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Time-space trade-offs in proof complexityLecture 2
Jakob Nordstrom
KTH Royal Institute of Technology
17th Estonian Winter School in Computer SciencePalmse, Estonia
February 26 – March 2, 2012
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 1 / 32
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Goal of Today’s Lecture
Focus on the resolution proof system
Quick recap of what was said last time
Brief overview of what is known for proof length and proof space
Prove length-space trade-offs for resolution (or rather: sketch proofs)
Discuss extensions to polynomial calculus
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 2 / 32
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Some Notation and Terminology
Literal a: variable x or its negation x
Clause C = a1 ∨ . . . ∨ ak: set of literalsAt most k literals: k-clause
CNF formula F = C1 ∧ . . . ∧ Cm: set of clausesk-CNF formula: CNF formula consisting of k-clauses
F D: semantical implication, α(F ) true ⇒ α(D) truefor all truth value assignments α
[n] = 1, 2, . . . , n
This course: focus on k-CNF formulas for k = O(1)(Avoids annoying technicalities, and can always convert to k-CNF anyway)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 3 / 32
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Resolution Revisited
Last time we talked about a resolution refutations as a sequence ofclause configurations D0, . . . , Dτ (snapshots of what’s on the board)
For all t, Dt obtained from Dt−1 by one of the following derivation steps:
Download Dt = Dt−1 ∪ C for axiom clause C ∈ F
Inference Dt = Dt−1 ∪ D for D inferred by resolution on clausesin Dt−1.
Erasure Dt = Dt−1 \ D for some D ∈ Dt−1.
But if we don’t care about space, then we can view a resolution refutationas simply a listing of the clauses (i.e., no erasures)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 4 / 32
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Resolution Proof System (Ignoring Space)
Resolution derivation π : F `A of clause A from F :Sequence of clauses π = D1, . . . , Ds such that Ds = A and each lineDi, 1 ≤ i ≤ s, is either
a clause C ∈ F (an axiom)
a resolvent derived from clauses Dj , Dk in π (with j, k < i) by theresolution rule
B ∨ x C ∨ x
B ∨ C
resolving on the variable x
Resolution refutation of CNF formula F :Derivation of empty clause ⊥ (clause with no literals) from F
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 5 / 32
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Example Resolution Refutation
F = (x ∨ z) ∧ (z ∨ y) ∧ (x ∨ y ∨ u) ∧ (y ∨ u)
∧ (u ∨ v) ∧ (x ∨ v) ∧ (u ∨ w) ∧ (x ∨ u ∨ w)
1. x ∨ z Axiom 9. x ∨ y Res(1, 2)2. z ∨ y Axiom 10. x ∨ y Res(3, 4)3. x ∨ y ∨ u Axiom 11. x ∨ u Res(5, 6)4. y ∨ u Axiom 12. x ∨ u Res(7, 8)5. u ∨ v Axiom 13. x Res(9, 10)6. x ∨ v Axiom 14. x Res(11, 12)7. u ∨ w Axiom 15. ⊥ Res(13, 14)8. x ∨ u ∨ w Axiom
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 6 / 32
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Resolution Sound and Complete
Resolution is sound and implicationally complete.
Sound If there is a resolution derivation π : F `Athen F A
Complete If F A then there is a resolution derivation π : F `A′ forsome A′ ⊆ A.
In particular:
F is unsatisfiable ⇔ ∃ resolution refutation of F
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 7 / 32
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Completeness of Resolution: Proof by Example
Decision tree:
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
0 1 0 1 0 1 0 1
0 1 0 1
0 1x
y u
z u v w
Resulting resolution refutation:
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
⊥
x x
x ∨ y x ∨ y x ∨ u x ∨ u
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 8 / 32
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Completeness of Resolution: Proof by Example
Decision tree:
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
0 1 0 1 0 1 0 1
0 1 0 1
0 1x
y u
z u v w
Resulting resolution refutation:
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
⊥
x x
x ∨ y x ∨ y x ∨ u x ∨ u
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 8 / 32
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Derivation Graph and Tree-Like Derivations
Derivation graph Gπ of a resolution derivation π:directed acyclic graph (DAG) with
vertices: clauses of the derivations
edges: from B ∨ x and C ∨ x to B ∨ C for each application of theresolution rule
A resolution derivation π is tree-like if Gπ is a tree(We can make copies of axiom clauses to make Gπ into a tree)
Example
Our example resolution proof is tree-like.(The derivation graph is on the previous slide.)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 9 / 32
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Derivation Graph and Tree-Like Derivations
Derivation graph Gπ of a resolution derivation π:directed acyclic graph (DAG) with
vertices: clauses of the derivations
edges: from B ∨ x and C ∨ x to B ∨ C for each application of theresolution rule
A resolution derivation π is tree-like if Gπ is a tree(We can make copies of axiom clauses to make Gπ into a tree)
Example
Our example resolution proof is tree-like.(The derivation graph is on the previous slide.)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 9 / 32
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Complexity Measures of Interest: Length and Space
Length: Lower bound on time for SAT solver(very straightforward connection)
Space: Lower bound on memory for SAT solver(requires more of an argument — will be happy to elaborate offline)
Length LR# clauses written on blackboard counted with repetitions
SpaceSeveral ways of measuring — will mainly be interested in two measures
1.
x
1
2.
y
2
∨ z
3
3.
v
4
∨ w
5
∨ y
6
Clause space SpR: 3Total space TotSpR: 6
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 10 / 32
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Complexity Measures of Interest: Length and Space
Length: Lower bound on time for SAT solver(very straightforward connection)
Space: Lower bound on memory for SAT solver(requires more of an argument — will be happy to elaborate offline)
Length LR# clauses written on blackboard counted with repetitions
SpaceSeveral ways of measuring — will mainly be interested in two measures
1.
x
1
2.
y
2
∨ z
3
3.
v
4
∨ w
5
∨ y
6
Clause space SpR: 3Total space TotSpR: 6
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 10 / 32
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Complexity Measures of Interest: Length and Space
Length: Lower bound on time for SAT solver(very straightforward connection)
Space: Lower bound on memory for SAT solver(requires more of an argument — will be happy to elaborate offline)
Length LR# clauses written on blackboard counted with repetitions
SpaceSeveral ways of measuring — will mainly be interested in two measures
1. x
1
2. y
2
∨ z
3
3. v
4
∨ w
5
∨ y
6
Clause space SpR: 3Total space TotSpR: 6
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 10 / 32
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Complexity Measures of Interest: Length and Space
Length: Lower bound on time for SAT solver(very straightforward connection)
Space: Lower bound on memory for SAT solver(requires more of an argument — will be happy to elaborate offline)
Length LR# clauses written on blackboard counted with repetitions
SpaceSeveral ways of measuring — will mainly be interested in two measures
1.
x1
2.
y2∨ z3
3.
v4∨ w5∨ y6
Clause space SpR: 3Total space TotSpR: 6
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 10 / 32
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Length and Space Bounds for Resolution (1 / 2)
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
⊥
x x
x ∨ y x ∨ y x ∨ u x ∨ uLet n = size of formula
≤ n variables ⇒decision tree size ≤ 2n+1 and height ≤ n
By induction: Clause at root of subtree of height h derivable in space h+2
Derive left child clause in space h + 1 and keep in memory
Derive right child clause in space 1 + (h + 1)
Resolve the two children clauses to get root clause
Hence:LR(F `⊥) = exp(O(n))
SpR(F `⊥) = O(n)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 11 / 32
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Length and Space Bounds for Resolution (1 / 2)
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
⊥
x x
x ∨ y x ∨ y x ∨ u x ∨ uLet n = size of formula
≤ n variables ⇒decision tree size ≤ 2n+1 and height ≤ n
By induction: Clause at root of subtree of height h derivable in space h+2
Derive left child clause in space h + 1 and keep in memory
Derive right child clause in space 1 + (h + 1)
Resolve the two children clauses to get root clause
Hence:LR(F `⊥) = exp(O(n))
SpR(F `⊥) = O(n)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 11 / 32
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Length and Space Bounds for Resolution (1 / 2)
x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w
⊥
x x
x ∨ y x ∨ y x ∨ u x ∨ uLet n = size of formula
≤ n variables ⇒decision tree size ≤ 2n+1 and height ≤ n
By induction: Clause at root of subtree of height h derivable in space h+2
Derive left child clause in space h + 1 and keep in memory
Derive right child clause in space 1 + (h + 1)
Resolve the two children clauses to get root clause
Hence:LR(F `⊥) = exp(O(n))
SpR(F `⊥) = O(n)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 11 / 32
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Length and Space Bounds for Resolution (2 / 2)
(n = size of formula)
Length: at most exponential in nMatching lower bounds up to constant factors in exponent[Urquhart ’87, Chvatal & Szemeredi ’88]
Clause space: at most linear in nMatching lower bounds up to constant factors[Toran ’99, Alekhnovich et al. ’00]
Total space: at most quadratic in nNo better lower bounds than linear in n!?
[Sidenote: space bounds hold even for “magic algorithms” always makingoptimal choices — so might be much stronger in practice]
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 12 / 32
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Length and Space Bounds for Resolution (2 / 2)
(n = size of formula)
Length: at most exponential in nMatching lower bounds up to constant factors in exponent[Urquhart ’87, Chvatal & Szemeredi ’88]
Clause space: at most linear in nMatching lower bounds up to constant factors[Toran ’99, Alekhnovich et al. ’00]
Total space: at most quadratic in nNo better lower bounds than linear in n!?
[Sidenote: space bounds hold even for “magic algorithms” always makingoptimal choices — so might be much stronger in practice]
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 12 / 32
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Comparing Length and Space
Some “rescaling” needed to get meaningful comparisons of length andspace
Length exponential in formula size in worst case
Clause space at most linear
So natural to compare space to logarithm of length
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 13 / 32
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Length-Space Correlations and/or Trade-offs?
∃ constant space refutation ⇒ ∃ polynomial length refutation[Atserias & Dalmau ’03]
For tree-like resolution: any polynomial length refutation can be carriedout in logarithmic space [Esteban & Toran ’99]
So essentially no trade-offs for tree-like resolution
Does short length imply small space for general resolution?Open for quite a while — even no consensus on likely “right answer”
Nothing known about length-space trade-offs for resolution refutations inthe general, unrestricted proof system
(Some trade-off results in restricted settings in [Ben-Sasson ’02,Nordstrom ’07])
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 14 / 32
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Length-Space Correlations and/or Trade-offs?
∃ constant space refutation ⇒ ∃ polynomial length refutation[Atserias & Dalmau ’03]
For tree-like resolution: any polynomial length refutation can be carriedout in logarithmic space [Esteban & Toran ’99]
So essentially no trade-offs for tree-like resolution
Does short length imply small space for general resolution?Open for quite a while — even no consensus on likely “right answer”
Nothing known about length-space trade-offs for resolution refutations inthe general, unrestricted proof system
(Some trade-off results in restricted settings in [Ben-Sasson ’02,Nordstrom ’07])
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 14 / 32
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Length-Space Correlations and/or Trade-offs?
∃ constant space refutation ⇒ ∃ polynomial length refutation[Atserias & Dalmau ’03]
For tree-like resolution: any polynomial length refutation can be carriedout in logarithmic space [Esteban & Toran ’99]
So essentially no trade-offs for tree-like resolution
Does short length imply small space for general resolution?Open for quite a while — even no consensus on likely “right answer”
Nothing known about length-space trade-offs for resolution refutations inthe general, unrestricted proof system
(Some trade-off results in restricted settings in [Ben-Sasson ’02,Nordstrom ’07])
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 14 / 32
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Length-Space Correlations and/or Trade-offs?
∃ constant space refutation ⇒ ∃ polynomial length refutation[Atserias & Dalmau ’03]
For tree-like resolution: any polynomial length refutation can be carriedout in logarithmic space [Esteban & Toran ’99]
So essentially no trade-offs for tree-like resolution
Does short length imply small space for general resolution?Open for quite a while — even no consensus on likely “right answer”
Nothing known about length-space trade-offs for resolution refutations inthe general, unrestricted proof system
(Some trade-off results in restricted settings in [Ben-Sasson ’02,Nordstrom ’07])
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 14 / 32
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1st result today: An Optimal Length-Space Separation
Length and space in resolution are “completely uncorrelated”
Theorem (Ben-Sasson & Nordstrom ’08)
There are k-CNF formula families of size n with
refutation length O(n)
refutation clause space Ω(n/ log n)
Optimal separation of length and space — given length O(n), alwayspossible to get clause space O(n/ log n)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 15 / 32
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2nd result today: Length-Space Trade-offs
There is a rich collection of length-space trade-offs
Results hold for
resolution
even stronger proof systems (which we won’t go into here)
Different trade-offs covering (almost) whole range of space from constantto linear
Simple, explicit formulas
(Also some very nice follow-up work in [Beame, Beck & Impagliazzo ’12]that we won’t have time to go into)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 16 / 32
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One Example: Robust Trade-offs for Small Space
Theorem (Ben-Sasson & Nordstrom ’11 (informal))
For any arbitrarily slowly growing function g there exist explicitk-CNF formulas of size n
refutable in resolution in space g(n) and
refutable in length linear in n and space ≈ 3√
n such that
any refutation in space 3√
n requires superpolynomial length
And an open problem:
Open Problem
Seems likely that 3√
n above should be possible to improve to√
n, butdon’t know how to prove this. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 17 / 32
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One Example: Robust Trade-offs for Small Space
Theorem (Ben-Sasson & Nordstrom ’11 (informal))
For any arbitrarily slowly growing function g there exist explicitk-CNF formulas of size n
refutable in resolution in space g(n) and
refutable in length linear in n and space ≈ 3√
n such that
any refutation in space 3√
n requires superpolynomial length
And an open problem:
Open Problem
Seems likely that 3√
n above should be possible to improve to√
n, butdon’t know how to prove this. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 17 / 32
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One Example: Robust Trade-offs for Small Space
Theorem (Ben-Sasson & Nordstrom ’11 (informal))
For any arbitrarily slowly growing function g there exist explicitk-CNF formulas of size n
refutable in resolution in space g(n) and
refutable in length linear in n and space ≈ 3√
n such that
any refutation in space 3√
n requires superpolynomial length
And an open problem:
Open Problem
Seems likely that 3√
n above should be possible to improve to√
n, butdon’t know how to prove this. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 17 / 32
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One Example: Robust Trade-offs for Small Space
Theorem (Ben-Sasson & Nordstrom ’11 (informal))
For any arbitrarily slowly growing function g there exist explicitk-CNF formulas of size n
refutable in resolution in space g(n) and
refutable in length linear in n and space ≈ 3√
n such that
any refutation in space 3√
n requires superpolynomial length
And an open problem:
Open Problem
Seems likely that 3√
n above should be possible to improve to√
n, butdon’t know how to prove this. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 17 / 32
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One Example: Robust Trade-offs for Small Space
Theorem (Ben-Sasson & Nordstrom ’11 (informal))
For any arbitrarily slowly growing function g there exist explicitk-CNF formulas of size n
refutable in resolution in space g(n) and
refutable in length linear in n and space ≈ 3√
n such that
any refutation in space 3√
n requires superpolynomial length
And an open problem:
Open Problem
Seems likely that 3√
n above should be possible to improve to√
n, butdon’t know how to prove this. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 17 / 32
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Plan for the Rest of This Lecture
Both of these theorems proved in the same way
Want to sketch intuition and main ideas in proofs
For details, see survey paper in course binder
To prove the theorems, need to go back to the early days ofcomputer science. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 18 / 32
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A Detour into Combinatorial Games
Want to find formulas that
can be quickly refuted but require large space
have space-efficient refutations requiring much time
Such time-space trade-off questions well-studied forpebble games modelling calculations described by DAGs([Cook & Sethi ’76] and many others)
Time needed for calculation: # pebbling moves
Space needed for calculation: max # pebbles required
Some quick graph terminology
DAGs consist of vertices with directed edges between them
vertices with no incoming edges: sources
vertices with no outgoing edges: sinks
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 19 / 32
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A Detour into Combinatorial Games
Want to find formulas that
can be quickly refuted but require large space
have space-efficient refutations requiring much time
Such time-space trade-off questions well-studied forpebble games modelling calculations described by DAGs([Cook & Sethi ’76] and many others)
Time needed for calculation: # pebbling moves
Space needed for calculation: max # pebbles required
Some quick graph terminology
DAGs consist of vertices with directed edges between them
vertices with no incoming edges: sources
vertices with no outgoing edges: sinks
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 19 / 32
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The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 0
Current # pebbles 0
Max # pebbles so far 0
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
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The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 1
Current # pebbles 1
Max # pebbles so far 1
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 38
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 2
Current # pebbles 2
Max # pebbles so far 2
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 39
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 3
Current # pebbles 3
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 40
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 4
Current # pebbles 2
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 41
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 5
Current # pebbles 1
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 42
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 6
Current # pebbles 2
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 43
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 7
Current # pebbles 3
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 44
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 8
Current # pebbles 2
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 45
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 8
Current # pebbles 2
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 46
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 9
Current # pebbles 3
Max # pebbles so far 3
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 47
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 10
Current # pebbles 4
Max # pebbles so far 4
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 48
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 11
Current # pebbles 3
Max # pebbles so far 4
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 49
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 12
Current # pebbles 2
Max # pebbles so far 4
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 50
The Black-White Pebble Game
Goal: get single black pebble on sink z of DAG G (with constant fan-in)
z
x y
u v w
# moves 13
Current # pebbles 1
Max # pebbles so far 4
1 Can place black pebble on (empty) vertex v if all predecessors(vertices with edges to v) have pebbles on them
2 Can always remove black pebble from vertex
3 Can always place white pebble on (empty) vertex
4 Can remove white pebble if all predecessors have pebbles
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 20 / 32
Page 51
Pebbling Contradiction
CNF formula encoding pebble game on DAG G
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
sources are true
truth propagatesupwards
but sink is false
Studied by [Bonet et al. ’98, Raz & McKenzie ’99, Ben-Sasson &Wigderson ’99] and others
We want to show that pebbling properties of DAGs somehow carry over toresolution refutations of pebbling contradictions
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 21 / 32
Page 52
Pebbling Contradiction
CNF formula encoding pebble game on DAG G
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
sources are true
truth propagatesupwards
but sink is false
Studied by [Bonet et al. ’98, Raz & McKenzie ’99, Ben-Sasson &Wigderson ’99] and others
We want to show that pebbling properties of DAGs somehow carry over toresolution refutations of pebbling contradictions
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 21 / 32
Page 53
Interpreting Refutations as Black-White Pebblings
Black-white pebbling models non-deterministic computation (where onecan guess partial results and verify later)
black pebbles ⇔ computed results
white pebbles ⇔ guesses needing to be verified
“Know z assuming v, w”
Corresponds to (v ∧ w) → z, i.e.,blackboard clause v ∨ w ∨ z
So translate clauses to pebbles by:unnegated variable⇒ black pebblenegated variable⇒white pebble
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 22 / 32
Page 54
Interpreting Refutations as Black-White Pebblings
Black-white pebbling models non-deterministic computation (where onecan guess partial results and verify later)
black pebbles ⇔ computed results
white pebbles ⇔ guesses needing to be verified
z
x y
u v w
“Know z assuming v, w”
Corresponds to (v ∧ w) → z, i.e.,blackboard clause v ∨ w ∨ z
So translate clauses to pebbles by:unnegated variable⇒ black pebblenegated variable⇒white pebble
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 22 / 32
Page 55
Interpreting Refutations as Black-White Pebblings
Black-white pebbling models non-deterministic computation (where onecan guess partial results and verify later)
black pebbles ⇔ computed results
white pebbles ⇔ guesses needing to be verified
z
x y
u v w
“Know z assuming v, w”
Corresponds to (v ∧ w) → z, i.e.,blackboard clause v ∨ w ∨ z
So translate clauses to pebbles by:unnegated variable⇒ black pebblenegated variable⇒white pebble
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 22 / 32
Page 56
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 57
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u Download axiom 1: u
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 58
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
vDownload axiom 1: uDownload axiom 2: v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 59
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
u ∨ v ∨ x
Download axiom 1: uDownload axiom 2: vDownload axiom 4: u ∨ v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 60
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
u ∨ v ∨ x
Download axiom 1: uDownload axiom 2: vDownload axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 61
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
u ∨ v ∨ x
v ∨ x
Download axiom 1: uDownload axiom 2: vDownload axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 62
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
u ∨ v ∨ x
v ∨ x
Download axiom 2: vDownload axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ xErase the clause u ∨ v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 63
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
v ∨ x
Download axiom 2: vDownload axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ xErase the clause u ∨ v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 64
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
u
v
v ∨ x
Download axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ xErase the clause u ∨ v ∨ xErase the clause u
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 65
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
v ∨ xDownload axiom 4: u ∨ v ∨ xInfer v ∨ x from
u and u ∨ v ∨ xErase the clause u ∨ v ∨ xErase the clause u
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 66
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
v ∨ xu and u ∨ v ∨ x
Erase the clause u ∨ v ∨ xErase the clause uInfer x from
v and v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 67
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
v ∨ x
x
u and u ∨ v ∨ xErase the clause u ∨ v ∨ xErase the clause uInfer x from
v and v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 68
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
v ∨ x
x
Erase the clause u ∨ v ∨ xErase the clause uInfer x from
v and v ∨ xErase the clause v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 69
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
xErase the clause u ∨ v ∨ xErase the clause uInfer x from
v and v ∨ xErase the clause v ∨ x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 70
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v
xErase the clause uInfer x from
v and v ∨ xErase the clause v ∨ xErase the clause v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 71
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x Erase the clause uInfer x from
v and v ∨ xErase the clause v ∨ xErase the clause v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 72
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
x ∨ y ∨ zInfer x from
v and v ∨ xErase the clause v ∨ xErase the clause vDownload axiom 6: x ∨ y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 73
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
x ∨ y ∨ zErase the clause v ∨ xErase the clause vDownload axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 74
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
x ∨ y ∨ z
y ∨ z
Erase the clause v ∨ xErase the clause vDownload axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 75
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
x ∨ y ∨ z
y ∨ z
Erase the clause vDownload axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ zErase the clause x ∨ y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 76
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
y ∨ zErase the clause vDownload axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ zErase the clause x ∨ y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 77
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
x
y ∨ zDownload axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ zErase the clause x ∨ y ∨ zErase the clause x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 78
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z Download axiom 6: x ∨ y ∨ zInfer y ∨ z from
x and x ∨ y ∨ zErase the clause x ∨ y ∨ zErase the clause x
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 79
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ yInfer y ∨ z from
x and x ∨ y ∨ zErase the clause x ∨ y ∨ zErase the clause xDownload axiom 5: v ∨ w ∨ y
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 80
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ yErase the clause x ∨ y ∨ zErase the clause xDownload axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ y
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 81
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ y
v ∨ w ∨ z
Erase the clause x ∨ y ∨ zErase the clause xDownload axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ y
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 82
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ y
v ∨ w ∨ z
Erase the clause xDownload axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ y
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 83
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ zErase the clause xDownload axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ y
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 84
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
y ∨ z
v ∨ w ∨ zDownload axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ yErase the clause y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 85
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z Download axiom 5: v ∨ w ∨ yInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ yErase the clause y ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 86
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
vInfer v ∨ w ∨ z from
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ yErase the clause y ∨ zDownload axiom 2: v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 87
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
v
w
y ∨ z and v ∨ w ∨ yErase the clause v ∨ w ∨ yErase the clause y ∨ zDownload axiom 2: vDownload axiom 3: w
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 88
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
v
w
z
Erase the clause v ∨ w ∨ yErase the clause y ∨ zDownload axiom 2: vDownload axiom 3: wDownload axiom 7: z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 89
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
v
w
z
Download axiom 2: vDownload axiom 3: wDownload axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 90
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
v
w
z
w ∨ z
Download axiom 2: vDownload axiom 3: wDownload axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 91
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
v
w
z
w ∨ z
Download axiom 3: wDownload axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ zErase the clause v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 92
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
w
z
w ∨ z
Download axiom 3: wDownload axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ zErase the clause v
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 93
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
v ∨ w ∨ z
w
z
w ∨ z
Download axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ zErase the clause vErase the clause v ∨ w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 94
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
w
z
w ∨ z
Download axiom 7: zInfer w ∨ z from
v and v ∨ w ∨ zErase the clause vErase the clause v ∨ w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 95
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
w
z
w ∨ z
v and v ∨ w ∨ zErase the clause vErase the clause v ∨ w ∨ zInfer z from
w and w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 96
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
w
z
w ∨ z
z
v and v ∨ w ∨ zErase the clause vErase the clause v ∨ w ∨ zInfer z from
w and w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 97
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
w
z
w ∨ z
z
Erase the clause vErase the clause v ∨ w ∨ zInfer z from
w and w ∨ zErase the clause w
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 98
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
z
w ∨ z
z
Erase the clause vErase the clause v ∨ w ∨ zInfer z from
w and w ∨ zErase the clause w
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 99
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
z
w ∨ z
z
Erase the clause v ∨ w ∨ zInfer z from
w and w ∨ zErase the clause wErase the clause w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 100
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
z
zErase the clause v ∨ w ∨ zInfer z from
w and w ∨ zErase the clause wErase the clause w ∨ z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 101
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
z
zw and w ∨ z
Erase the clause wErase the clause w ∨ zInfer ⊥ from
z and z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 102
Example of Refutation-Pebbling Correspondence
1. u2. v3. w4. u ∨ v ∨ x5. v ∨ w ∨ y6. x ∨ y ∨ z7. z
z
x y
u v w
z
z
⊥
w and w ∨ zErase the clause wErase the clause w ∨ zInfer ⊥ from
z and z
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 23 / 32
Page 103
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 104
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 105
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 106
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 107
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 108
From Resolution to Pebbling
Theorem (Adapted from [Ben-Sasson ’02])
Any resolution refutation translates into black-white pebbling with
# moves = O(refutation length)
# pebbles = O(# variables on board)
Proof sketch.
For every clause configuration Dt
black-pebble vertices with positive literals
white-pebble vertices with negativt but no positive literals
Argue that for Dt−1 Dt, pebbling placements and removals are legal
Download: Always pebbles below new black pebble
Inference: No change in pebbles
Erasure: Only erase after resolution step; only variable resolved overdisappears ⇒ corresponds to black vertex — OK
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 24 / 32
Page 109
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 110
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 111
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 112
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 113
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 114
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 115
From Pebbling to Resolution
Observation (Ben-Sasson et al. ’00)
Any black-pebbles-only pebbling translates into resolution refutation with
refutation length = O(# moves)
total space = O(# pebbles)
Proof sketch.
Invariant: keep clause u in memory for all black-pebbled vertices u
When source vertex v pebbled, can download source axiom v
When non-source v is pebbled, all predecessors u ∈ pred(v) are black
Download∨
u∈pred(v) u ∨ v and resolve with all clauses u foru ∈ pred(v) to derive v
At end of pebbling, z is black-pebbled
Download sink axiom z and resolve with clause z to derive ⊥
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 25 / 32
Page 116
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 117
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 118
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 119
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 120
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 121
But Unfortunately This Totally Doesn’t Work. . .
Unfortunately pebbling contradictions extremely easy w.r.t. clause space!
Theorem (Ben-Sasson ’02)
Any pebbling contradiction can be refuted in resolution in linear lengthand constant clause space simultaneously
Proof sketch.
Start by resolving z and∨
u∈pred(z) u ∨ z
Then, in reverse topological order of vertices v, resolve with pebblingaxioms
∨u∈pred(v) u ∨ v
Invariant: One clause in memory; only negative literals; only forvertices preceding v in topological order
Finally, have one wide clause with negative literals over all sources
Use source axioms to resolve away these literals one by one
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 26 / 32
Page 122
Key New Idea: Variable Substitution
Make formula harder by substituting exclusive or x1 ⊕ x2 of two newvariables x1 and x2 for every variable x (also works for other Booleanfunctions with “right” properties):
x ∨ y
⇓
¬(x1 ⊕ x2) ∨ (y1 ⊕ y2)
⇓
(x1 ∨ x2 ∨ y1 ∨ y2)
∧ (x1 ∨ x2 ∨ y1 ∨ y2)
∧ (x1 ∨ x2 ∨ y1 ∨ y2)
∧ (x1 ∨ x2 ∨ y1 ∨ y2)
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 27 / 32
Page 123
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 124
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 125
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 126
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 127
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
x1 ∨ x2
x1 ∨ x2
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 128
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
x1 ∨ x2
x1 ∨ x2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 129
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
x1 ∨ x2
x1 ∨ x2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
y1 ∨ y2
y1 ∨ y2
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 130
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
x1 ∨ x2
x1 ∨ x2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
y1 ∨ y2
y1 ∨ y2
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 131
Key Technical Result: Substitution Theorem
Let F [⊕] denote formula with XOR x1 ⊕ x2 substituted for x
Obvious approach for refuting F [⊕]: mimic refutation of F
x
x ∨ y
y
For such refutation of F [⊕]:
length ≥ length for F
clause space ≥ # variables onboard in proof for F
x1 ∨ x2
x1 ∨ x2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
x1 ∨ x2 ∨ y1 ∨ y2
y1 ∨ y2
y1 ∨ y2
Prove that this is (sort of) best one can do for F [⊕]!
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 28 / 32
Page 132
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 133
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 134
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 135
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 136
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 137
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 138
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 139
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 140
Sketch of Proof of Substitution Theorem
Given refutation of F [⊕], extract “shadow refutation” of F
XOR formula F [⊕] Original formula F
If XOR blackboard implies e.g.¬(x1 ⊕ x2) ∨ (y1 ⊕ y2). . .
write x ∨ y on shadow blackboard
For consecutive XOR blackboardconfigurations. . .
can get between correspondingshadow blackboards by legal reso-lution derivation steps
. . . (sort of) upper-bounded byXOR derivation length
Length of shadow blackboardderivation . . .
. . . is at most # clauses on XORblackboard
# variables mentioned on shadowblackboard. . .
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 29 / 32
Page 141
Putting the Pieces Together
Making variable substitutions in pebbling formulas
lifts lower bound from number of variables to clause space
maintains upper bound in terms of total space and length
Get our results by
using known pebbling results from literature of 70s and 80s
proving a couple of new pebbling results [Nordstrom ’10]
to get tight trade-offs, showing that resolution proofs can sometimesdo better than black-only pebblings [Nordstrom ’10]
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 30 / 32
Page 142
Putting the Pieces Together
Making variable substitutions in pebbling formulas
lifts lower bound from number of variables to clause space
maintains upper bound in terms of total space and length
Get our results by
using known pebbling results from literature of 70s and 80s
proving a couple of new pebbling results [Nordstrom ’10]
to get tight trade-offs, showing that resolution proofs can sometimesdo better than black-only pebblings [Nordstrom ’10]
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 30 / 32
Page 143
Extension to Polynomial Calculus
Using somewhat different techniques, can extend trade-offs topolynomial calculus [Beck, Nordstrom & Tang ’12]
Same formulas and much simpler proof, but lose a bit in parameters
Also, can’t get unconditional space lower bounds for polynomialcalculus this way
Will discuss space in polynomial calculus in final two lectures
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 31 / 32
Page 144
An Intriguing Open Problem
Recall key technical theorem: amplify space lower bounds through variablesubstitution
Almost completely oblivious to proof system under study, and has beenextended to strictly stronger k-DNF resolution proof systems — maybecan be made to work for other stronger systems as well?
Open Problem
Can the Substitution Theorem be proven for, say, cutting planes orpolynomial calculus, thus yielding space lower bounds and time-spacetrade-offs for these proof systems as well?
Approach in previous papers provably will not work
Partial progress with different techniques in [Huynh & Nordstrom ’12] and[Beck, Nordstrom & Tang ’12] indicate that answer should be “yes”
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 32 / 32
Page 145
An Intriguing Open Problem
Recall key technical theorem: amplify space lower bounds through variablesubstitution
Almost completely oblivious to proof system under study, and has beenextended to strictly stronger k-DNF resolution proof systems — maybecan be made to work for other stronger systems as well?
Open Problem
Can the Substitution Theorem be proven for, say, cutting planes orpolynomial calculus, thus yielding space lower bounds and time-spacetrade-offs for these proof systems as well?
Approach in previous papers provably will not work
Partial progress with different techniques in [Huynh & Nordstrom ’12] and[Beck, Nordstrom & Tang ’12] indicate that answer should be “yes”
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 32 / 32
Page 146
An Intriguing Open Problem
Recall key technical theorem: amplify space lower bounds through variablesubstitution
Almost completely oblivious to proof system under study, and has beenextended to strictly stronger k-DNF resolution proof systems — maybecan be made to work for other stronger systems as well?
Open Problem
Can the Substitution Theorem be proven for, say, cutting planes orpolynomial calculus, thus yielding space lower bounds and time-spacetrade-offs for these proof systems as well?
Approach in previous papers provably will not work
Partial progress with different techniques in [Huynh & Nordstrom ’12] and[Beck, Nordstrom & Tang ’12] indicate that answer should be “yes”
Jakob Nordstrom (KTH) Proof complexity: Lecture 2 EWSCS ’12 32 / 32