Double universal quantifiers at skolemization step of first-order logic to CNF conversion
$begingroup$
I am trying to convert the following formula
$$
(forall_{X}forall_{Y}((forall_{Z}p(X,Y,Z))rightarrow (exists_{P}q(Y,P))))wedgeexists_{S}r(S)
$$
to CNF. After eliminating the implication, moving negations inwards, and moving quantifiers outwards, I end up with
$$
forall_{X}forall_{Y}(exists_{S}r(S) wedge (exists_{P}(exists_{Z}neg p(X,Y,Z)vee q(Y,P))))
$$
How are skolem functions for the double universal quantifiers? My assumption is that one could do this:
replace
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};exists z;P(y)}
$$
with
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};P(f(x_{1},ldots ,x_{n}, y_{1}ldots y_{n}))}
$$
Is this correct? If not, how to handle the double universal quantifiers?
Thanks in advanve for help
first-order-logic
asked Dec 5 '18 at 15:30
MutageneMutagene
1
$endgroup$
add a comment |
$begingroup$
I am trying to convert the following formula
$$
(forall_{X}forall_{Y}((forall_{Z}p(X,Y,Z))rightarrow (exists_{P}q(Y,P))))wedgeexists_{S}r(S)
$$
to CNF. After eliminating the implication, moving negations inwards, and moving quantifiers outwards, I end up with
$$
forall_{X}forall_{Y}(exists_{S}r(S) wedge (exists_{P}(exists_{Z}neg p(X,Y,Z)vee q(Y,P))))
$$
How are skolem functions for the double universal quantifiers? My assumption is that one could do this:
replace
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};exists z;P(y)}
$$
with
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};P(f(x_{1},ldots ,x_{n}, y_{1}ldots y_{n}))}
$$
Is this correct? If not, how to handle the double universal quantifiers?
Thanks in advanve for help
first-order-logic
asked Dec 5 '18 at 15:30
MutageneMutagene
1
$endgroup$
$begingroup$
Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55
add a comment |
$begingroup$
I am trying to convert the following formula
$$
(forall_{X}forall_{Y}((forall_{Z}p(X,Y,Z))rightarrow (exists_{P}q(Y,P))))wedgeexists_{S}r(S)
$$
to CNF. After eliminating the implication, moving negations inwards, and moving quantifiers outwards, I end up with
$$
forall_{X}forall_{Y}(exists_{S}r(S) wedge (exists_{P}(exists_{Z}neg p(X,Y,Z)vee q(Y,P))))
$$
How are skolem functions for the double universal quantifiers? My assumption is that one could do this:
replace
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};exists z;P(y)}
$$
with
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};P(f(x_{1},ldots ,x_{n}, y_{1}ldots y_{n}))}
$$
Is this correct? If not, how to handle the double universal quantifiers?
Thanks in advanve for help
first-order-logic
asked Dec 5 '18 at 15:30
MutageneMutagene
1
$endgroup$
I am trying to convert the following formula
$$
(forall_{X}forall_{Y}((forall_{Z}p(X,Y,Z))rightarrow (exists_{P}q(Y,P))))wedgeexists_{S}r(S)
$$
to CNF. After eliminating the implication, moving negations inwards, and moving quantifiers outwards, I end up with
$$
forall_{X}forall_{Y}(exists_{S}r(S) wedge (exists_{P}(exists_{Z}neg p(X,Y,Z)vee q(Y,P))))
$$
How are skolem functions for the double universal quantifiers? My assumption is that one could do this:
replace
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};exists z;P(y)}
$$
with
$$
{displaystyle forall x_{1}ldots forall x_{n} ,forall y_{1}ldots forall y_{n};P(f(x_{1},ldots ,x_{n}, y_{1}ldots y_{n}))}
$$
Is this correct? If not, how to handle the double universal quantifiers?
Thanks in advanve for help
first-order-logic
first-order-logic
asked Dec 5 '18 at 15:30
MutageneMutagene
1
asked Dec 5 '18 at 15:30
MutageneMutagene
1
asked Dec 5 '18 at 15:30
MutageneMutagene
1
asked Dec 5 '18 at 15:30
MutageneMutagene
1
asked Dec 5 '18 at 15:30
MutageneMutagene
1
1
$begingroup$
Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55
add a comment |
$begingroup$
Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55
$begingroup$
Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55
$begingroup$
Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55
add a comment |
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Correct. See Skolemization : "Skolemization is performed by replacing every existentially quantified variable $y$ with a term $f(x_{1},ldots ,x_{n})$ whose function symbol $f$ is new. The variables of this term are as follows. If the formula is in prenex normal form, $x_{1},ldots ,x_{n}$ are the variables that are universally quantified and whose quantifiers precede that of $y$. "
$endgroup$
– Mauro ALLEGRANZA
Dec 5 '18 at 15:55