The document appears to be a compilation of various logical expressions and formal representations, possibly related to propositional logic or formal proofs. It includes references to clausal forms, horn clauses, and relationships between variables using logical operators. The content is heavily fragmented and filled with symbolic notations and mathematical phrases.

1. 31
2:
2.2:
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!
.
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. A. $"% &
1.
2. " )& " .'
/# ( " * &% " ) ( " " .' " + ) *"& ,
"& " ) ( " '" " &)" ) +"*& "
" .' .
+ *" # ( " + ) ) :
+ "& "& % :
+" %:
>!""
A. $"% &
2. )"%
+ *" !" " wff !+ + "& " +"& )" & +
+ + % + )"%
?!""
* :
• + )"% "& (+ + " ) + &" + "&
" +"& ( " ) *! ) " ' )# ) + ) '
(wff).
• $ ( " 3 + )"% :
(,"( ( )
,"( ( )
clausal form (CF)
• * ! " + " + ( " + * + " !+ ' )"
)(,"( .
A. $"% &
2. )"%
1. (,"(
@!""
* :
/ !+ "& )" # " (# ), "& :
+ ( " "& :
" !" "& (literal) * ' * + ) ) + )
*"& :
.#. + )" :
#$% &'# ($) %& &'# ($# #$% *'* ($) %& ($# *'*
+ $ , - - - + -
"& )" (,"( ( )
A. $"% &
2. )"%
2. ,"(
A!""
* :
/ !+ "& )" ! " (! ), "& :
+ ( " "& :
" !" "& ( ' (literal) * ' * + ) ) + )
*"& :
.#. + )" :
#$% &'# ($) %& &'# ($# #$% *'* ($) %& ($# *'*
+ $ , - - - + -
"& )" ,"( ( )

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. . $"% &
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. . $"% &
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. !""
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. >!""
& + % + )" )" () '1)) :
1: & "&
2: & + & ( ) * ('" " ( &
3: + & " ) * ('" " '"% "&
4: # ' "& ,%
5: )( + "& + + & " ) * ('" " ( &
6: & )( + "& + )( + "&
( ) * (+% ! + + % + )" () '1)) )" + )" .
: 5 ) + "&) " /1,+ & " _) _* ('" /2, ,% /1 )( + "&/2
. ) )"
3
?!""
( ) " + ! + + % + )" )"