My presentation for RuFiDim

1. Graph Expansion, Tseitin formulas and resolution
proofs for CSP
Dmitriy Itsykson Vsevolod Oparin
St. Petersburg Department of Steklov
Institute of Mathematics of
Russian Academy of Sciences
St. Petersburg University of
Russian Academy of Science
September 26, 2012
Dmitriy Itsykson, Vsevolod Oparin 1/10 | Tseitin formulas and resolution proofs for CSP

2. Outline
SAT and DPLL
Tseitin formula
Lower bound in propositional case
Upper bound
CSP
Main results
Dmitriy Itsykson, Vsevolod Oparin 2/10 | Tseitin formulas and resolution proofs for CSP

3. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method?
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

4. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

5. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

6. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

7. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

8. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

9. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

10. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

11. NP-hard problem
Problem
Given boolean formula φ(x1 , x2 , · · · , xn ) nd a1 , a2 , · · · , an ∈ {0, 1}
s.t. φ(a1 , a2 , · · · , an ) = 1.
Method? DPLL algorithm!
φ(x , y , z ) := (x ∨ y ) ∧ (¬y ∨ z ) ∧ (¬z ) ∧ (¬x )
φ(1, y , z ) := 0 φ(0, y , z ) := (y ) ∧ (¬y ∨ z ) ∧ (¬z )
φ(0, y , 1) := 0 φ(0, y , 0) := (y ) ∧ (¬y )
φ(0, 0, 0) := 0 φ(0, 1, 0) := 0
The tree is the proof. The size of the proof ST is the number of
vertecies.
Dmitriy Itsykson, Vsevolod Oparin 3/10 | Tseitin formulas and resolution proofs for CSP

12. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1
x2 x3 ⇒
0 x1 0
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

13. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x2 x3 ⇒
0 x1 0
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

14. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

15. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

16. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

17. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

18. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

19. Tseitin formula
Denition
Consider d -regular graph G = V , E and function f : V → {0, 1}.
Associate every edge e ∈ E with a variable xe . For every vertex v
write constraint (v ,u)∈E x(v ,u) = f (v ) (mod 2).
1 E → {x1 , x2 , x3 }
x3 ⇒  x1 + x2 = 0 (mod 2)

x2
x2 + x3 = 1 (mod 2)
0 0 x1 + x3 = 0 (mod 2)

x1
Theorem
φ ∈ SAT ⇐⇒ v ∈V f (v ) = 1 (mod 2).
Dmitriy Itsykson, Vsevolod Oparin 4/10 | Tseitin formulas and resolution proofs for CSP

20. Lower bounds for propositional formula
First bound
G.S. Tseitin, 1968.
⇒ superpolynomial lower bound on
the size of proof
Dmitriy Itsykson, Vsevolod Oparin 5/10 | Tseitin formulas and resolution proofs for CSP

21. Lower bounds for propositional formula
First bound
G.S. Tseitin, 1968.
⇒ superpolynomial lower bound on
the size of proof
Strongest result
E. Ben-Sasson and A. Wigderson, 2001.
Dmitriy Itsykson, Vsevolod Oparin 5/10 | Tseitin formulas and resolution proofs for CSP

22. Lower bounds for propositional formula
First bound
G.S. Tseitin, 1968.
⇒ superpolynomial lower bound on
the size of proof
Strongest result
E. Ben-Sasson and A. Wigderson, 2001.
Expansion
e (G ) = min E (U , U ),
U U
U
where U ⊆ V s.t. |U |/|V | ∈ 1
3
,2 .
3
Dmitriy Itsykson, Vsevolod Oparin 5/10 | Tseitin formulas and resolution proofs for CSP

23. Lower bounds for propositional formula
First bound
G.S. Tseitin, 1968.
⇒ superpolynomial lower bound on
the size of proof
Strongest result
E. Ben-Sasson and A. Wigderson, 2001.
ST ≥ 2e (φ)−d
Expansion
e (G ) = min E (U , U ),
U U
U
where U ⊆ V s.t. |U |/|V | ∈ 1
3
,2 .
3
Dmitriy Itsykson, Vsevolod Oparin 5/10 | Tseitin formulas and resolution proofs for CSP

24. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

25. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
Dene substitution operation.
G → G1 G2
φ(G )
φ(G1 ) φ(G2 ) φ(G2 ) φ(G1 )
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

26. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
Dene substitution operation.
Divide graph G = V , E into
subgraphs G = V , E ,1 1 1 G → G1 G2
G = V , E s.t. E (V , V ) is
2 2 2 1 2
minimal and V diers from V at
1 2 φ(G )
most twice.
φ(G1 ) φ(G2 ) φ(G2 ) φ(G1 )
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

27. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
Dene substitution operation.
Divide graph G = V , E into
subgraphs G = V , E ,1 1 1G → G G 1 2
G = V , E s.t. E (V , V ) is
2 2 2 1 2
minimal and V diers from V at
1 φ(G ) 2
most twice.
Substitute all possible values to
variables corresponding in cut. φ(G ) φ(G ) φ(G ) φ(G ) 1 2 2 1
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

28. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
Dene substitution operation.
Divide graph G = V , E into
subgraphs G = V , E ,1 1 1G → G G 1 2
G = V , E s.t. E (V , V ) is
2 2 2 1 2
minimal and V diers from V at
1 φ(G ) 2
most twice.
Substitute all possible values to
variables corresponding in cut. φ(G ) φ(G ) φ(G ) φ(G ) 1 2 2 1
Run operation recursively.
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

29. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
Dene substitution operation.
Divide graph G = V , E into
subgraphs G = V , E ,1 1 1G → G G 1 2
G = V , E s.t. E (V , V ) is
2 2 2 1 2
minimal and V diers from V at
1 φ(G ) 2
most twice.
Substitute all possible values to
variables corresponding in cut. φ(G ) φ(G ) φ(G ) φ(G ) 1 2 2 1
Run operation recursively.
Dmitriy Itsykson, Vsevolod Oparin 6/10 | Tseitin formulas and resolution proofs for CSP

30. Upper bound
Is graph expansion necessary for large proof size? That's easy!
Theorem
e (H )·log V
There is subgraph H of graph G s.t. ST ≤ 2 .
3
2
Proof.
≤ e (H )
log V3
2
operations
h ≤ e (H ) · log V ⇒ 3
2
e (H )·log V
ST ≤ 2
3
2
Dmitriy Itsykson, Vsevolod Oparin 7/10 | Tseitin formulas and resolution proofs for CSP

31. CSP
Comparison
Constraint satisfaction problem. Given a set of variables and a set
of functions from subset of these variables to {0, 1}, so called
contraints, nd substituion from a set D s.t. satisfy all constraints.
SAT D = {0, 1} Set of clauses
CSP D = {0, 1, · · · , W − 1} Set of constraints
DPLL: W branches.
··· ···
x
Tseitin formula: D = ZW u v
−x
Cv : (v ,u )∈E x(v ,u) = f (v ) (mod W )
Dmitriy Itsykson, Vsevolod Oparin 8/10 | Tseitin formulas and resolution proofs for CSP

32. CSP
Comparison
Constraint satisfaction problem. Given a set of variables and a set
of functions from subset of these variables to {0, 1}, so called
contraints, nd substituion from a set D s.t. satisfy all constraints.
SAT D = {0, 1} Set of clauses
CSP D = {0, 1, · · · , W − 1} Set of constraints
DPLL: W branches.
··· ···
x
Tseitin formula: D = ZW u v
−x
Cv : (v ,u )∈E x(v ,u) = f (v ) (mod W )
Dmitriy Itsykson, Vsevolod Oparin 8/10 | Tseitin formulas and resolution proofs for CSP

33. CSP
Comparison
Constraint satisfaction problem. Given a set of variables and a set
of functions from subset of these variables to {0, 1}, so called
contraints, nd substituion from a set D s.t. satisfy all constraints.
SAT D = {0, 1} Set of clauses
CSP D = {0, 1, · · · , W − 1} Set of constraints
DPLL: W branches.
··· ···
x
Tseitin formula: D = ZW u v
−x
Cv : (v ,u )∈E x(v ,u) = f (v ) (mod W )
Dmitriy Itsykson, Vsevolod Oparin 8/10 | Tseitin formulas and resolution proofs for CSP

34. Results
For generalized Tseitin formula builded on ZW and graph G we
obtained next results.
e (H )· V
Upper bound: ST ≤ W for the subgraph H .
log 3
2
Dmitriy Itsykson, Vsevolod Oparin 9/10 | Tseitin formulas and resolution proofs for CSP

35. Results
For generalized Tseitin formula builded on ZW and graph G we
obtained next results.
e (H )· V
Upper bound: ST ≤ W for the subgraph H .
log 3
2
Lower bound (method of BW): ST ≥ 2e (G )−d .
Dmitriy Itsykson, Vsevolod Oparin 9/10 | Tseitin formulas and resolution proofs for CSP

36. Results
For generalized Tseitin formula builded on ZW and graph G we
obtained next results.
e (H )· V
Upper bound: ST ≤ W for the subgraph H .
log 3
2
Lower bound (method of BW): ST ≥ 2e (G )−d .
Lower bound (main result): ST ≥ W e (G )−d .
Dmitriy Itsykson, Vsevolod Oparin 9/10 | Tseitin formulas and resolution proofs for CSP

37. Thank you!
Thank you! Questions?
Dmitriy Itsykson, Vsevolod Oparin 10/10 | Tseitin formulas and resolution proofs for CSP