1 ' 1 ) *" 1 5. " & ) % ' 1 ) *" 1 ) ) " 6. " & ) % * ,"!2"% ) "+&+"* % ( '" % 7. + '" - ( ' ! ) *"& ( AND . $"% & 3. " + wff )" 1. 2 '" - % ( "+ % 1 !"" 1 : ,-% . 1 . * 1# ( " (# )( "+ % + ( (+ # ( " , " + )( "+ % )" OR: 2 98 . *" : " +"& )" + ) : 2 : : : 2 ; !) : " + ( " « ) " » )( "+ % )" OR: : : : 2 ; " + ( " «*"2 » )( "+ % )" OR: : : : ;
  4. 4. . $"% & 3. " + wff )" 2. 2 '" - % ( "+ % 1 !"" 2 : % 2 . " ( " )" + ( (+ # ( ) "+&+"* % 1 + )"% . $ ! # ) ) *( & " + ) *"& " De Morgan: (…)( #" …) : : : ; ,% ) *( & " + ) *"& " : : : : ; ,% De Morgan: : : : ; 6 6 6 6 < = < = < = < = . $"% & 3. " + wff )" 3. 2 '" - % 4+ 2 1 ) *" 1 ( '" + & ) ) <!"" 3 : ,-% . / - $ (# ) (+ 2 + ) *"& *" "& ) +"*& " + ( ' ! + ) *"& , " ) ! " " .' ( (+ 2 ! + ) *"& " + ) " *"& : • + + • + + • + + (+ 2 + ) *"& "& ) +"*& " + ( ' ! + ) *"& , " ) ! " " .' " & )( ) " , " ) " .' ( ' ! + ) *"& *"& : • ($) %& ($) %& >?@ABC • + + D • + + D 8 • + + D + D E . $"% & 3. " + wff )" 3. 2 '" - % 4+ 2 1 ) *" 1 ( '" + & ) ) >!"" (…)( #" …) : : : ; 2 '"& ( " ( (+ 2 ! + ) *"& " : : D D : : ; . $"% & 3. " + wff )" 4. + ) " .' 1 ' 1 ) *" 1 ?!"" 4 : , ) $ 4 . '' , ( " % " .' 1 % ' 1 + ) *" 1 , ) 1) " " ' + ) *"& #" 2"#% ) " .' ) # ! '' , ( " &) # % " &)"% " .' ) +"*& " ( + ) *"& (…)( #" …) : D D : : ; " )& ) 2 " ) y )" + ) *"& : : D D : : ;
  5. 5. . $"% & 3. " + wff )" 5. " & ) % ' 1 + ) *" 1 ) " @!"" 5 : ) % 5 . " ! " ( ' ! + ) *"& " ) " +" * " " .' #" '' 2" , # ) + 1 ( . . # ( " * &% "2 ( " )% ( ' ! + ) *"& " + ) + ' + ) . (…)( #" …) : D D : : ; " & ) % ' 1 + ) *" 1 ) " : : D D : : ; . $"% & 3. " + wff )" 6. " & ) % * ,"!2"% ) "+&+"* % ( '" % A!"" 6 : " +- % 6 . " ! " OR 1) " )( * ( ( '" ( + )" )" 1 + )"% ) 5 ) "& "+ " ) !: < = F G < = < F 1 " & "() (: < = F H G < = < F < H (…)( #" …) : D D : : ; 88888888888888888888888888888888888888888<888888 8888888888888888888888888=88888888888888888888888888 88888888888888888888888888888F888888888888888888 888888888888888888H88888888888888888888 ,% "+ " ) !: : D D : : ; 888888888888888888888888888888888<888888 8888888=88888888888 8888888888F888888888 ,% "+ " ) !: : D D : : ; " . ) ( + )" ) ! # ( " "' : : D D : : ; . $"% & 3. " + wff )" 7. + '" - ( ' ! + ) *"& ( AND D!"" 7 : % . ) + AND 7 . * 1# ( " ( ' ! + ) *"& " )+ " + )" " . ) ( )( * ) ( AND "' + )" *" + +" # ( " )" 2 + )" &* " .' 1 ( '' , ( " )" + )" + ( # ( &* " .' 1 ). (…)( #" …) : D D : : ; + '" - + ) *" 1 AND. " )& " .' 1 + ( # ( &* )" + )" : IJ : D KJ D LJ M : M : M NJ O ; O E!"" . " "& . . ( !+ (: 2 P + Q 2 % : 0 1: 2 '" - % )( "+ % 1 88888 2 P + Q 2 ("2 '" - )( "+ % 1 ) G P + Q (" . .* +' ) ) G P + Q 0 2: )" ) + )" " + "& 0 3: 2 '" - 4+ 2 1 ) *" 1 G8 P D + Q 0 4: + ) " .' 1 ' 1 ) *" 1 " + "& 0 5: " & ) % + ) *" 1 ) " 8 P D + Q 0. " * ' & 1. ()& ) + ) 1. " ' 6) ) . . % # • +" &+ %) + ( ) ) + "& + '' . + ) " ) + & ) % . % ( ' & (
  6. 6. !"" 0. " * ' & 1. ()& ) + ) 1. " ' 6) ) . . % ( 0 ): 0 6: " & ) % * ,"!2"% ) "+&+"* % ( '" % 8 P D + Q ( "+ " ) !) G8 P D + Q ( "+ " ) !) G8 P D + Q G8 P D + Q 0 7: + '" - ( ' ! + ) *"& ( AND IJ P D KJ + LJ M Q M !"" 0. " * ' & 1. ()& ) + ) 2. )" Horn ( + '' ) % : *" 1: 2 /#" . . .: 8 *" 2: + 8 2 /#" . . .: + % Horn • " + '' ) )" + !+ ( + )" + ( "& + )" Horn ( + '' ( ). +" &+ %) ( , + ! " "2 ( " ") . . . 5 ) + 1 ") ' ( " +" : R R RM S R R R RM R R S R ( '" . ) )" 1 !"" & " + % )! ' ' 1 + )"% : (Q ∧ T) R, (Q ∨ P) R, S (T ∨ P), Q, S, ¬T " - " + )" )" . . ) )" 2 <!"" . " "& . . . ( + % !+ (: ( ) ( ) ( ) ( )( )( )( )wuRwvRwvtRvutRut ,,,, ∧∃¬∀∧¬∃∀
  7. 7. >!"" & + % + )" )" () '1)) : 1: & "& 2: & + & ( ) * ('" " ( & 3: + & " ) * ('" " '"% "& 4: # ' "& ,% 5: )( + "& + + & " ) * ('" " ( & 6: & )( + "& + )( + "& ( ) * (+% ! + + % + )" () '1)) )" + )" . : 5 ) + "&) " /1,+ & " _) _* ('" /2, ,% /1 )( + "&/2 . ) )" 3 ?!"" ( ) " + ! + + % + )" )"

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