Scandinavian Logic Symposium 2014, Tampere, August 25th, 2014
The Expressive Power of k-ary
Inclusion and Exclusion Atoms
Raine Rönnholm
School of Information Sciences,
University of Tampere, Finland
Inclusion and Exclusion Logics
I denote Inclusion Logic by INC and Exclusion Logic by EXC.
Inclusion-Exclusion Logic is denoted by INEX.
Inclusion and Exclusion Logics
I denote Inclusion Logic by INC and Exclusion Logic by EXC.
Inclusion-Exclusion Logic is denoted by INEX.
Ýt1 , Ñ
Ýt2 be k-tuples of terms.
Let Ñ
Ýt1 „ Ñ
Ýt2 have the following truth condition:
k-ary inclusion atoms Ñ
Ýt1 „ Ñ
Ýt2 , iff for all s
M (X Ñ
PX
Ýt1 q s 1pÑ
Ýt2 q.
there exists s 1 P X , s.t. s pÑ
Inclusion and Exclusion Logics
I denote Inclusion Logic by INC and Exclusion Logic by EXC.
Inclusion-Exclusion Logic is denoted by INEX.
Ýt1 , Ñ
Ýt2 be k-tuples of terms.
Let Ñ
Ýt1 „ Ñ
Ýt2 have the following truth condition:
k-ary inclusion atoms Ñ
Ýt1 „ Ñ
Ýt2 , iff for all s
M (X Ñ
PX
Ýt1 q s 1pÑ
Ýt2 q.
there exists s 1 P X , s.t. s pÑ
Ýt1 | Ñ
Ýt2 have the following truth condition:
k-ary exclusion atoms Ñ
Ýt1 | Ñ
Ýt2 , iff for all s, s 1
M (X Ñ
PX
Ýt1 q s 1pÑ
Ýt2 q.
it holds that s pÑ
Inclusion and Exclusion Logics
I denote Inclusion Logic by INC and Exclusion Logic by EXC.
Inclusion-Exclusion Logic is denoted by INEX.
Ýt1 , Ñ
Ýt2 be k-tuples of terms.
Let Ñ
Ýt1 „ Ñ
Ýt2 have the following truth condition:
k-ary inclusion atoms Ñ
Ýt1 „ Ñ
Ýt2 , iff for all s
M (X Ñ
PX
Ýt1 q s 1pÑ
Ýt2 q.
there exists s 1 P X , s.t. s pÑ
Ýt1 | Ñ
Ýt2 have the following truth condition:
k-ary exclusion atoms Ñ
Ýt1 | Ñ
Ýt2 , iff for all s, s 1
M (X Ñ
PX
Ýt1 q s 1pÑ
Ýt2 q.
it holds that s pÑ
I denote Inclusion Logic containing at most k-ary atoms by INC[k]. Similarly I use
EXC[k] and INEX[k] for Exclusion Logic and Inclusion-Exclusion Logic.
Dependence and Nondependence Logics
I denote Dependence Logic by DEP and Nondependence Logic by NDEP.
Dependence and Nondependence Logics
I denote Dependence Logic by DEP and Nondependence Logic by NDEP.
pt1, . . . , tk q have the following truth condition:
M (X pt1 , . . . , tk q, iff for all s, s 1 P X for which
s pt1 . . . tk 1 q s 1 pt1 . . . tk 1 q, also s ptk q s 1 ptk q.
k-ary dependence atoms
Dependence and Nondependence Logics
I denote Dependence Logic by DEP and Nondependence Logic by NDEP.
pt1, . . . , tk q have the following truth condition:
M (X pt1 , . . . , tk q, iff for all s, s 1 P X for which
s pt1 . . . tk 1 q s 1 pt1 . . . tk 1 q, also s ptk q s 1 ptk q.
k-ary nondependence atoms pt1 , . . . , tk q have the following truth condition:
M (X pt1 , . . . , tk q, iff for all s P X there exists an s 1 P X ,
s.t. s pt1 . . . tk 1 q s 1 pt1 . . . tk 1 q, but s ptk q s 1 ptk q.
k-ary dependence atoms
Dependence and Nondependence Logics
I denote Dependence Logic by DEP and Nondependence Logic by NDEP.
pt1, . . . , tk q have the following truth condition:
M (X pt1 , . . . , tk q, iff for all s, s 1 P X for which
s pt1 . . . tk 1 q s 1 pt1 . . . tk 1 q, also s ptk q s 1 ptk q.
k-ary nondependence atoms pt1 , . . . , tk q have the following truth condition:
M (X pt1 , . . . , tk q, iff for all s P X there exists an s 1 P X ,
s.t. s pt1 . . . tk 1 q s 1 pt1 . . . tk 1 q, but s ptk q s 1 ptk q.
k-ary dependence atoms
I use DEP[k] and NDEP[k] for Dependence and Nondependence Logics
containing at most k-ary atoms.
Some known facts about the expressive power of logics
Fact 1 (Galliani) On the level of formulas:
EXC DEP
INC NDEP
INEX INDEP (Independence Logic).
Some known facts about the expressive power of logics
Fact 1 (Galliani) On the level of formulas:
EXC DEP
INC NDEP
INEX INDEP (Independence Logic).
Fact 2 (Galliani) On the level of sentences:
DEP[1] FO NDEP[1].
Some known facts about the expressive power of logics
Fact 1 (Galliani) On the level of formulas:
EXC DEP
INC NDEP
INEX INDEP (Independence Logic).
Fact 2 (Galliani) On the level of sentences:
DEP[1] FO NDEP[1].
Fact 3 (Väänänen) On the level of sentences:
DEP ESO (Existential Second Order Logic).
Some known facts about the expressive power of logics
Fact 1 (Galliani) On the level of formulas:
EXC DEP
INC NDEP
INEX INDEP (Independence Logic).
Fact 2 (Galliani) On the level of sentences:
DEP[1] FO NDEP[1].
Fact 3 (Väänänen) On the level of sentences:
DEP ESO (Existential Second Order Logic).
Fact 4 (Galliani and Hella) On the level of sentences:
INC GFP (Positive Greatest Fixed Point Logic).
The expressive power of logics on the level of formulas
INEX
¡
EXC
DEP
NDEP INC
INDEP
NDEP[1]
DEP[1]
¡
FO
The expressive power of logics on the level of sentences
INC
NDEP
INDEP DEP
EXC INEX
¡
NDEP[1]
FO DEP[1]
The expressive power of logics on the level of sentences
NDEP
INC
GFP
INDEP DEP
EXC INEX
¡
NDEP[1]
FO DEP[1]
ESO
Translation between DEP and EXC
By using translations in Galliani’s proof for "DEP
we get the following results:
EXC",
Translation between DEP and EXC
By using translations in Galliani’s proof for "DEP
we get the following results:
EXC",
k-ary dependence atoms can be expressed in EXC[k]:
M (X pt1 , . . . , tk q, iff M (X @ x px
tk _ t1 . . . tk 1x | t1 . . . tk q.
Translation between DEP and EXC
By using translations in Galliani’s proof for "DEP
we get the following results:
EXC",
k-ary dependence atoms can be expressed in EXC[k]:
M (X pt1 , . . . , tk q, iff M (X @ x px
tk _ t1 . . . tk 1x | t1 . . . tk q.
k-ary exclusion atoms can be expressed in DEP[k
Ýt1 | Ñ
Ýt2 , iff M (X @ Ñ
Ýy D w1 D w2
M (X Ñ
1]:
pw1q ^ py1, . . . , yk , w2q
Ýt1 q _ pw1 w2 ^ Ñ
Ýt2 q.
Ýy Ñ
Ýy Ñ
^ pw1 w2 ^ Ñ
Translation between DEP and EXC
By using translations in Galliani’s proof for "DEP
we get the following results:
EXC",
k-ary dependence atoms can be expressed in EXC[k]:
M (X pt1 , . . . , tk q, iff M (X @ x px
tk _ t1 . . . tk 1x | t1 . . . tk q.
k-ary exclusion atoms can be expressed in DEP[k
Ýt1 | Ñ
Ýt2 , iff M (X @ Ñ
Ýy D w1 D w2
M (X Ñ
1]:
pw1q ^ py1, . . . , yk , w2q
Ýt1 q _ pw1 w2 ^ Ñ
Ýt2 q.
Ýy Ñ
Ýy Ñ
^ pw1 w2 ^ Ñ
Corollary DEP[k]
¤ EXC[k] ¤ DEP[k
1].
Expressing EXC[k] with ESO[k] (k-ary Existential Second Order Logic)
Let ϕ be EXC[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn ϕ1 ,
Expressing EXC[k] with ESO[k] (k-ary Existential Second Order Logic)
Let ϕ be EXC[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn ϕ1 ,
where ϕ1 is defined recursively:
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing EXC[k] with ESO[k] (k-ary Existential Second Order Logic)
Let ϕ be EXC[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn ϕ1 ,
where ϕ1 is defined recursively:
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing EXC[k] with ESO[k] (k-ary Existential Second Order Logic)
Let ϕ be EXC[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn ϕ1 ,
where ϕ1 is defined recursively:
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Corollary EXC[k] ¤ ESO[k].
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The following properties of graphs can be defined in EXC[1]:
Ÿ
Disconnectivity
Ÿ
k-colorability
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The following properties of graphs can be defined in EXC[1]:
Ÿ
Disconnectivity
Ÿ
k-colorability
Corollary EXC[1]
¦ DEP[1], and thus DEP[1] EXC[1].
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The following properties of graphs can be defined in EXC[1]:
Ÿ
Disconnectivity
Ÿ
k-colorability
Corollary EXC[1]
¦ DEP[1], and thus DEP[1] EXC[1].
Väänänen has shown that there are properties that can be expressed in DEP[2],
but cannot be expressed in ESO[1]:
Ÿ
Infinity of a model
Ÿ
Even cardinality of a model
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The following properties of graphs can be defined in EXC[1]:
Ÿ
Disconnectivity
Ÿ
k-colorability
Corollary EXC[1]
¦ DEP[1], and thus DEP[1] EXC[1].
Väänänen has shown that there are properties that can be expressed in DEP[2],
but cannot be expressed in ESO[1]:
Ÿ
Infinity of a model
Ÿ
Even cardinality of a model
Corollary On the level of sentences DEP[2]
¦ ESO[1], and thus EXC[1] DEP[2].
The expressive power of unary exclusion logic
We have shown before that DEP[1]
¤ EXC[1] ¤ DEP[2].
The following properties of graphs can be defined in EXC[1]:
Ÿ
Disconnectivity
Ÿ
k-colorability
Corollary EXC[1]
¦ DEP[1], and thus DEP[1] EXC[1].
Väänänen has shown that there are properties that can be expressed in DEP[2],
but cannot be expressed in ESO[1]:
Ÿ
Infinity of a model
Ÿ
Even cardinality of a model
Corollary On the level of sentences DEP[2]
So we have shown that: DEP[1]
¦ ESO[1], and thus EXC[1] DEP[2].
EXC[1] DEP[2].
Translation between NDEP and INC
k-ary nondependence atoms can be expressed in INC[k] (Galliani’s translation):
M (X pt1 , . . . , tk q, iff M (X D x px
tk ^ t1 . . . tk 1x „ t1 . . . tk q,
Translation between NDEP and INC
k-ary nondependence atoms can be expressed in INC[k] (Galliani’s translation):
M (X pt1 , . . . , tk q, iff M (X D x px
tk ^ t1 . . . tk 1x „ t1 . . . tk q,
k-ary inclusion atoms can be expressed in NDEP[k
Ýt1 „ Ñ
Ýt2 , iff M (X @ w1 @ w2 pw1
M (X Ñ
1]:
w2q
Ýt1 ^ z y1q
Ýy Ñ
Ýy D z ppw1 w2 ^ Ñ
_ @ w1 @ w2 D Ñ
Ýt2 qq ^ py1, . . . , yk , z q.
Ýy Ñ
_ pw1 w2 ^ Ñ
Translation between NDEP and INC
k-ary nondependence atoms can be expressed in INC[k] (Galliani’s translation):
M (X pt1 , . . . , tk q, iff M (X D x px
tk ^ t1 . . . tk 1x „ t1 . . . tk q,
k-ary inclusion atoms can be expressed in NDEP[k
Ýt1 „ Ñ
Ýt2 , iff M (X @ w1 @ w2 pw1
M (X Ñ
1]:
w2q
Ýt1 ^ z y1q
Ýy Ñ
Ýy D z ppw1 w2 ^ Ñ
_ @ w1 @ w2 D Ñ
Ýt2 qq ^ py1, . . . , yk , z q.
Ýy Ñ
_ pw1 w2 ^ Ñ
Corollary NDEP[k]
¤ INC[k] ¤ NDEP[k
1].
Expressing INC[k] with ESO[k]
Let ϕ be INC[k]-sentence. We label all the instances of inclusion atoms
Ýt1 „ Ñ
Ýt2 q1, . . . , pÑ
Ýt1 „ Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn
©
Ýu
ϕ1 ^
@Ñ
i 1
n
Ýu _ ϕ1i pÑ
Ýu q ,
Pi Ñ
Expressing INC[k] with ESO[k]
Let ϕ be INC[k]-sentence. We label all the instances of inclusion atoms
Ýt1 „ Ñ
Ýt2 q1, . . . , pÑ
Ýt1 „ Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn
where ϕ1 and ϕ1i (i
©
Ýu
ϕ1 ^
@Ñ
i 1
n
Ýu _ ϕ1i pÑ
Ýu q ,
Pi Ñ
P t1, . . . , nu) are defined recursively:
pψq1 ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 for all i P t1, . . . , nu
ppÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing INC[k] with ESO[k]
Let ϕ be INC[k]-sentence. We label all the instances of inclusion atoms
Ýt1 „ Ñ
Ýt2 q1, . . . , pÑ
Ýt1 „ Ñ
Ýt2 qn occuring in ϕ. Now it holds:
pÑ
M ( ϕ, iff M ( D P1 . . . D Pn
where ϕ1 and ϕ1i (i
©
Ýu
ϕ1 ^
@Ñ
i 1
n
Ýu _ ϕ1i pÑ
Ýu q ,
Pi Ñ
P t1, . . . , nu) are defined recursively:
pψq1 ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 for all i P t1, . . . , nu
ppÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing INC[k] with ESO[k]
where ϕ1 and ϕ1i (i
P t1, . . . , nu) are defined recursively:
pψq1i ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qj q1i Pj Ñ
Ýt1 , if j i
ppÑ
Ýt1 „ Ñ
Ýt2 qj q1i pÑ
Ýt2 q ^ Pj Ñ
Ýt1 , if j i
Ýu Ñ
ppÑ
pψ ^ θq1i ψi1 ^ θi1
$
1
'
if pt1 „ t2 qi is a subformula of ψ
&ψi ,
1
1
pψ _ θqi 'θi ,
if pt1 „ t2 qi is a subformula of θ
% 1
1
ψi _ θi , if pt1 „ t2 qi is not a subformula of ψ _ θ
1
pD x ψqi D x ψi1
p@ x ψq1i D x ψi1 ^ @ x ψ1
Expressing INC[k] with ESO[k]
where ϕ1 and ϕ1i (i
P t1, . . . , nu) are defined recursively:
pψq1i ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qj q1i Pj Ñ
Ýt1 , if j i
ppÑ
Ýt1 „ Ñ
Ýt2 qj q1i pÑ
Ýt2 q ^ Pj Ñ
Ýt1 , if j i
Ýu Ñ
ppÑ
pψ ^ θq1i ψi1 ^ θi1
$
1
'
if pt1 „ t2 qi is a subformula of ψ
&ψi ,
1
1
pψ _ θqi 'θi ,
if pt1 „ t2 qi is a subformula of θ
% 1
1
ψi _ θi , if pt1 „ t2 qi is not a subformula of ψ _ θ
1
pD x ψqi D x ψi1
p@ x ψq1i D x ψi1 ^ @ x ψ1
Expressing INC[k] with ESO[k]
where ϕ1 and ϕ1i (i
P t1, . . . , nu) are defined recursively:
pψq1i ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qj q1i Pj Ñ
Ýt1 , if j i
ppÑ
Ýt1 „ Ñ
Ýt2 qj q1i pÑ
Ýt2 q ^ Pj Ñ
Ýt1 , if j i
Ýu Ñ
ppÑ
pψ ^ θq1i ψi1 ^ θi1
$
1
'
if pt1 „ t2 qi is a subformula of ψ
&ψi ,
1
1
pψ _ θqi 'θi ,
if pt1 „ t2 qi is a subformula of θ
% 1
1
ψi _ θi , if pt1 „ t2 qi is not a subformula of ψ _ θ
1
pD x ψqi D x ψi1
p@ x ψq1i D x ψi1 ^ @ x ψ1
Expressing INC[k] with ESO[k]
where ϕ1 and ϕ1i (i
P t1, . . . , nu) are defined recursively:
pψq1i ψ, if ψ is a literal
Ýt1 „ Ñ
Ýt2 qj q1i Pj Ñ
Ýt1 , if j i
ppÑ
Ýt1 „ Ñ
Ýt2 qj q1i pÑ
Ýt2 q ^ Pj Ñ
Ýt1 , if j i
Ýu Ñ
ppÑ
pψ ^ θq1i ψi1 ^ θi1
$
1
'
if pt1 „ t2 qi is a subformula of ψ
&ψi ,
1
1
pψ _ θqi 'θi ,
if pt1 „ t2 qi is a subformula of θ
% 1
1
ψi _ θi , if pt1 „ t2 qi is not a subformula of ψ _ θ
1
pD x ψqi D x ψi1
p@ x ψq1i D x ψi1 ^ @ x ψ1
Corollary INC[k] ¤ ESO[k].
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
Galliani and Hella have noted that the following property of graphs
can be defined in INC[1]:
Ÿ
Finite directed graph contains a cycle
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
Galliani and Hella have noted that the following property of graphs
can be defined in INC[1]:
Ÿ
Finite directed graph contains a cycle
Corollary INC[1]
¦ NDEP[1], and thus NDEP[1] INC[1].
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
Galliani and Hella have noted that the following property of graphs
can be defined in INC[1]:
Ÿ
Finite directed graph contains a cycle
Corollary INC[1]
¦ NDEP[1], and thus NDEP[1] INC[1].
By using the Ehrenfeucht-Fraïssé game for the inclusion logic (presented by
Galliani and Hella) it can be shown that there exists no INC[1]-formula ϕ
which would be equivalent with the NDEP[2]-formula px , y q.
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
Galliani and Hella have noted that the following property of graphs
can be defined in INC[1]:
Ÿ
Finite directed graph contains a cycle
Corollary INC[1]
¦ NDEP[1], and thus NDEP[1] INC[1].
By using the Ehrenfeucht-Fraïssé game for the inclusion logic (presented by
Galliani and Hella) it can be shown that there exists no INC[1]-formula ϕ
which would be equivalent with the NDEP[2]-formula px , y q.
Corollary NDEP[2]
¦ INC[1], and thus INC[1] NDEP[2].
The expressive power of unary inclusion logic
We have shown before that NDEP[1]
¤ INC[1] ¤ NDEP[2].
Galliani and Hella have noted that the following property of graphs
can be defined in INC[1]:
Ÿ
Finite directed graph contains a cycle
Corollary INC[1]
¦ NDEP[1], and thus NDEP[1] INC[1].
By using the Ehrenfeucht-Fraïssé game for the inclusion logic (presented by
Galliani and Hella) it can be shown that there exists no INC[1]-formula ϕ
which would be equivalent with the NDEP[2]-formula px , y q.
Corollary NDEP[2]
¦ INC[1], and thus INC[1] NDEP[2].
So we have shown that: NDEP[1]
INC[1] NDEP[2].
Expressing INEX[k] with ESO[k]
Let ϕ be INEX[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ, and define ϕ1:
pÑ
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
Ýt1 „ Ñ
Ýt2 q1 Ñ
Ýt1 „ Ñ
Ýt2
pÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing INEX[k] with ESO[k]
Let ϕ be INEX[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ, and define ϕ1:
pÑ
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
Ýt1 „ Ñ
Ýt2 q1 Ñ
Ýt1 „ Ñ
Ýt2
pÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Expressing INEX[k] with ESO[k]
Let ϕ be INEX[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ, and define ϕ1:
pÑ
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
Ýt1 „ Ñ
Ýt2 q1 Ñ
Ýt1 „ Ñ
Ýt2
pÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Because ϕ1 is an INC[k]-sentence, it is equivalent with some ESO[k]-sentence µ.
Now it holds: M ( ϕ, iff M ( D P1 . . . D Pn µ.
Expressing INEX[k] with ESO[k]
Let ϕ be INEX[k]-sentence. We label all the instances of exclusion atoms
Ýt1 | Ñ
Ýt2 q1, . . . , pÑ
Ýt1 | Ñ
Ýt2 qn occuring in ϕ, and define ϕ1:
pÑ
pψq1 ψ, if ψ is a literal
Ýt1 | Ñ
Ýt2 qi q1 Pi Ñ
Ýt1 ^ Pi Ñ
Ýt2 for all i P t1, . . . , nu
ppÑ
Ýt1 „ Ñ
Ýt2 q1 Ñ
Ýt1 „ Ñ
Ýt2
pÑ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θ q1 ψ 1 _ θ 1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Because ϕ1 is an INC[k]-sentence, it is equivalent with some ESO[k]-sentence µ.
Now it holds: M ( ϕ, iff M ( D P1 . . . D Pn µ.
Corollary INEX[k]
¤ ESO[k].
Term value preserving disjunction
Ýt1 , . . . , Ñ
Ýtn be k-tuples of terms.
Let Ñ
The following connective can be expressed in INEX[k]:
1
Ñ
Ñ
M (X ϕ _Ý
t1 ...Ý
tn ψ, iff there exists Y , Y
„ X s.t.
Y Y 1 X , M (Y ϕ and M (Y ψ,
Ýti q Y 1pÑ
Ýti q X pÑ
Ýti q
and if Y , Y 1 H, then Y pÑ
for all i P t1, . . . , nu.
Y
1
Term value preserving disjunction
Expressing term value preserving disjunction in INEX[k]:
Ñ
Ñ
ϕ _Ý
t1 ...Ý
tn ψ : pϕ \ ψ q \ D y1 D y2
^ Dx
py1q ^ py2q ^ y1 y2
n
©
pθi \ θi1 q ^ px y1 ^ ϕq _ px y2 ^ ψq
i 1
P t1, . . . , nu:
Ýti qq _ px y2 ^ @ Ñ
Ýti qq
Ýz pÑ
Ýz „ Ñ
Ýz pÑ
Ýz „ Ñ
θi : px y1 ^ @ Ñ
Ýti ^ ÝwÑ2 Ñ
Ýti ^ D ÝwÑ1 D ÝwÑ2 px y1 ^ Ý
Ýu Ñ
Ñ1 Ñ
Ýu q
Ýu | Ñ
θi1 : D Ñ
w
Ýti q ^ Ñ
Ýti „ wÝÑ1 ^ Ñ
Ýti „ wÝÑ2
Ýu ^ wÝÑ2 Ñ
_ px y2 ^ wÝÑ1 Ñ
where for each i
,
Term value preserving disjunction
Expressing term value preserving disjunction in INEX[k]:
Ñ
Ñ
ϕ _Ý
t1 ...Ý
tn ψ : pϕ \ ψ q \ D y1 D y2
^ Dx
py1q ^ py2q ^ y1 y2
n
©
pθi \ θi1 q ^ px y1 ^ ϕq _ px y2 ^ ψq
i 1
P t1, . . . , nu:
Ýti qq _ px y2 ^ @ Ñ
Ýti qq
Ýz pÑ
Ýz „ Ñ
Ýz pÑ
Ýz „ Ñ
θi : px y1 ^ @ Ñ
Ýti ^ ÝwÑ2 Ñ
Ýti ^ D ÝwÑ1 D ÝwÑ2 px y1 ^ Ý
Ýu Ñ
Ñ1 Ñ
Ýu q
Ýu | Ñ
θi1 : D Ñ
w
Ýti q ^ Ñ
Ýti „ wÝÑ1 ^ Ñ
Ýti „ wÝÑ2
Ýu ^ wÝÑ2 Ñ
_ px y2 ^ wÝÑ1 Ñ
(Connective \ is the intuitionistic disjunction which can be expressed in DEP[1])
where for each i
,
Expressing ESO[k] with INEX[k]
Let D P1 . . . D Pn ϕ be ESO[k]-sentence that is satisfied, iff it is satisfied with
non-empty interpretations for P1 , . . . , Pn . Now it holds:
M ( D P1 . . . D Pn ϕ, iff
Ñ1 . . . D ÝwÑn ϕ1,
DÝ
w
Expressing ESO[k] with INEX[k]
Let D P1 . . . D Pn ϕ be ESO[k]-sentence that is satisfied, iff it is satisfied with
non-empty interpretations for P1 , . . . , Pn . Now it holds:
M ( D P1 . . . D Pn ϕ, iff
Ñ1 . . . D ÝwÑn ϕ1,
DÝ
w
where ϕ1 is defined recursively:
ψ1
ψ, if ψ is literal and Pi does not occur in ψ
Ñ
Ý
Ýt „ Ñ
1
Ýwi
pPi t q Ñ
Ñ
Ý
Ñ
Ý
Ýwi
p Pi t q1 t | Ñ
pψ ^ θ q1 ψ 1 ^ θ 1
pψ _ θq1 ψ1 _wÝÑ...wÝÑ θ1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
1
n
Expressing ESO[k] with INEX[k]
Let D P1 . . . D Pn ϕ be ESO[k]-sentence that is satisfied, iff it is satisfied with
non-empty interpretations for P1 , . . . , Pn . Now it holds:
M ( D P1 . . . D Pn ϕ, iff
Ñ1 . . . D ÝwÑn ϕ1,
DÝ
w
where ϕ1 is defined recursively:
ψ1
ψ, if ψ is literal and Pi does not occur in ψ
Ñ
Ý
Ýt „ Ñ
1
Ýwi
pPi t q Ñ
Ñ
Ý
Ñ
Ý
Ýwi
p Pi t q1 t | Ñ
pψ ^ θ q 1 ψ 1 ^ θ 1
pψ _ θq1 ψ1 _wÝÑ...wÝÑ θ1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
1
n
Expressing ESO[k] with INEX[k]
Let D P1 . . . D Pn ϕ be ESO[k]-sentence that is satisfied, iff it is satisfied with
non-empty interpretations for P1 , . . . , Pn . Now it holds:
M ( D P1 . . . D Pn ϕ, iff
Ñ1 . . . D ÝwÑn ϕ1,
DÝ
w
where ϕ1 is defined recursively:
ψ1
ψ, if ψ is literal and Pi does not occur in ψ
Ñ
Ý
Ýt „ Ñ
1
Ýwi
pPi t q Ñ
Ñ
Ý
Ñ
Ý
Ýwi
p Pi t q1 t | Ñ
pψ ^ θ q 1 ψ 1 ^ θ 1
pψ _ θq1 ψ1 _wÝÑ...wÝÑ θ1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
1
n
Expressing ESO[k] with INEX[k]
Let D P1 . . . D Pn ϕ be ESO[k]-sentence that is satisfied, iff it is satisfied with
non-empty interpretations for P1 , . . . , Pn . Now it holds:
M ( D P1 . . . D Pn ϕ, iff
Ñ1 . . . D ÝwÑn ϕ1,
DÝ
w
where ϕ1 is defined recursively:
ψ1
ψ, if ψ is literal and Pi does not occur in ψ
Ñ
Ý
Ýt „ Ñ
1
Ýwi
pPi t q Ñ
Ñ
Ý
Ñ
Ý
Ýwi
p Pi t q1 t | Ñ
pψ ^ θ q 1 ψ 1 ^ θ 1
pψ _ θq1 ψ1 _wÝÑ...wÝÑ θ1
pD x ψq1 D x ψ1
p@ x ψq1 @ x ψ1
Corollary ESO[k] ¤ INEX[k], and thus INEX[k] ESO[k].
1
n
The expressive power of logics on the level of formulas
INEX
INC
¡
DEP
EXC
NDEP
INDEP
NDEP[1]
DEP[1]
¡
FO
The expressive power of logics on the level of formulas
INEX
INC
¡
EXC
DEP
NDEP
INDEP
DEP[2]
NDEP[2]
NDEP[1]
DEP[1]
¡
FO
The expressive power of logics on the level of formulas
INEX
¡
INC
EXC
DEP
NDEP
INDEP
NDEP[2]
DEP[2]
EXC[1]
¡
NDEP[1]
DEP[1]
¡
FO
The expressive power of logics on the level of formulas
INDEP
¡
INC
EXC
DEP
NDEP
INEX
¡
INC[1]
NDEP[1]
DEP[2]
NDEP[2]
EXC[1]
¡
DEP[1]
¡
FO
The expressive power of logics on the level of formulas
INDEP
¡
INC
DEP
¡
¡
INEX[1]
NDEP[2]
INC[1]
NDEP[1]
EXC
DEP[2]
NDEP
INEX
EXC[1]
¡
DEP[1]
¡
FO
The expressive power of logics on the level of formulas
INEX
INC
¡
DEP
1]
DEP[k
1]
¤
NDEP[k
EXC
NDEP
INDEP
NDEP[k]
DEP[k]
¡
FO
The expressive power of logics on the level of formulas
INEX
¡
INC
DEP
1]
DEP[k
1]
¤
¤
NDEP[k
EXC
NDEP
INDEP
EXC[k]
¥
NDEP[k]
DEP[k]
¡
FO
The expressive power of logics on the level of formulas
INDEP
¡
INC
1]
DEP[k
¤
¥
INC[k]
NDEP[k]
1]
¤
NDEP[k
EXC
DEP
NDEP
INEX
EXC[k]
¥
¤
DEP[k]
¡
FO
The expressive power of logics on the level of formulas
INDEP
DEP[k
¤
¡
¥
INC[k]
EXC[k]
¥
¤
DEP[k]
¡
FO
1]
¤
1]
NDEP[k]
EXC
DEP
INEX[k]
NDEP[k
¡
INC
NDEP
INEX
The expressive power of logics on the level of sentences
INDEP DEP
EXC INEX
NDEP
INC
GFP
¡
FO
ESO
The expressive power of logics on the level of sentences
INDEP DEP
EXC INEX
INC
NDEP
GFP
INC[k]
EXC[k]
¡
FO
ESO
The expressive power of logics on the level of sentences
INDEP DEP
EXC INEX
INC
NDEP
INEX[k]
GFP
¥
¤
INC[k]
EXC[k]
¡
FO
ESO
The expressive power of logics on the level of sentences
INDEP DEP
EXC INEX
INC
NDEP
INEX[k]
GFP
ESO[k]
¥
¤
INC[k]
EXC[k]
¡
FO
ESO
The expressive power of logics on the level of sentences
INDEP DEP
EXC INEX
NDEP
INC
EXC[k]
INEX[k]
ESO[k]
?
INC[k]
GFP
ESO
¡
FO
Open questions and further recearch
Ÿ
On the level of sentences does INC[1]
NDEP[2]?
Open questions and further recearch
Ÿ
On the level of sentences does INC[1]
Ÿ
Does it hold for all k
PZ
#
NDEP[2]?
, that
DEP[k] EXC[k] DEP[k+1]
NDEP[k] INC[k] NDEP[k+1]
Open questions and further recearch
Ÿ
On the level of sentences does INC[1]
Ÿ
Does it hold for all k
PZ
#
Ÿ
NDEP[2]?
, that
DEP[k] EXC[k] DEP[k+1]
NDEP[k] INC[k] NDEP[k+1]
If the latter is true, is it true on the level of sentences?
Open questions and further recearch
Ÿ
On the level of sentences does INC[1]
Ÿ
Does it hold for all k
PZ
#
NDEP[2]?
, that
DEP[k] EXC[k] DEP[k+1]
NDEP[k] INC[k] NDEP[k+1]
Ÿ
If the latter is true, is it true on the level of sentences?
Ÿ
What kind of fragments of ESO[k] are INC[k] and EXC[k]?
Appendix: Some operators expressible in EXC[1]
The following operators can be expressed in EXC[1]:
Appendix: Some operators expressible in EXC[1]
The following operators can be expressed in EXC[1]:
Universal quantification over the values of a given term:
M (X p@ x
„ t q ϕ,
iff M (X rX pt q{x s ϕ.
Appendix: Some operators expressible in EXC[1]
The following operators can be expressed in EXC[1]:
Universal quantification over the values of a given term:
M (X p@ x
„ t q ϕ,
iff M (X rX pt q{x s ϕ.
Existential quantification over the complement
of the values of a given set of terms:
M (X pD x
|
n
¤
ti q ϕ,
i 1
iff there exists f : X
Ñ
n
¤
i 1
X pti q, s.t. M (X rf {x s ϕ.
Appendix: Some properties of graphs expressible in EXC[1]
pV , E q is disconnected, iff
G ( @ x D y1 pD y2 | y1 q px y1 _ x y2 q
^ p@ z1 „ y1qp@ z2 „ y2q
Undirected graph G
Ez1 z2 .
Appendix: Some properties of graphs expressible in EXC[1]
pV , E q is disconnected, iff
G ( @ x D y1 pD y2 | y1 q px y1 _ x y2 q
^ p@ z1 „ y1qp@ z2 „ y2q
Undirected graph G
Ez1 z2 .
pV , E q is k-colorable, iff
G ( @ x D y1 pD y2 | y1 q . . . pD yk | y1 Y Y yk 1 q
Undirected graph G
k
ª
i 1
x
yi ^
k
©
i 1
p@ z1 „ yi qp@ z2 „ yi q
Ez1 z2 .
Appendix: A NDEP[2]-formula which is not expressiple in INC[1]
Let L H and M be an L-model, s.t. dompMq t0, 1u.
Let X ts1 , s2 u and Y ts1 , s2 , s3 u, where
tpx , 0q, py , 0qu
s2 tpx , 0q, py , 1qu
s3 tpx , 1q, py , 1qu
s1
Appendix: A NDEP[2]-formula which is not expressiple in INC[1]
Let L H and M be an L-model, s.t. dompMq t0, 1u.
Let X ts1 , s2 u and Y ts1 , s2 , s3 u, where
tpx , 0q, py , 0qu
s2 tpx , 0q, py , 1qu
s3 tpx , 1q, py , 1qu
Clearly now M (X px , y q, but M *Y px , y q.
s1
Appendix: A NDEP[2]-formula which is not expressiple in INC[1]
Let L H and M be an L-model, s.t. dompMq t0, 1u.
Let X ts1 , s2 u and Y ts1 , s2 , s3 u, where
tpx , 0q, py , 0qu
s2 tpx , 0q, py , 1qu
s3 tpx , 1q, py , 1qu
Clearly now M (X px , y q, but M *Y px , y q.
s1
By using the Ehrenfeucht-Fraïssé game for inclusion logic we can show
that there exists no INC[1]-formula ϕ so that
M (X ϕ, but M *Y ϕ.