Descriptive complexity linear algebra of Bjarki Holm Logical - - PowerPoint PPT Presentation
Descriptive complexity linear algebra of Bjarki Holm Logical Approaches to Barriers in Computing & Complexity II Isaac Newton Institute Overview Study definability of natural problems in linear algebra and expressiveness
Logical Approaches to Barriers in Computing & Complexity II
Bjarki Holm
Isaac Newton Institute
Study definability of natural problems in linear algebra and expressiveness of logics with algebraic operators.
method
Study definability of natural problems in linear algebra and expressiveness of logics with algebraic operators.
method
Existential second-order logic Second-order variables existentially quantified, followed by a first-order formula:
∃R1, . . . , Rk . ϕ(R1, . . . , Rk)
ESO —
Existential second-order logic Second-order variables existentially quantified, followed by a first-order formula:
∃R1, . . . , Rk . ϕ(R1, . . . , Rk)
ESO —
Fagin (1974)
A decision problem is in NP if and only if it can be defined in ESO.
Existential second-order logic Second-order variables existentially quantified, followed by a first-order formula:
∃R1, . . . , Rk . ϕ(R1, . . . , Rk)
“guess”
ESO —
Fagin (1974)
A decision problem is in NP if and only if it can be defined in ESO.
Existential second-order logic Second-order variables existentially quantified, followed by a first-order formula:
∃R1, . . . , Rk . ϕ(R1, . . . , Rk)
“guess” “verify”
ESO —
Fagin (1974)
A decision problem is in NP if and only if it can be defined in ESO.
Existential second-order logic Second-order variables existentially quantified, followed by a first-order formula:
∃R1, . . . , Rk . ϕ(R1, . . . , Rk)
“guess” “verify”
ESO —
Fagin (1974)
A decision problem is in NP if and only if it can be defined in ESO. Is there a logic for PTIME?
FP is first-order logic with an inflationary fixed-point
A property P of ordered structures can be decided in PTIME if and only if P can be defined by a sentence of FP.
Immerman-Vardi (1982)
FP is first-order logic with an inflationary fixed-point
A property P of ordered structures can be decided in PTIME if and only if P can be defined by a sentence of FP.
Immerman-Vardi (1982)
Ordered structure: Vocabulary contains a binary symbol interpreted as a total ordering of the vertices.
“6”
FP is first-order logic with an inflationary fixed-point
A property P of ordered structures can be decided in PTIME if and only if P can be defined by a sentence of FP.
Immerman-Vardi (1982)
FP is first-order logic with an inflationary fixed-point
A property P of ordered structures can be decided in PTIME if and only if P can be defined by a sentence of FP.
Immerman-Vardi (1982)
graph has an even or odd number of vertices.
FP is first-order logic with an inflationary fixed-point
A property P of ordered structures can be decided in PTIME if and only if P can be defined by a sentence of FP.
Immerman-Vardi (1982)
graph has an even or odd number of vertices.
terms that count the number of solutions to formulas.
FO FP FPC PTIME
Ordered structures—1982
FO FP FPC PTIME
Ordered structures—1982
FO FP FPC PTIME
Trees—1986
Ordered structures—1982
FO FP FPC PTIME
Trees—1986 Planar graphs—1998
Ordered structures—1982
FO FP FPC PTIME
Trees—1986 Planar graphs—1998 Graphs of bounded treewidth—1999
Ordered structures—1982
FO FP FPC PTIME
Trees—1986 Planar graphs—1998 Graphs of bounded treewidth—1999 Minor-closed classes
Ordered structures—1982
FO FP FPC PTIME
Trees—1986 Planar graphs—1998 Graphs of bounded treewidth—1999 Minor-closed classes
“Almost all” graphs—1996
Ordered structures—1982
FO FP FPC PTIME
Trees—1986 Planar graphs—1998 Graphs of bounded treewidth—1999 Minor-closed classes
“Almost all” graphs—1996
???
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
G and H agree on all sentences of Ck Duplicator has a winning strategy in the k-pebble bijection game on G and H iff
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck — To show that a property P is not definable in FPC:
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck — To show that a property P is not definable in FPC: For each k, exhibit a pair of graphs Gk and Hk for which
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck — To show that a property P is not definable in FPC: For each k, exhibit a pair of graphs Gk and Hk for which
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck — To show that a property P is not definable in FPC: For each k, exhibit a pair of graphs Gk and Hk for which
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
Facts
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
Facts
game in polynomial time.
first-order logic with variables x1, ..., xk and counting quantifiers of the form
⇠ 9≥ix . ϕ
Ck —
equivalence, for some k.
bijection game
Facts
game in polynomial time.
isomorphism: Weisfeiler-Lehman method.
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992) “CFI property”
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992) “CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
“CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
(not even degree 3) “CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
(not even degree 3) “CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
(not even degree 3) (not even size 4) “CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
Still, the CFI query is hardly a natural graph property...
(not even degree 3) (not even size 4) “CFI property”
Corollary FPC does not capture PTIME on
There is a polynomial-time decidable property of finite graphs that is not definable in FPC.
Cai, Fürer and Immerman (1992)
Still, the CFI query is hardly a natural graph property... More recently: See which problems in linear algebra can be expressed in FPC
(not even degree 3) (not even size 4) “CFI property”
Corollary FPC does not capture PTIME on
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
A = (aij)
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
A = (aij) Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
rows and columns
all PTIME matrix properties can be defined in FP
A = (aij) Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
rows and columns
all PTIME matrix properties can be defined in FP
A = (aij) Many natural matrix properties invariant under permutation of rows and columns Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
rows and columns
all PTIME matrix properties can be defined in FP
A = (aij) Many natural matrix properties invariant under permutation of rows and columns Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
rows and columns
all PTIME matrix properties can be defined in FP
A = (aij) Many natural matrix properties invariant under permutation of rows and columns Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
— an m-by-n rectangular array of elements
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
rows and columns
all PTIME matrix properties can be defined in FP
A = (aij) Many natural matrix properties invariant under permutation of rows and columns (rank, determinant, etc.) Recall: Over ordered structures FP (and hence FPC) can define all polynomial-time properties.
I, J — finite and non-empty sets D — a group, a ring or a field
I, J — finite and non-empty sets D — a group, a ring or a field
I J
A : I ⇥ J ! D
I, J — finite and non-empty sets D — a group, a ring or a field
I J
“an I-by-J matrix over D”
A : I ⇥ J ! D
I J t = I J
I, J — finite and non-empty sets D — a group, a ring or a field
I J t = A x b = I J
I, J — finite and non-empty sets D — a group, a ring or a field
I J t = A x b = I J
As a relational structure over a fixed domain D:
I J t = A x b = I J
As a relational structure over a fixed domain D:
I J t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
As a relational structure over a fixed domain D:
I J t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
A0
As a relational structure over a fixed domain D:
I J t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
A0
As a relational structure over a fixed domain D:
I J t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
A1
As a relational structure over a fixed domain D:
I J
1 1 1 1 1 1 1 1 1 1 1 1 1
t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
A1
As a relational structure over a fixed domain D:
I J
1 1 1 1 1 1 1 1 1 1 1 1 1
In this talk: Focus on I = J
t = A x b = I J
) Ad ✓ I ⇥ J
d bd ✓ I
⇥ ! S = (I, J; (Ad)d∈D, (bd)d∈D)
where and
A1
Solvability of systems of linear equations over any fixed finite Abelian group is not definable in FPC.
Atserias, Bulatov and Dawar (2007)
Corollary Solvability of systems of linear equations over any fixed finite field is not definable in FPC.
Atserias, Bulatov and Dawar (2007)
Corollary Solvability of systems of linear equations over any fixed finite field is not definable in FPC. Recall: A linear system Ax = b over a field k is solvable if and only if the matrices A and (A|b) have the same rank
Atserias, Bulatov and Dawar (2007)
Corollary Solvability of systems of linear equations over any fixed finite field is not definable in FPC. Recall: A linear system Ax = b over a field k is solvable if and only if the matrices A and (A|b) have the same rank
Atserias, Bulatov and Dawar (2007)
Corollary Matrix rank over finite fields is not definable in FPC.
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field.
Dawar, H., Grohe, Laubner (2009)
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field. 2. The inverse to any invertible square matrix over Z, Q and any finite field.
Dawar, H., Grohe, Laubner (2009)
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field. 2. The inverse to any invertible square matrix over Z, Q and any finite field. 3. Rank of a matrix over Q.
Dawar, H., Grohe, Laubner (2009)
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field. 2. The inverse to any invertible square matrix over Z, Q and any finite field. 3. Rank of a matrix over Q. 4. Minimal polynomial of a square matrix over Q and any finite field.
Dawar, H., Grohe, Laubner (2009) H.-Pakusa (2010)
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field. 2. The inverse to any invertible square matrix over Z, Q and any finite field. 3. Rank of a matrix over Q. 4. Minimal polynomial of a square matrix over Q and any finite field.
Dawar, H., Grohe, Laubner (2009)
Fundamental linear-algebraic property over fields that separates FPC from PTIME: rank over finite fields
H.-Pakusa (2010)
1. Characteristic polynomial and determinant of a square matrix over Z, Q and any finite field. 2. The inverse to any invertible square matrix over Z, Q and any finite field. 3. Rank of a matrix over Q. 4. Minimal polynomial of a square matrix over Q and any finite field.
Dawar, H., Grohe, Laubner (2009)
Fundamental linear-algebraic property over fields that separates FPC from PTIME: rank over finite fields
H.-Pakusa (2010)
(Next talk: solvability problems over groups and rings)
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
ϕ(x, y)
formula graph G = (V, E)
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
V V
ϕ(x, y)
formula graph G = (V, E)
) MG
ϕ : (over GF(2))
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
V V
ϕ(x, y)
formula graph G = (V, E)
) MG
ϕ : (over GF(2))
u v
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
V V
(u, v) 7! ( 1 if G | = ϕ[u, v],
ϕ(x, y)
formula graph G = (V, E)
) MG
ϕ : (over GF(2))
u v
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
V V
(u, v) 7! ( 1 if G | = ϕ[u, v],
ϕ(x, y)
formula graph G = (V, E)
) MG
ϕ : (over GF(2))
u v
Example:
) MG
ϕ = adjacency matrix of G
I ϕ(x, y) := E(x, y)
Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
V V
(u, v) 7! ( 1 if G | = ϕ[u, v],
ϕ(x, y)
formula graph G = (V, E)
) MG
ϕ : (over GF(2))
u v
Example:
) MG
ϕ = adjacency matrix of G
I ϕ(x, y) := E(x, y)
More generally: formalise matrices over GF(p), p prime Recall: View any A in I x I as a matrix over GF(2).
) A ✓ I ⇥ I
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
vertex variables: range
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
vertex variables: range
number variables: range over N
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
vertex variables: range
number variables: range over N
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
(p prime)
vertex variables: range
number variables: range over N
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
(p prime)
Semantics: over GF(p)
(rkp(x, y).ϕ)G := rank(MG
ϕ )
vertex variables: range
number variables: range over N
Variables are typed:
1 2 3 4 5 6 7 ...
N
G = (V, E)
(p prime)
Semantics: over GF(p)
(rkp(x, y).ϕ)G := rank(MG
ϕ )
Logics FPRp, FPR and similarly FORp, FOR
vertex variables: range
number variables: range over N
For any prime p, FPRp can express solvability of linear equations over GF(p).
Dawar, Grohe, H., Laubner (2009)
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
t
=
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
t
=
Represent each element of GF(pm) as an m-by-m matrix over GF(p)
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
t
=
equivalent system over GF(p)
t
=
Represent each element of GF(pm) as an m-by-m matrix over GF(p)
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
t
=
equivalent system over GF(p)
t
=
Corollary For any prime p, FPC ⊊ FPRp ⊆ PTIME.
Represent each element of GF(pm) as an m-by-m matrix over GF(p)
For any prime p, FPRp can express solvability of linear equations over GF(pm) for any m.
t
=
equivalent system over GF(p)
t
=
Corollary For any prime p, FPC ⊊ FPRp ⊆ PTIME.
(we can simulate counting by expressing rank of diagonal matrices)
Represent each element of GF(pm) as an m-by-m matrix over GF(p)
Non-isomorphic CFI graphs can be distinguished by a sentence of FOR2.
Dawar, Grohe, H., Laubner (2009)
Non-isomorphic CFI graphs can be distinguished by a sentence of FOR2.
Dawar, Grohe, H., Laubner (2009)
Recall: FPC does not capture PTIME on graphs of bounded colour-class size not even size 4
Non-isomorphic CFI graphs can be distinguished by a sentence of FOR2.
Dawar, Grohe, H., Laubner (2009)
Isomorphism of graphs of colour class size 4 can be expressed in FOR2.
Dawar, H. (2011)
Recall: FPC does not capture PTIME on graphs of bounded colour-class size not even size 4
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method.
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list:
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list: A pebble game for finite-variable rank logics for which...
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list: A pebble game for finite-variable rank logics for which...
polynomial time, and
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list: A pebble game for finite-variable rank logics for which...
polynomial time, and
algorithm”, like for the counting game on graphs.
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list: A pebble game for finite-variable rank logics for which...
polynomial time, and
algorithm”, like for the counting game on graphs.
matrix-rank game
Recall: Proofs of inexpressibility in FPC are generally formulated using a game method. Our wish list: A pebble game for finite-variable rank logics for which...
polynomial time, and
algorithm”, like for the counting game on graphs.
matrix-rank game invertible- map game
first-order logic with variables x1, ..., xk and rank quantifiers of the form — [ rk≥i
p (x, y) . (')
[ Rk
p
first-order logic with variables x1, ..., xk and rank quantifiers of the form — [ rk≥i
p (x, y) . (')
[ Rk
p
equivalence, for some k.
[ Rk
p
first-order logic with variables x1, ..., xk and rank quantifiers of the form — [ rk≥i
p (x, y) . (')
[ Rk
p
matrix-rank game (over GF(p))
[ Rk
p
equivalence, for some k.
[ Rk
p
first-order logic with variables x1, ..., xk and rank quantifiers of the form —
G and H agree on all sentences of k-variable rank logic over GF(p) Duplicator has a winning strategy in the k-pebble matrix- rank game on G and H iff
[ rk≥i
p (x, y) . (')
[ Rk
p
matrix-rank game (over GF(p))
[ Rk
p
equivalence, for some k.
[ Rk
p
Game played on finite graphs G and H
Game played on finite graphs G and H
into disjoint {0,1}-matrices (“partition matrices”).
Game played on finite graphs G and H
into disjoint {0,1}-matrices (“partition matrices”).
has to ensure that every linear combination of partition matrices over G has the same GF(p)-rank as the corresponding linear combination over H.
Game played on finite graphs G and H
into disjoint {0,1}-matrices (“partition matrices”).
has to ensure that every linear combination of partition matrices over G has the same GF(p)-rank as the corresponding linear combination over H.
Problem: Not known if we can decide in polynomial time which player wins the game.
Game played on finite graphs G and H
into disjoint {0,1}-matrices (“partition matrices”).
has to ensure that every linear combination of partition matrices over G has the same GF(p)-rank as the corresponding linear combination over H.
Problem: Not known if we can decide in polynomial time which player wins the game.
Two tuples (A1, A2, ..., Am) and (B1, B2, ..., Bm) of n-by-n matrices over a field k are simultaneously similar if there is an invertible S such that S Ai S-1 = Bi for all i.
Two tuples (A1, A2, ..., Am) and (B1, B2, ..., Bm) of n-by-n matrices over a field k are simultaneously similar if there is an invertible S such that S Ai S-1 = Bi for all i. There is a deterministic algorithm that, given two m- tuples A and B of n-by-n matrices over a finite field GF(q), determines in time poly(n, m, q) whether A and B are simultaneously similar.
Chistov, Karpinsky and Ivanyov (1997)
Invertible-map game on G and H over GF(p):
disjoint {0,1}-matrices (“partition matrices”).
Invertible-map game on G and H over GF(p):
disjoint {0,1}-matrices (“partition matrices”).
ensure that the two tuples of partition matrices (over G and H) are simultaneously similar over GF(p).
Invertible-map game on G and H over GF(p):
disjoint {0,1}-matrices (“partition matrices”).
ensure that the two tuples of partition matrices (over G and H) are simultaneously similar over GF(p). Facts:
Invertible-map game on G and H over GF(p):
disjoint {0,1}-matrices (“partition matrices”).
ensure that the two tuples of partition matrices (over G and H) are simultaneously similar over GF(p). Facts:
Invertible-map game on G and H over GF(p):
disjoint {0,1}-matrices (“partition matrices”).
ensure that the two tuples of partition matrices (over G and H) are simultaneously similar over GF(p). Facts:
pebble invertible-map game on G and H then she also wins the k-pebble matrix rank game on G and H.
[ Rk
p
Recall: Our wish list: A pebble game for finite-variable rank logics for which...
polynomial time, and
algorithm”, like for the counting game on graphs.
matrix-rank game invertible- map game
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
u u1 u2 u3 v v1 v2 v3
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
u u1 u2 u3 v v1 v2 v3
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
Initial:
u ∼0 v
∼ ⊇ ∼ ⊇ ⊇ ∼ v deg(u) = deg(v)
iff
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
Initial:
u ∼0 v
∼ ⊇ ∼ ⊇ ⊇ ∼ v deg(u) = deg(v)
iff
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
3 2 1 1 1 1 3 3 2 1 1 1 2
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
Initial:
u ∼0 v
∼ ⊇ ∼ ⊇ ⊇ ∼ v deg(u) = deg(v)
iff Refine: iff
⇡ ⇠ ◆ ⇠ ◆ ◆ ⇠ kN(u) \ αk = kN(v) \ αk
and for all α 2 V/ ⇠i:
u ⇠i+1 v u ⇠i v
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
3 2 1 1 1 1 3 3 2 1 1 1 2
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
Initial:
u ∼0 v
∼ ⊇ ∼ ⊇ ⊇ ∼ v deg(u) = deg(v)
iff Refine: iff
⇡ ⇠ ◆ ⇠ ◆ ◆ ⇠ kN(u) \ αk = kN(v) \ αk
and for all α 2 V/ ⇠i:
u ⇠i+1 v u ⇠i v
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
3 2 1 1 1 1 3 3 2 1 1 1 2
⇠ ↵ = {w | deg(w) = 2}
Input: Graph G = (V, E) Output: Equivalence relation on V.
C ≈
“colour refinement”
Initial:
u ∼0 v
∼ ⊇ ∼ ⊇ ⊇ ∼ v deg(u) = deg(v)
iff Refine: iff
⇡ ⇠ ◆ ⇠ ◆ ◆ ⇠ kN(u) \ αk = kN(v) \ αk
and for all α 2 V/ ⇠i:
u ⇠i+1 v u ⇠i v
d ⇠0 ◆ ⇠1 ◆ . . . ◆ ⇠m = ⇠m+1 =: ⇡
Inductively define:
u u1 u2 u3 v v1 v2 v3
3 2 1 1 1 1 3 3 2 1 1 1 2
⇠ ↵ = {w | deg(w) = 2}
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
such that ; else “isomorphic”.
G ˙ [ H ⇠ ↵ 2 G ˙ [ H/ ⇡ s.t. k↵ \ VGk 6= k↵ \ VHk
C ≈
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
such that ; else “isomorphic”.
G ˙ [ H ⇠ ↵ 2 G ˙ [ H/ ⇡ s.t. k↵ \ VGk 6= k↵ \ VHk
C ≈
Some facts:
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
such that ; else “isomorphic”.
G ˙ [ H ⇠ ↵ 2 G ˙ [ H/ ⇡ s.t. k↵ \ VGk 6= k↵ \ VHk
C ≈
Some facts:
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
such that ; else “isomorphic”.
G ˙ [ H ⇠ ↵ 2 G ˙ [ H/ ⇡ s.t. k↵ \ VGk 6= k↵ \ VHk
C ≈
Some facts:
Input: Graphs G = (VG, EG) and H = (VH, EH) Output: “isomorphic” or “not isomorphic”
such that ; else “isomorphic”.
G ˙ [ H ⇠ ↵ 2 G ˙ [ H/ ⇡ s.t. k↵ \ VGk 6= k↵ \ VHk
C ≈
Some facts:
One-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
k 0 i < k
Γi(~ u, ↵) := {w 2 V | ~ u w
i 2 ↵}
˙
One-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
k 0 i < k
Γi(~ u, ↵) := {w 2 V | ~ u w
i 2 ↵}
˙
u a v b w
Example: Let k = 3 and
2 ↵ := {(x, y, z) 2 V 3 | (x, y, z) = }
One-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
k 0 i < k
Γi(~ u, ↵) := {w 2 V | ~ u w
i 2 ↵}
˙
u a v b w
Example: Let k = 3 and
2 ↵ := {(x, y, z) 2 V 3 | (x, y, z) = }
{ 2 Γ0(uvw, ↵) = {a, b} Γ1(uvw, ↵) = ;
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Refine: iff and for all
k 0 i < k
for all there is a bijection
2 ⇠ f : V ! V s.t. ! f : Γi(~ u, ↵) 7! Γi(~ v, ↵)
⇠ ↵ 2 V k/ ⇠m
⇠ ~ u ⇠m+1 ~ v
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Γi(~ u, ↵) := {w 2 V | ~ u w
i 2 ↵}
˙
Refine: iff and for all
k 0 i < k
for all there is a bijection
2 ⇠ f : V ! V s.t. ! f : Γi(~ u, ↵) 7! Γi(~ v, ↵)
⇠ ↵ 2 V k/ ⇠m
⇠ ~ u ⇠m+1 ~ v
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Theorem: iff they agree on all Ck-formulas in G.
Γi(~ u, ↵) := {w 2 V | ~ u w
i 2 ↵}
˙
Refine: iff and for all
k 0 i < k
for all there is a bijection
2 ⇠ f : V ! V s.t. ! f : Γi(~ u, ↵) 7! Γi(~ v, ↵)
⇠ ↵ 2 V k/ ⇠m
⇠ ~ u ⇠m+1 ~ v
As before: compute k-dimensional WL* refinement and compare across the two graphs. PTIME for fixed k: k-dim WL* runs in time O(nk+1 log(n)).
As before: compute k-dimensional WL* refinement and compare across the two graphs. PTIME for fixed k: k-dim WL* runs in time O(nk+1 log(n)). There exists a sequence of pairs {(Gn, Hn)}n of non- isomorphic graphs for which it holds that:
algorithm.
Cai, Fürer and Immerman (1992)
Two-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
2 [ ⇡ k \ k 6 k \ k Γij(~ u, ↵) := {(a, b) 2 V ⇥ V | ~ u a
i b j 2 ↵} ✓ V ⇥ V
˙
k 0 i 6= j < k
Two-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
2 [ ⇡ k \ k 6 k \ k Γij(~ u, ↵) := {(a, b) 2 V ⇥ V | ~ u a
i b j 2 ↵} ✓ V ⇥ V
˙
{0,1}-matrix
k 0 i 6= j < k
Two-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
2 [ ⇡ k \ k 6 k \ k Γij(~ u, ↵) := {(a, b) 2 V ⇥ V | ~ u a
i b j 2 ↵} ✓ V ⇥ V
˙
{0,1}-matrix
Example: Let k = 3 and
2 ↵ := {(x, y, z) 2 V 3 | (x, y, z) = }
u a v b w c d e
k 0 i 6= j < k
Two-element extensions in G = (V, E) For , a k-tuple and , let:
[ ↵ ✓ V k
[ ~ u 2 V k
2 [ ⇡ k \ k 6 k \ k Γij(~ u, ↵) := {(a, b) 2 V ⇥ V | ~ u a
i b j 2 ↵} ✓ V ⇥ V
˙
{0,1}-matrix
Example: Let k = 3 and
2 ↵ := {(x, y, z) 2 V 3 | (x, y, z) = }
u a v b w c d e
⇡ Γ12:
u v w a b c d e u v w a b c d e 1 1 1 1 1 1 1 1
k 0 i 6= j < k
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Input: Graph G = (V, E) Output: Equivalence relation on Vk.
C ≈
Initial: iff
atpG(~ u) = atpG(~ v)
G
~ u ⇠0 ~ v
Refine: iff and for all for all there is s.t.
k 0 i 6= j < k
S 2 GLV (GF(p))
2 S · Γij(~ u, ↵) · S−1 = Γij(~ v, ↵)
⇠ ↵ 2 V k/ ⇠m
⇠ ~ u ⇠m+1 ~ v
Similar to WL: compute k-dimensional IM refinement and compare across the two graphs (here over GF(p))
Similar to WL: compute k-dimensional IM refinement and compare across the two graphs (here over GF(p))
◆ ◆
Similar to WL: compute k-dimensional IM refinement and compare across the two graphs (here over GF(p)) For each k and prime p, there is a pair of non-isomorphic graphs that can be distinguished by 3-dim IMp but not by k-dim WL*.
Dawar and H. (2012)
◆ ◆
Similar to WL: compute k-dimensional IM refinement and compare across the two graphs (here over GF(p)) For each k and prime p, there is a pair of non-isomorphic graphs that can be distinguished by 3-dim IMp but not by k-dim WL*.
Dawar and H. (2012)
◆ ◆ For each k and distinct primes p and q, there is a pair of non-isomorphic graphs that can be distinguished by 3- dim IMp but not by k-dim IMq.
Consider the invertible-map algorithm for larger matrices (higher arity) and finite sets of primes. Can we give instances where the general algorithm fails to express graph isomorphism?
For formula , integer n and prime p, let denote the GF(p)-rank of the matrix defined by
ϕ(x, y) ϕ(x, y)
C rp
ϕ(n)
For formula , integer n and prime p, let denote the GF(p)-rank of the matrix defined by
ϕ(x, y) ϕ(x, y)
C rp
ϕ(n)
Polynomial-rank conjecture For each and each prime p, there are unary polynomials f0, ..., fp-1 such that for all (sufficiently large) n congruent to i modulo p.
ϕ(x, y)
C rp
ϕ(n) = fi(n)
For formula , integer n and prime p, let denote the GF(p)-rank of the matrix defined by
ϕ(x, y) ϕ(x, y)
C rp
ϕ(n)
Polynomial-rank conjecture For each and each prime p, there are unary polynomials f0, ..., fp-1 such that for all (sufficiently large) n congruent to i modulo p.
ϕ(x, y)
C rp
ϕ(n) = fi(n)
True for:
(x1, x2)
(y1, y2)
For formula , integer n and prime p, let denote the GF(p)-rank of the matrix defined by
ϕ(x, y) ϕ(x, y)
C rp
ϕ(n)
Polynomial-rank conjecture For each and each prime p, there are unary polynomials f0, ..., fp-1 such that for all (sufficiently large) n congruent to i modulo p.
ϕ(x, y)
C rp
ϕ(n) = fi(n)
True for:
(x1, x2) (y1, y2, y2, ..., yn)
Kirsten (2012)
For formula , integer n and prime p, let denote the GF(p)-rank of the matrix defined by
ϕ(x, y) ϕ(x, y)
C rp
ϕ(n)
Polynomial-rank conjecture For each and each prime p, there are unary polynomials f0, ..., fp-1 such that for all (sufficiently large) n congruent to i modulo p.
ϕ(x, y)
C rp
ϕ(n) = fi(n)
???
(x1, x2, ..., xn) (y1, y2, y2, ..., yn)
Kirsten (2012)
Consider classes on which we know that FPC does not capture PTIME:
to a natural logic? More open problems to come in the next talk!