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Topics in Logic and Automata TheoryLogic and Automata over Graphs
Abdullah Abdul Khadir
Chennai Mathematical [email protected]
May 14, 2010
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Notations and Symbols
Henceforth we assume the following :-
• σ is the vocabulary σ=(R1,...,Rm,c1,...,cs ) where,∀i ∈ {1, ..,m},Ri is a relation symbol of arity ki , for someki ∈ N and ∀i ∈ {1, .., s}, ci is a unique constant symbol.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Notations and Symbols
Henceforth we assume the following :-
• σ is the vocabulary σ=(R1,...,Rm,c1,...,cs ) where,∀i ∈ {1, ..,m},Ri is a relation symbol of arity ki , for someki ∈ N and ∀i ∈ {1, .., s}, ci is a unique constant symbol.
• A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ) and B=(B,RB1 ,...,RB
m,cB1 ,...,cB
s ) aretwo structures interpreting σ over the domains A and B
respectively.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Notations and Symbols
Henceforth we assume the following :-
• σ is the vocabulary σ=(R1,...,Rm,c1,...,cs ) where,∀i ∈ {1, ..,m},Ri is a relation symbol of arity ki , for someki ∈ N and ∀i ∈ {1, .., s}, ci is a unique constant symbol.
• A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ) and B=(B,RB1 ,...,RB
m,cB1 ,...,cB
s ) aretwo structures interpreting σ over the domains A and B
respectively.
• GA and GB are the gaiffman graphs (explained in the nextslide) for A and B respectively.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Notations and Symbols
Henceforth we assume the following :-
• σ is the vocabulary σ=(R1,...,Rm,c1,...,cs ) where,∀i ∈ {1, ..,m},Ri is a relation symbol of arity ki , for someki ∈ N and ∀i ∈ {1, .., s}, ci is a unique constant symbol.
• A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ) and B=(B,RB1 ,...,RB
m,cB1 ,...,cB
s ) aretwo structures interpreting σ over the domains A and B
respectively.
• GA and GB are the gaiffman graphs (explained in the nextslide) for A and B respectively.
• Given an element a ∈ A, N(A,a)↾d is the neighbourhood orsphere or subgraph of the Gaiffman graph of A, GA, with a ascenter and a radius of d.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
• A is the domain of the structure A.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
• A is the domain of the structure A.
• E is a binary relation on A such that for any two elementsa,b ∈ A, E(a,b) holds iff ∃ a relation RA
i of arity ki and ki
variables {x1, ..., xki} ∈ A such that R(x1, ..., xki
) holds.Moreover a and b are two of the ki variables.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
• A is the domain of the structure A.
• E is a binary relation on A such that for any two elementsa,b ∈ A, E(a,b) holds iff ∃ a relation RA
i of arity ki and ki
variables {x1, ..., xki} ∈ A such that R(x1, ..., xki
) holds.Moreover a and b are two of the ki variables.
Example : A = ({1,2,3,4},≤)
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
• A is the domain of the structure A.
• E is a binary relation on A such that for any two elementsa,b ∈ A, E(a,b) holds iff ∃ a relation RA
i of arity ki and ki
variables {x1, ..., xki} ∈ A such that R(x1, ..., xki
) holds.Moreover a and b are two of the ki variables.
Example : A = ({1,2,3,4},≤)
1 2 3 4
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Gaiffman Graph
Given a structure A=(A,RA1 ,...,RA
m,cA1 ,...,cA
s ), the Gaiffman graph isthe undirected graph GA = (A,E) where
• A is the domain of the structure A.
• E is a binary relation on A such that for any two elementsa,b ∈ A, E(a,b) holds iff ∃ a relation RA
i of arity ki and ki
variables {x1, ..., xki} ∈ A such that R(x1, ..., xki
) holds.Moreover a and b are two of the ki variables.
Example : A = ({1,2,3,4},≤)
1 2 3 4
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Local Equivalence
Some points and terms to note related to graphs :
• For any d ∈ N, the number of spheres of radius d is finite.
• Let n ∈ N, be the number of spheres for a fixed radius d.
• Then we can talk of a type signature of a graph given by(#Type1,...,#Typen) which is the number of spheres of radiusd that are of Type1,Type2 ... Typen.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Local Equivalence
Some points and terms to note related to graphs :
• For any d ∈ N, the number of spheres of radius d is finite.
• Let n ∈ N, be the number of spheres for a fixed radius d.
• Then we can talk of a type signature of a graph given by(#Type1,...,#Typen) which is the number of spheres of radiusd that are of Type1,Type2 ... Typen.
Definition (Local d-Equivalence)
Two structures A and B are said to be locally d-equivalent forsome d ∈ N, iff both A and B have the same type signature ofradius d. Let it be denoted by A ∼d B.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Equivalence
Definition (Logical r-Equivalence)
Two structures A and B are said to be logically r-equivalent forsome r ∈ N, iff they satisfy the same first order formulae ofquantifier depth r. Let it be denoted by A ≡r B.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Equivalence
Definition (Logical r-Equivalence)
Two structures A and B are said to be logically r-equivalent forsome r ∈ N, iff they satisfy the same first order formulae ofquantifier depth r. Let it be denoted by A ≡r B.
We recall that :
• If A ≡r B then the Duplicator has a winning strategy for ther-round EF-game.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Equivalence
Definition (Logical r-Equivalence)
Two structures A and B are said to be logically r-equivalent forsome r ∈ N, iff they satisfy the same first order formulae ofquantifier depth r. Let it be denoted by A ≡r B.
We recall that :
• If A ≡r B then the Duplicator has a winning strategy for ther-round EF-game.
• The above statement holds in both directions namely,A ≡r B ⇐⇒ Duplicator has a winning strategy for ther-round EF game.
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
Essentially, what Hanf’s theorem states is that for sufficiently largeradius d, local equivalence is the same as logical equivalence.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
Essentially, what Hanf’s theorem states is that for sufficiently largeradius d, local equivalence is the same as logical equivalence.
Proof sketch :-
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
Essentially, what Hanf’s theorem states is that for sufficiently largeradius d, local equivalence is the same as logical equivalence.
Proof sketch :-
• The duplicator’s strategy in Round 1
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
Essentially, what Hanf’s theorem states is that for sufficiently largeradius d, local equivalence is the same as logical equivalence.
Proof sketch :-
• The duplicator’s strategy in Round 1
• The duplicator’s strategy in Round i
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Hanf’s theorem
Theorem (Hanf’s)
Let d,r ∈ N such that d ≥ 3r−1.
Then, A ∼d B =⇒ A ≡r B .
Essentially, what Hanf’s theorem states is that for sufficiently largeradius d, local equivalence is the same as logical equivalence.
Proof sketch :-
• The duplicator’s strategy in Round 1
• The duplicator’s strategy in Round i
• Variations of hanf’s theorem
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The EF-Game GraphsGA GB
0
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0
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89
10
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Round 1GA GB
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• The Spoiler chooses a vertex from any graph (here, A)
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Round 1GA GB
a1
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8 9
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89
10
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• The Spoiler chooses a vertex from any graph (here, A)
• The d-neighbourhood of a in A, denoted N(A,a)↾d is of one of the types{Type1, ... Typen}
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Round 1GA GB
a1
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8 9
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b1
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89
10
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• The Spoiler chooses a vertex from any graph (here, A)
• The d-neighbourhood of a in A, denoted N(A,a)↾d is of one of the types{Type1, ... Typen}
• The Duplicator picks an element from the other structure (here, b ∈ B) suchthat N(A,a)↾d ∼= N(B,b)↾d
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Round i, Case 1GA GB
a1
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b1
2
3
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89
10
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• In Round i, the first case is when the Spoiler picks a vertex that is within 2*3i−2
of any previously selected point (maybe more than one).
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Round i, Case 1GA GB
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b1
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89
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• In Round i, the first case is when the Spoiler picks a vertex that is within 2*3i−2
of any previously selected point (maybe more than one).
• Then the Duplicator will use the isomorphism of the d-radius sphere around anyone of the centres to obtain a similar vertex on the other graph.
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Round i, Case 1GA GB
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bb1
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89
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• The reason why it works is because, as shown in the figure, in subsequent i-1rounds the Spoiler will not be able to get out of the isomorphism of thed-sphere around the previously selected points.
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Round i, Case 1GA GB
a
9
3i−2 2*3i−2
3*3i−2
b
9
• The reason why it works is because, as shown in the figure, in subsequent i-1rounds the Spoiler will not be able to get out of the isomorphism of thed-sphere around the respective previously selected points.
• This is due to the fact that ∀ i ∈ N, 3i−2 ≥ (i-1).
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Round i, Case 2GA GB
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• Now, the other case is if the Spoiler picks a vertex that is outside 2*3i−2 of allpreviously selected points.
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Round i, Case 2GA GB
a12
b12
• Now, the other case is if the Spoiler picks a vertex that is outside 2*3i−2 of allpreviously selected points.
• Then, as in Round 1, the Duplicator will be able to pick a vertex b ∈ B suchthat N(A,a)↾d ∼= N(B,b)↾d. Also, this particular point b is not in the range of2*3i−2 distance of any other previously selected point in GB.
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Variations of Hanf’s theorem
Definition
Given d,t ∈ N, we can define the concept of type signatures of radius d withthreshold t such that the values (#Type1,...,#Typen) are counted only upto athreshold t and anything ≥ t is considered ∞. Two structures A and B, aresaid to be d-equivalent with threshold t if their type signatures with radius dare equal. It is denoted A ∼d,t B.
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Variations of Hanf’s theorem
Definition
Given d,t ∈ N, we can define the concept of type signatures of radius d withthreshold t such that the values (#Type1,...,#Typen) are counted only upto athreshold t and anything ≥ t is considered ∞. Two structures A and B, aresaid to be d-equivalent with threshold t if their type signatures with radius dare equal. It is denoted A ∼d,t B.
Theorem
Given d ∈ N and two structures A and B, if A ∼d B then there exists a fixedt∈ N such that A ∼d,t B.
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Variations of Hanf’s theorem
Definition
Given d,t ∈ N, we can define the concept of type signatures of radius d withthreshold t such that the values (#Type1,...,#Typen) are counted only upto athreshold t and anything ≥ t is considered ∞. Two structures A and B, aresaid to be d-equivalent with threshold t if their type signatures with radius dare equal. It is denoted A ∼d,t B.
Theorem
Given d ∈ N and two structures A and B, if A ∼d B then there exists a fixedt∈ N such that A ∼d,t B.
Theorem (Hanf’s theorem for EMSO)
• Let φ be an EMSO formula with n second-order quantifiers given by,φ = ∃X1 ... ∃Xnψ(X1, ...,Xn), where ψ is a pure first order sentence.
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Variations of Hanf’s theorem
Definition
Given d,t ∈ N, we can define the concept of type signatures of radius d withthreshold t such that the values (#Type1,...,#Typen) are counted only upto athreshold t and anything ≥ t is considered ∞. Two structures A and B, aresaid to be d-equivalent with threshold t if their type signatures with radius dare equal. It is denoted A ∼d,t B.
Theorem
Given d ∈ N and two structures A and B, if A ∼d B then there exists a fixedt∈ N such that A ∼d,t B.
Theorem (Hanf’s theorem for EMSO)
• Let φ be an EMSO formula with n second-order quantifiers given by,φ = ∃X1 ... ∃Xnψ(X1, ...,Xn), where ψ is a pure first order sentence.
• If we consider the extended models of A,A′=(A × 2{0,1,...k},RA
1 ,...,RAm ,cA
1 ,...,cAs ), then we can reuse Hanf’s theorem as only the ψ part
remains to be interpreted over these modified structures.
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MSO and automata over pictures
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MSO and automata over pictures
Definition (Pictures, Picture Languages)
• A picture, p, over an alphabet Σ is basically a function of the formp : {1,2,..., n} × {1,2,...,m} → Σ, for any n,m ∈ N
• The set of all pictures (over Σ) is the set of all possible functions p forevery n,m ∈ N. It is denoted by Σ∗∗
• A picture language is a subset of Σ∗∗
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MSO and automata over pictures
Definition (Pictures, Picture Languages)
• A picture, p, over an alphabet Σ is basically a function of the formp : {1,2,..., n} × {1,2,...,m} → Σ, for any n,m ∈ N
• The set of all pictures (over Σ) is the set of all possible functions p forevery n,m ∈ N. It is denoted by Σ∗∗
• A picture language is a subset of Σ∗∗
p :
q :
r :
a b
b a
a a b
a b a
a a a a a a a
b b b b b b b
a a a a a a a
b b b b b b b
a a a a a a a
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MSO and automata over pictures
Definition (Pictures, Picture Languages)
• A picture, p, over an alphabet Σ is basically a function of the formp : {1,2,..., n} × {1,2,...,m} → Σ, for any n,m ∈ N
• The set of all pictures (over Σ) is the set of all possible functions p forevery n,m ∈ N. It is denoted by Σ∗∗
• A picture language is a subset of Σ∗∗
• If p is a picture of size (m,n), then bp is the picture p surrounded by aspecial boundary symbol # /∈ Σ
p : r :
#
#
#
#
#
#
#
#
# #
# #
a b
b a
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# # # # # # #
# # # # # # #
a a a a a a a
b b b b b b b
a a a a a a a
b b b b b b b
a a a a a a a
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Row Concatenation of 2 pictures
p : q :r :
p ⊖ r :p ⊖ q : Undefinedq ⊖ r : Undefined
(As the number of columns are incompatible)
a b a
b a a
a b
b a
a b b
b a b
b b a
a b a
b a a
a b b
b a b
b b a
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Column Concatenation of 2 pictures
p : q :r :
p ⊘ r : Undefinedq ⊘ r : Undefined
(As the number of rows are incompatible)
p ⊘ q:
a b a
b a a
a b
b a
a b b
b a b
b b a
a b a a b
b a a b a
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Row and Column Concatenation of 2 pictures
p : q :r :
p ⊖ r :p ⊘ q:
a b a
b a a
a b
b a
a b b
b a b
b b a
a b a a b
b a a b a
a b a
b a a
a b b
b a b
b b a
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
• Column Kleene Closure of L, L∗⊘ = ∪iL⊘i
• Row Kleene Closure of L, L∗⊖ = ∪iL⊖i
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
• Column Kleene Closure of L, L∗⊘ = ∪iL⊘i
• Row Kleene Closure of L, L∗⊖ = ∪iL⊖i
Definition (Projections)
• Let Σ1 and Σ2 be two finite alphabets such that | Σ1 | ≥ | Σ2 | andπ : Σ1 → Σ2 is a mapping.
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
• Column Kleene Closure of L, L∗⊘ = ∪iL⊘i
• Row Kleene Closure of L, L∗⊖ = ∪iL⊖i
Definition (Projections)
• Let Σ1 and Σ2 be two finite alphabets such that | Σ1 | ≥ | Σ2 | andπ : Σ1 → Σ2 is a mapping.
• Then given p ∈ Σ∗∗1 , π(p) is the picture p’ ∈ Σ∗∗
2 such thatp’(i,j) = π(p(i , j)) ∀1 ≤ i ≤ l1(p), 1 ≤ j ≤ l2(p)
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
• Column Kleene Closure of L, L∗⊘ = ∪iL⊘i
• Row Kleene Closure of L, L∗⊖ = ∪iL⊖i
Definition (Projections)
• Let Σ1 and Σ2 be two finite alphabets such that | Σ1 | ≥ | Σ2 | andπ : Σ1 → Σ2 is a mapping.
• Then given p ∈ Σ∗∗1 , π(p) is the picture p’ ∈ Σ∗∗
2 such thatp’(i,j) = π(p(i , j)) ∀1 ≤ i ≤ l1(p), 1 ≤ j ≤ l2(p)
• Similarly, given a picture language L ⊆ Σ∗∗1 , the projection of L by
π : Σ1 → Σ2 is defined as π(L) = {π(p) | p ∈ L} ⊆ Σ∗∗2
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Operations on Picture Languages
Definition
• Let L,L1 and L2 be 3 picture languages (subsets of Σ∗∗)
• Then, L1 ⊘ L2 = {x⊘y ‖ x ∈ L1 and y ∈ L2}. Similarly for L1 ⊖ L2.
• L⊘1 = L ; L⊘n = L⊘(n−1)⊘ L. Similarly for L⊖n.
• Column Kleene Closure of L, L∗⊘ = ∪iL⊘i
• Row Kleene Closure of L, L∗⊖ = ∪iL⊖i
Definition (Projections)
• Let Σ1 and Σ2 be two finite alphabets such that | Σ1 | ≥ | Σ2 | andπ : Σ1 → Σ2 is a mapping.
• Then given p ∈ Σ∗∗1 , π(p) is the picture p’ ∈ Σ∗∗
2 such thatp’(i,j) = π(p(i , j)) ∀1 ≤ i ≤ l1(p), 1 ≤ j ≤ l2(p)
• Similarly, given a picture language L ⊆ Σ∗∗1 , the projection of L by
π : Σ1 → Σ2 is defined as π(L) = {π(p) | p ∈ L} ⊆ Σ∗∗2
• Given a picture p of size (m,n), if h ≤ m, k ≤ n, we denote by Th,k(p) theset of all subpictures (contiguous rectangular subblocks) of p of size (h,k).
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
• If L = {p∈ Γ∗∗ | T2,2(bp) ⊆ ∆}, then we call ∆ alocal representation by tiles for the language L.
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
• If L = {p∈ Γ∗∗ | T2,2(bp) ⊆ ∆}, then we call ∆ alocal representation by tiles for the language L.
• We denote by LOC the family of local picturelanguages.
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
• If L = {p∈ Γ∗∗ | T2,2(bp) ⊆ ∆}, then we call ∆ alocal representation by tiles for the language L.
• We denote by LOC the family of local picturelanguages.
• An example of a picture language in LOC, considerL0 ⊆ {0, 1}∗∗ of square pictures (of size at least(2,2)) in which all nondiagonal positions carry symbol0 whereas the diagonal positions carry symbol 1.
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
• If L = {p∈ Γ∗∗ | T2,2(bp) ⊆ ∆}, then we call ∆ alocal representation by tiles for the language L.
• We denote by LOC the family of local picturelanguages.
• An example of a picture language in LOC, considerL0 ⊆ {0, 1}∗∗ of square pictures (of size at least(2,2)) in which all nondiagonal positions carry symbol0 whereas the diagonal positions carry symbol 1.
• An appropriate set of tiles for L0 consists of the 16different (2,2)-subblocks of the picture on the right.
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Local Picture Languages(LOC)
Definition
A picture language L ⊆ Γ∗∗ is local if there exists a set ∆ ofpictures (or “tiles”) of size (2,2) over Γ∪ {#}, such thatL={p∈ Γ∗∗ | T2,2(p) ⊆ ∆}
• If L = {p∈ Γ∗∗ | T2,2(bp) ⊆ ∆}, then we call ∆ alocal representation by tiles for the language L.
• We denote by LOC the family of local picturelanguages.
• An example of a picture language in LOC, considerL0 ⊆ {0, 1}∗∗ of square pictures (of size at least(2,2)) in which all nondiagonal positions carry symbol0 whereas the diagonal positions carry symbol 1.
• An appropriate set of tiles for L0 consists of the 16different (2,2)-subblocks of the picture on the right.
#
#
#
#
#
#
#
#
#
#
#
#
# # # #
# # # #
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
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Recognizable Picture Languages(REC)
Definition
A picture language L ⊆ Σ∗∗ is recognizable if there exists a locallanguage L’ over an alphabet Γ and a mapping
π : Γ → Σ such that L=π(L′)
• As an example of such a language is the set of squares over Σ = {a} anda suitable local language would be L0 considered previously and themapping, π : {0, 1} → {a}
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Recognizable Picture Languages(REC)
Definition
A picture language L ⊆ Σ∗∗ is recognizable if there exists a locallanguage L’ over an alphabet Γ and a mapping
π : Γ → Σ such that L=π(L′)
• As an example of such a language is the set of squares over Σ = {a} anda suitable local language would be L0 considered previously and themapping, π : {0, 1} → {a}
• We consider only Γ = Σ× Q and π : Σ× Q → Σ as the alphabet andmapping respectively for the local language L’ of L.
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Recognizable Picture Languages(REC)
Definition
A picture language L ⊆ Σ∗∗ is recognizable if there exists a locallanguage L’ over an alphabet Γ and a mapping
π : Γ → Σ such that L=π(L′)
• As an example of such a language is the set of squares over Σ = {a} anda suitable local language would be L0 considered previously and themapping, π : {0, 1} → {a}
• We consider only Γ = Σ× Q and π : Σ× Q → Σ as the alphabet andmapping respectively for the local language L’ of L.
• It is sufficient to consider Local languages of the type above as everylocal language L’ given in the definition, with an arbitrary alphabet maybe modified into a local language with the alphabet, Γ = Σ× Q.
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Recognizable Picture Languages(REC)
Definition
A picture language L ⊆ Σ∗∗ is recognizable if there exists a locallanguage L’ over an alphabet Γ and a mapping
π : Γ → Σ such that L=π(L′)
• As an example of such a language is the set of squares over Σ = {a} anda suitable local language would be L0 considered previously and themapping, π : {0, 1} → {a}
• We consider only Γ = Σ× Q and π : Σ× Q → Σ as the alphabet andmapping respectively for the local language L’ of L.
• It is sufficient to consider Local languages of the type above as everylocal language L’ given in the definition, with an arbitrary alphabet maybe modified into a local language with the alphabet, Γ = Σ× Q.
• Under the above considerations, the tiling System is denoted by the triple(Σ,Q,∆).
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
Abdullah Abdul Khadir Topics in Logic and Automata Theory
![Page 66: Topics in Logic and Automata Theory - Logic and Automata ...abdullah/thesis/abdullah_thesis...Notations and Symbols Henceforth we assume the following :-• σ is the vocabulary σ=(R](https://reader034.vdocuments.pub/reader034/viewer/2022051911/60016b3946ef5c7fb1139308/html5/thumbnails/66.jpg)
Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,• Boolean intersection.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,• Boolean intersection.
• REC is not closed with respect to Boolean complementation.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,• Boolean intersection.
• REC is not closed with respect to Boolean complementation.Proof idea :
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,• Boolean intersection.
• REC is not closed with respect to Boolean complementation.Proof idea :
• Let Σ be an alphabet and let L be a language over Σ given byL={p∈ Σ∗∗ | p=s⊖s where s is a square }
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Automata Theoretic Approach to Picture Languages
• Row and column concatenation.
• Row and column closure.
• Projections of picture languages.
• Local Picture languages (LOC).
• Recognizable picture languages (REC).
• REC is closed with respect to• projection,• row and column concatenation,• row and column closure,• Boolean union,• Boolean intersection.
• REC is not closed with respect to Boolean complementation.Proof idea :
• Let Σ be an alphabet and let L be a language over Σ given byL={p∈ Σ∗∗ | p=s⊖s where s is a square }
• The claim is that L /∈ REC while L ∈ REC.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x)
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
• Formulas are built up from atomic formulas by means of the Booleanconnectives and the quantifiers ∃ and ∀ applicable to first-order as well assecond-order variables.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
• Formulas are built up from atomic formulas by means of the Booleanconnectives and the quantifiers ∃ and ∀ applicable to first-order as well assecond-order variables.
• If φ(X1, ..., Xn) is a formula with at most X1, ..., Xn occurring free in φ,p is a picture, and Q1, ...,Qn are subsets of dom(p), we write
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
• Formulas are built up from atomic formulas by means of the Booleanconnectives and the quantifiers ∃ and ∀ applicable to first-order as well assecond-order variables.
• If φ(X1, ..., Xn) is a formula with at most X1, ..., Xn occurring free in φ,p is a picture, and Q1, ...,Qn are subsets of dom(p), we write
((p),Q1, ...,Qn) |= φ(X1, ...,Xn)
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
• Formulas are built up from atomic formulas by means of the Booleanconnectives and the quantifiers ∃ and ∀ applicable to first-order as well assecond-order variables.
• If φ(X1, ..., Xn) is a formula with at most X1, ..., Xn occurring free in φ,p is a picture, and Q1, ...,Qn are subsets of dom(p), we write
((p),Q1, ...,Qn) |= φ(X1, ...,Xn)
if p satisfies φ under the above mentioned interpretation where Qi istaken as interpretation of Xi .
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical definability of Picture Languages
A few notations and terminologies :
• Given a picture p ∈ Σ∗∗, we can identify the structurep = (dom(p),S1,S2,(Pa)a∈Σ),
• x,y,z,x1 ,x2, ..., are first-order variables for points of dom(p) whileX,Y,Z,X1 ,X2, ..., are MSO variables denoting sets of positions.
• Atomic formulas are of the form x=y, xSiy, X(x) and Pa(x) interpreted asequality between x and y, (x,y) ∈ Si , x∈ X ,x∈ Pa respectively.
• Formulas are built up from atomic formulas by means of the Booleanconnectives and the quantifiers ∃ and ∀ applicable to first-order as well assecond-order variables.
• If φ(X1, ..., Xn) is a formula with at most X1, ..., Xn occurring free in φ,p is a picture, and Q1, ...,Qn are subsets of dom(p), we write
((p),Q1, ...,Qn) |= φ(X1, ...,Xn)
if p satisfies φ under the above mentioned interpretation where Qi istaken as interpretation of Xi .
• If φ is a sentence we write p |= φ.
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Definability of Picture Languages
Definition (MSO-definable)
A picture language L is monadic second-order definable (L ∈ MSO), if there isa monadic second-order sentence φ with L = L(φ).
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Definability of Picture Languages
Definition (MSO-definable)
A picture language L is monadic second-order definable (L ∈ MSO), if there isa monadic second-order sentence φ with L = L(φ).
Definition (FO-definable)
A picture language L is first-order definable (L ∈ FO), if there is a sentence φconatining only first-order quantifiers such that L = L(φ).
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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Logical Definability of Picture Languages
Definition (MSO-definable)
A picture language L is monadic second-order definable (L ∈ MSO), if there isa monadic second-order sentence φ with L = L(φ).
Definition (FO-definable)
A picture language L is first-order definable (L ∈ FO), if there is a sentence φconatining only first-order quantifiers such that L = L(φ).
Definition (EMSO-definable)
Finally, A picture language L is existential monadic second-order definable(L ∈ EMSO), if there is a sentence of the form
φ = ∃X1 ... ∃Xnψ(X1, ...,Xn) where ψ contains only first-order
quantifiers such that L = L(φ).
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
Proof Idea :
• The direction (REC =⇒ EMSO) is the easy one, and all we have to do isprove the following lemma
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
Proof Idea :
• The direction (REC =⇒ EMSO) is the easy one, and all we have to do isprove the following lemma
Definition
p∈ L iff ∃ picture c ∈ Q∗∗ of the same size as p such that p × c is tilable by ∆
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
Proof Idea :
• The direction (REC =⇒ EMSO) is the easy one, and all we have to do isprove the following lemma
Definition
p∈ L iff ∃ picture c ∈ Q∗∗ of the same size as p such that p × c is tilable by ∆
• We capture the tiling of the picture p × c by the EMSO formulaφ = ∃X1...∃Xk (φpartition ∧ ∀ x1...x4
(χm ∧ χt ∧ χb ∧ χl ∧ χr ∧ χtl ∧ χtr ∧ χbl ∧ χbr))
Abdullah Abdul Khadir Topics in Logic and Automata Theory
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
Proof Idea :
• The direction (REC =⇒ EMSO) is the easy one, and all we have to do isprove the following lemma
Definition
p∈ L iff ∃ picture c ∈ Q∗∗ of the same size as p such that p × c is tilable by ∆
• We capture the tiling of the picture p × c by the EMSO formulaφ = ∃X1...∃Xk (φpartition ∧ ∀ x1...x4
(χm ∧ χt ∧ χb ∧ χl ∧ χr ∧ χtl ∧ χtr ∧ χbl ∧ χbr))
• φpartition(X1,...,Xk ) :∀z(X1(z) ∨...∨ Xk(z)) ∧
V
i 6=j¬(Xi (z) ∧ Xj(z)).
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REC ⇐⇒ EMSO
Now, the main theorem concerning picture languages
Theorem
For any picture language L:L∈ REC iff L ∈ EMSO.
Proof Idea :
• The direction (REC =⇒ EMSO) is the easy one, and all we have to do isprove the following lemma
Definition
p∈ L iff ∃ picture c ∈ Q∗∗ of the same size as p such that p × c is tilable by ∆
• We capture the tiling of the picture p × c by the EMSO formulaφ = ∃X1...∃Xk (φpartition ∧ ∀ x1...x4
(χm ∧ χt ∧ χb ∧ χl ∧ χr ∧ χtl ∧ χtr ∧ χbl ∧ χbr))
• φpartition(X1,...,Xk ) :∀z(X1(z) ∨...∨ Xk(z)) ∧
V
i 6=j¬(Xi (z) ∧ Xj(z)).
• While χm,χt ,χb,χl ,χr ,χtl ,χtr ,χbl ,χbr refer to the formulae describing (2,2)local neighbourhoods.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
• For the same parameters above, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j = d , then we say that p1 is exactly d,t-equivalent to p2 denotedp1 ≃d,t p2.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
• For the same parameters above, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j = d , then we say that p1 is exactly d,t-equivalent to p2 denotedp1 ≃d,t p2.
• A picture language,L, is called locally d-testable with threshold t if L isa union of ∼d,t -equivalence classes
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
• For the same parameters above, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j = d , then we say that p1 is exactly d,t-equivalent to p2 denotedp1 ≃d,t p2.
• A picture language,L, is called locally d-testable with threshold t if L isa union of ∼d,t -equivalence classes
• If it holds for some t,we say that L is locally threshold d-testable.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
• For the same parameters above, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j = d , then we say that p1 is exactly d,t-equivalent to p2 denotedp1 ≃d,t p2.
• A picture language,L, is called locally d-testable with threshold t if L isa union of ∼d,t -equivalence classes
• If it holds for some t,we say that L is locally threshold d-testable.
• If it holds for some d and t then we say L is locally threshold testable.
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EMSO =⇒ REC
For this direction, we use the variation of hanf’s theorem for EMSO.
• Given two pictures, p1,p2, d,t ∈ N, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j ≤ d , then we say that p1 is d,t-equivalent to p2 denoted p1 ∼d,t p2.
• For pictures, we consider only rectangles and not spheres.
• For the same parameters above, if T(i,j)(p1)↾(d,t)=T(i,j)(p2)↾(d,t)∀ i,j = d , then we say that p1 is exactly d,t-equivalent to p2 denotedp1 ≃d,t p2.
• A picture language,L, is called locally d-testable with threshold t if L isa union of ∼d,t -equivalence classes
• If it holds for some t,we say that L is locally threshold d-testable.
• If it holds for some d and t then we say L is locally threshold testable.
• Finally, if L is a union of ≃d,t-classes for some t, L is called locallystrictly threshold d-testable.
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EMSO =⇒ REC
Theorem (Theorem 1)
A picture language is first-order definable iff it is locally threshold testable.
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EMSO =⇒ REC
Theorem (Theorem 1)
A picture language is first-order definable iff it is locally threshold testable.
Theorem (Theorem 2)
Using theorem 1, we claim that if L ∈ EMSO then L is a projection of a locallythreshold testable picture language.
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EMSO =⇒ REC
Theorem (Theorem 1)
A picture language is first-order definable iff it is locally threshold testable.
Theorem (Theorem 2)
Using theorem 1, we claim that if L ∈ EMSO then L is a projection of a locallythreshold testable picture language.
Thus, Theorem 1 =⇒ Theorem 2 =⇒ (EMSO =⇒ REC)
We only need to prove Theorem 1 now.
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EMSO =⇒ REC
Theorem (Theorem 1)
A picture language is first-order definable iff it is locally threshold testable.
Theorem (Theorem 2)
Using theorem 1, we claim that if L ∈ EMSO then L is a projection of a locallythreshold testable picture language.
Thus, Theorem 1 =⇒ Theorem 2 =⇒ (EMSO =⇒ REC)
We only need to prove Theorem 1 now.
Proof sketch of Theorem 1 :-
• The proof for the direction ⇐= is by an adaptation of Hanf’s theorem topictures. We can use the bound as d=2*3n+1 and t=n*32n forn-equivalence.
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EMSO =⇒ REC
Theorem (Theorem 1)
A picture language is first-order definable iff it is locally threshold testable.
Theorem (Theorem 2)
Using theorem 1, we claim that if L ∈ EMSO then L is a projection of a locallythreshold testable picture language.
Thus, Theorem 1 =⇒ Theorem 2 =⇒ (EMSO =⇒ REC)
We only need to prove Theorem 1 now.
Proof sketch of Theorem 1 :-
• The proof for the direction ⇐= is by an adaptation of Hanf’s theorem topictures. We can use the bound as d=2*3n+1 and t=n*32n forn-equivalence.
• So that completes the proof for ⇐= of theorem 1 and we only need toprove the reverse direction.
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First-order definable =⇒ Locally threshold testable
Proof Sketch :-
Theorem
Each locally threshold d-testable language L can be decomposed into L0∪ L1∪... ∪ Ld−2 where Li ⊆ Σ∗∗
i (0≤i≤d-2) is locally strictly threshold (i+2)-testable.
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First-order definable =⇒ Locally threshold testable
Proof Sketch :-
Theorem
Each locally threshold d-testable language L can be decomposed into L0∪ L1∪... ∪ Ld−2 where Li ⊆ Σ∗∗
i (0≤i≤d-2) is locally strictly threshold (i+2)-testable.
Definition
Let d≥2 be a positive integer. A picture language L⊆ Σ∗∗d−2 is d-local if there
exists a set ∆(d) of pictures of size (d,d) (or “d-tiles”) over Σ∪ {#}, such that
L={p∈ Σ∗∗ | Td,d(bp) ⊆ ∆(d)}
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First-order definable =⇒ Locally threshold testable
Proof Sketch :-
Theorem
Each locally threshold d-testable language L can be decomposed into L0∪ L1∪... ∪ Ld−2 where Li ⊆ Σ∗∗
i (0≤i≤d-2) is locally strictly threshold (i+2)-testable.
Definition
Let d≥2 be a positive integer. A picture language L⊆ Σ∗∗d−2 is d-local if there
exists a set ∆(d) of pictures of size (d,d) (or “d-tiles”) over Σ∪ {#}, such that
L={p∈ Σ∗∗ | Td,d(bp) ⊆ ∆(d)}
Theorem
Let d ≥ 3 be a positive integer. A locally strictly threshold d-testable picturelanguage L ⊆ Σ∗∗
d−2 is the projection of a d-local language.
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First-order definable =⇒ Locally threshold testable
Proof Sketch :-
Theorem
Each locally threshold d-testable language L can be decomposed into L0∪ L1∪... ∪ Ld−2 where Li ⊆ Σ∗∗
i (0≤i≤d-2) is locally strictly threshold (i+2)-testable.
Definition
Let d≥2 be a positive integer. A picture language L⊆ Σ∗∗d−2 is d-local if there
exists a set ∆(d) of pictures of size (d,d) (or “d-tiles”) over Σ∪ {#}, such that
L={p∈ Σ∗∗ | Td,d(bp) ⊆ ∆(d)}
Theorem
Let d ≥ 3 be a positive integer. A locally strictly threshold d-testable picturelanguage L ⊆ Σ∗∗
d−2 is the projection of a d-local language.
Theorem
A d-local picture language is a projection of a local language.
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Alternation hierarchy of MSO over grids and graphs
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Alternation hierarchy of MSO over grids and graphs
• The signature of grids is given by τGrid = ([m,n],Sm,n1 ,Sm,n
2 )where [m,n] = [m] × [n]
• The signature of t-bit grids for some t∈ N is given byτt−Grid = ([m,n],Sm,n
1 ,Sm,n2 ,X1,...,Xt )
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Alternation hierarchy of MSO over grids and graphs
• The signature of grids is given by τGrid = ([m,n],Sm,n1 ,Sm,n
2 )where [m,n] = [m] × [n]
• The signature of t-bit grids for some t∈ N is given byτt−Grid = ([m,n],Sm,n
1 ,Sm,n2 ,X1,...,Xt )
Theorem
∀ k ≥ 1, B(Σk)(Grids) $ ∆k+1(Grids)
The inclusion results are as shown thediagram in the right with undirectededges indicating strict inclusion.
Σk (Grid) Πk(Grid)
B(Σk )(Grid)
∆k+1(Grid)
Σk (Grid) Πk(Grid)
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B(Σk)(Grids) $ ∆k+1(Grids)
The basis of the theorem is definability results for sets of grids.• For a function f :N → N we denote by Lf the set of grids whose size is
(m,f(m)) for m ≥ 1.
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B(Σk)(Grids) $ ∆k+1(Grids)
The basis of the theorem is definability results for sets of grids.• For a function f :N → N we denote by Lf the set of grids whose size is
(m,f(m)) for m ≥ 1.
• A formula φ over τGrid defines the function f:N → N iff ModGrid (φ) = Lf .
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B(Σk)(Grids) $ ∆k+1(Grids)
The basis of the theorem is definability results for sets of grids.• For a function f :N → N we denote by Lf the set of grids whose size is
(m,f(m)) for m ≥ 1.
• A formula φ over τGrid defines the function f:N → N iff ModGrid (φ) = Lf .
• A function is at most k-fold exponential if f(m) is sk(O(m)),where s0(m) = m and sk+1(m) = 2sk (m).
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B(Σk)(Grids) $ ∆k+1(Grids)
The basis of the theorem is definability results for sets of grids.• For a function f :N → N we denote by Lf the set of grids whose size is
(m,f(m)) for m ≥ 1.
• A formula φ over τGrid defines the function f:N → N iff ModGrid (φ) = Lf .
• A function is at most k-fold exponential if f(m) is sk(O(m)),where s0(m) = m and sk+1(m) = 2sk (m).
Theorem
Every B(Σk )-definable function is at most k-fold exponential.
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B(Σk)(Grids) $ ∆k+1(Grids)
The basis of the theorem is definability results for sets of grids.• For a function f :N → N we denote by Lf the set of grids whose size is
(m,f(m)) for m ≥ 1.
• A formula φ over τGrid defines the function f:N → N iff ModGrid (φ) = Lf .
• A function is at most k-fold exponential if f(m) is sk(O(m)),where s0(m) = m and sk+1(m) = 2sk (m).
Theorem
Every B(Σk )-definable function is at most k-fold exponential.
Theorem
Let f1(m) = 2m, fk+1(m) = fk (m)2fk (m) for m,k ≥ 1.∀k≥ 1,the function fk is definable in Σk and Πk over τGrid
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Complexity of a B(Σk)-definable function
Proof Sketch:
• For a picture language L over alphabet Γ and an integer m≥ 1, we denoteby L(m) the word language L restricted to Γm,1.
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Complexity of a B(Σk)-definable function
Proof Sketch:
• For a picture language L over alphabet Γ and an integer m≥ 1, we denoteby L(m) the word language L restricted to Γm,1.
• ∀ t ≥ 0 and for every φ ∈ Σ1 with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with 2cm states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
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Complexity of a B(Σk)-definable function
Proof Sketch:
• For a picture language L over alphabet Γ and an integer m≥ 1, we denoteby L(m) the word language L restricted to Γm,1.
• ∀ t ≥ 0 and for every φ ∈ Σ1 with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with 2cm states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
• ∀ k ≥ 1 and for every φ ∈ Σk with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with sk (cm) states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
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Complexity of a B(Σk)-definable function
Proof Sketch:
• For a picture language L over alphabet Γ and an integer m≥ 1, we denoteby L(m) the word language L restricted to Γm,1.
• ∀ t ≥ 0 and for every φ ∈ Σ1 with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with 2cm states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
• ∀ k ≥ 1 and for every φ ∈ Σk with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with sk (cm) states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
• Let N ⊆ N be recognizable by some n-state NFA. Then ∃ k ≤ (n+2)2
and an integer p such that N is recognized by a DFA A with states0,...,k+(p-1) such that A reaches the state k+((l-k)mod p) after readingan input of length l≥k.( =⇒ N is (n+2)2-periodic).
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Complexity of a B(Σk)-definable function
Proof Sketch:
• For a picture language L over alphabet Γ and an integer m≥ 1, we denoteby L(m) the word language L restricted to Γm,1.
• ∀ t ≥ 0 and for every φ ∈ Σ1 with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with 2cm states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
• ∀ k ≥ 1 and for every φ ∈ Σk with free variables among X1, ... ,Xt
∃c≥1 such that for all m≥1, there is an NFA with sk (cm) states thatrecognises the word language Modt(φ)(m) over ({0,1}t)m,1.
• Let N ⊆ N be recognizable by some n-state NFA. Then ∃ k ≤ (n+2)2
and an integer p such that N is recognized by a DFA A with states0,...,k+(p-1) such that A reaches the state k+((l-k)mod p) after readingan input of length l≥k.( =⇒ N is (n+2)2-periodic).
• Let φ be a B(Σk)-sentence. There is a constant c≥1 such that for everym≥1 the set Mod0(φ)(m) is sk(cm)-periodic. Thus from all the abovestatements we get that every B(Σk )-definable function is at most k-foldexponential.
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fk(m) ∈ ∆k(Grid)
m
f1(m)=2m
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The Complete column-numbering of a grid
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fk(m) ∈ ∆k(Grid)
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The Complete column-numbering of a grid
fi (m)
fi+1(m) = fi .2fi (m)
0 0 ... 0 0 1 1 ... 1 1 0 . . . . . 1 0 0 ... 0 0 1 1 ... 1 1
Complete fi -numbering along the top row of the grid
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MSCs, MPA and EMSOMSC
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MSCs, MPA and EMSOMSC
Message Sequence Charts :-
1!2
2?1
1?4 4!1
1!2 2!3 3?2
1!3 3?1
2?33!2
2?1
4?5 5!4
4!5 5?4
5!3
3?5c
c
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Abdullah Abdul Khadir Topics in Logic and Automata Theory
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MSCs, MPA and EMSOMSC
Message Sequence Charts :-
1!2
2?1
1?4 4!1
1!2 2!3 3?2
1!3 3?1
2?33!2
2?1
4?5 5!4
4!5 5?4
5!3
3?5c
c
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4 5
53
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MSC (over P) is a graph M = (E,{∆p}p∈P ,∆c ,λ)
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MSCs, MPA and EMSOMSC
Message Passing Automata :-
A1 A2
1!2, o
1?2, o
1!2, x 1?2, o
2?1, o 2!1, o
2!1, x2?1, x
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MSCs, MPA and EMSOMSC
Message Passing Automata :-
A1 A2
1!2, o
1?2, o
1!2, x 1?2, o
2?1, o 2!1, o
2!1, x2?1, x
MPA (over P) is a structure (A) = (((A)p)p∈P ,D,s−in,F)
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Definitions
Definition
An MSC (over P) is a graph M = (E,{∆p}p∈P ,∆c ,λ) ∈ DG(Act,Pc) such that
• ∆p is a total order.
• ∆c ⊆ E × E is the set of edges connecting messages.
• Number of messages sent is equal to the number of messages received
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Definitions
Definition
An MSC (over P) is a graph M = (E,{∆p}p∈P ,∆c ,λ) ∈ DG(Act,Pc) such that
• ∆p is a total order.
• ∆c ⊆ E × E is the set of edges connecting messages.
• Number of messages sent is equal to the number of messages received
Definition
MPA (over P) is a structure (A) = (((A)p)p∈P ,D,s−in,F) such that
• D is a set of synchronization messages.
• for each p∈P ,A is a pair (Sp , δp) where
• Sp is a set of local states• δp ⊆ Sp× Actp ×D × Sp
• s−in ∈ Πp∈PSp is the global initial state.
• F ⊆ Πp∈PSp is the set of global final states.
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Logical Definability
Definition (MSO(Σ,C))
MSO(Σ,C) over the class DG are built up from the atomicformulas λ(x) = a (for a ∈ Σ), x∆cy (for c∈ C), x∈ X and x=y.
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Logical Definability
Definition (MSO(Σ,C))
MSO(Σ,C) over the class DG are built up from the atomicformulas λ(x) = a (for a ∈ Σ), x∆cy (for c∈ C), x∈ X and x=y.
Definition
A graph acceptor over (Σ,C) is a structure B = (Q,R, S ,Occ)where
• Q is its nonempty finite set of states
• B ∈ N is the radius
• S is a finite set of R-spheres over (Σ × Q,C) and
• Occ is a boolean combinations of conditions of the form“sphere H ∈ S occurs at least n times” where n ∈ N.
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The Theorems
Theorem
MPA ≡ EMSOMSC
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The Theorems
Theorem
MPA ≡ EMSOMSC
Theorem
The monadic quantifier alternation hierarchy over MSC is infinite.
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Summary
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Summary
• Hanf’s theorem
• Picture languages• Local picture Languages• Recognizable picture languages• REC ⇐⇒ EMSO
• Monadic quantifier Alternation hierarchy over Grids andgraphs
• MSC ⇐⇒ EMSOMSC
Abdullah Abdul Khadir Topics in Logic and Automata Theory