Transcript A Tutorial Introduction to Proof Complexity

A Tutorial Introduction to Proof
Complexity
Paul Beame
University of Washington
Proof Systems and Their
Complexity
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NP, proofs, and proof systems
L∊NP: there is a polynomial time computable V s.t.
 x ∊ L ⇔ y . |y|≤|x|O(1). V(x,y)
 V is a verifier for L
 y is a proof that x ∊ L
Defn: A proof system* for L is a polytime (verifier) V s.t.
 x ∊ L ⇔ y . V(x,y)
*Cook-Reckhow
defn: polytime computable map f from Σ* onto L≠∅
(f(x,y)=x if V(x,y) and = a fixed x0 ∊ L otherwise)
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Complexity of a proof system
Defn: A proof system for L is a polytime (verifier) V s.t.
 x ∊ L ⇔ y . V(x,y)
Defn: The complexity of proof system V is smallest fnctn
S: ℕ →ℕ s.t. x ∊ L ⇔ y . |y|≤ S(|x|). V(x,y)
Defn: V is polynomially bounded iff S(n) is nO(1)
Cor: L∊NP iff L has a polynomially bounded proof system
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Propositional proof systems
Defn: A propositional proof system is a proof system V
for the set TAUT of propositional logic tautologies
 F ∊ TAUT ⇔  proof P . V(F,P)
  direction is usually called soundness
  direction is usually called completeness
Alt Defn: A propositional proof system is a proof system
V for the set UNSAT of propositional logic contradictions
Since TAUT and UNSAT are coNP-complete…
Thm: NP=coNP iff there is a polynomially bounded
propositional proof system
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Proof complexity vs search
 Proof complexity only measures how large proofs
must be, not how easy they are to find
 Lower bounds on proof complexity imply lower bounds on
nondeterministic algorithms
 Stronger than lower bounds on deterministic search
 Any complete SAT solver yields a propositional proof
system
 Proof that F∊UNSAT is transcript of failed search on input F
 Size of proof ≈ running time of SAT solver on input F
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p-simulation
Defn: Proof system U polynomially simulates proof
system V iff
 they prove the same language L
 proofs in V can be efficiently converted into proofs in U
Defn: U and V are polynomially equivalent iff they
polynomially simulate each other
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Proof systems using CNF input
By same trick [Tseitin 68, Cook 71] that reduces SAT to CNFSAT, can
assume w.l.o.g. that propositional proof systems are for the
language CNF-UNSAT
 Given propositional formula F produce CNF formula F’
 Add an extra variable yG corresponding to each
sub-formula G of F
 F’ has clauses expressing the fact that yG takes on the
value of G determined by the inputs to F
 Add clause yF to express that F is must be true
Thm: F’ ∊ CNF-UNSAT iff F ∊ UNSAT
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Clauses
 if F = G  H, include clauses
 yG  yF
 yH  yF
 yF  yG  yH
 if F = G  H, include clauses
 yF  yG
 yF  yH
 yG  yH  yF
 if F = G, include clauses
 yF  yG
 yF  yG
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Sample propositional proof systems
 Truth tables
 proof is a fully filled out truth table
 easy to verify that it is filled out correctly and all truth
assignments yield T
 Axiom/Inference systems
 inference rules: e.g. modus ponens A, (A  B) | B
 axioms: e.g. excluded middle | (A  A)
 axioms & inference rules are schemas
 can make consistent substitution of arbitrary formulas for
variables in schema
 e.g. excluded middle yields ((xy)  (xy))
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Resolution
 Refutation system using CNF clauses only
 Start with original input clauses of CNF F
 Resolution rule
 (A  x), (B  x) | (A  B)
 Goal: derive empty clause ⊥
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Frege Systems
 Finite, implicationally complete set R of axioms/inference rules
 Refutation version:
 Proof of unsatisfiability of F: sequence F1,…,Fr of formulas
(called lines) s.t.
 F1 = F
 each Fj follows from an axiom in R or follows from
previous ones via an inference rule in R
 Fr = ⊥ trivial falsehood, e.g. (x x)
 Positive version:
 Start with nothing, end with tautology F
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All Frege systems are p-equivalent
 Key idea: Every use of a rule of one system can be
derived in the other system in a constant # of steps
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𝓒-Frege proof systems
 Many circuit complexity classes 𝓒 are defined as follows:
 𝓒 = {f: f is computed by polynomial-size circuits
with structural property P𝓒}
 Define 𝓒-Frege to be the p-equivalence class of Frege-style
proof systems s.t.
 each line has structural property P𝓒
 it has a finite set of axioms/inference rules that is complete
for circuits with property P𝓒
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Some circuit classes 𝓒





P/poly - polysize circuits
NC1 - polysize formulas = O(log n)-depth fan-in 2 circuits
Clauses
k-DNF - k-DNF formulas
AC0 - constant-depth unbounded fan-in polysize
circuits using AND/OR/NOT gates
 TC0 - threshold gates instead
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Examples
 Frege = NC1-Frege
 NC1 (logarithmic depth fan-in 2) circuits can be expanded
into trees (formulas) of polynomial size
 Formulas can always be re-balanced so they have
logarithmic depth (which are automatically polynomial size)
 Resolution = Clauses-Frege
 Every line is a clause
 Res(k) = k-DNF-Frege
 Every line is a k-DNF formula
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Extended Frege Proofs
 Like Frege proofs plus extra extension steps
 Each extension step defines a new propositional variable to
stand for an arbitrary formula on the current set of variables
 Using extension variables, each line represents a circuit in
the original variables
 Extended-Frege = P/poly-Frege
 Equivalent to Substitution-Frege in which each inferred
formula immediately is available as an axiom schema
 Equivalent to Extended-Resolution which adds extension
clauses that define new variable y≡C using clauses y  C
and z  y for each z in C.
 Idea: SAT to CNF-SAT conversion also works for circuits
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The DAG of a proof
F1
Axioms/inputs
are sources
F5
F2
F3
F10
F8
F7
F6
F11
F4
F12
F9
F13
Sink labelled by tautology
(or ⊥ for a refutation)
Inference rule
associated with
each node
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Proof structure
 For any axiom/inference system can consider
 Tree-like proofs
 Restricted version in which the proof DAG must be a
tree. (Source formulas may be repeated.)
 General (DAG-like) proofs
 No restriction on the proof DAG
 Other systems can be static (in which the entire proof
string can only be seen as one large inference)
 E.g. Truth tables
 Can sometimes use a similar idea for all three kinds of
proofs. Proof size: static ≥ tree-like ≥ general
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Note: Special structure for resolution proofs
So far
 Resolution
 Proof structure is an unrestricted DAG
 Tree-like resolution
 Proof structure is a tree
Also
 Regular resolution
 A clause that is derived by resolving on variable x can
never lead to a clause in which variable x is re-introduced
 Generalizes tree resolution which has this property wlog.
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Davis-Putnam-Logemann-Loveland
(DPLL) Procedure
 A family of complete SAT solvers
 a collection of algorithms for finding SAT
assignments/proofs
 Its traces form proofs of unsatifiability
 ≡ tree-like resolution refutations
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Simple DPLL Algorithm
 DPLL(F)
 While (F contains a clause of size 1)
unit
 set variable to make that clause true
propagation
 simplify all clauses using this assignment
 If F = ∅ then
 output current truth assignment and HALT
 If F does not contain ⊥ (empty clause) then
 Choose unset literal x
 Run DPLL(Fx0)
 Run DPLL(Fx1 )
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DPLL Refutation Trace
a
Clauses
0
1. a b c
2. ac
3. b
4. a d
5. d b
1
b
0
1
c
0
1
b
0
d
3
1
2
1
0
4
3
1
5
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DPLL Refutation Trace
a
Clauses
0
1. a b c
2. ac
3. b
4. a d
5. d b
1
b
0
1
c
0
b
0
d
3
1
b
1
0
1
2
4
a b c
ac
a d
3
1
b
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. d b
b
c
b
d
3
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. d b
b
c
b
d
3
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. a b
b
b
d:a b
3
c: a b
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. a b
b
b
d:a b
3
c: a b
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. a b
b: a
b: a
c: a b
d:a b
3
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution
a
Clauses
1. a b c
2. ac
3. b
4. a d
5. a b
b: a
b: a
c: a b
d:a b
3
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Tree Resolution Refutation
a:⊥
Clauses
1. a b c
2. ac
3. b
4. a d
5. a b
b: a
b: a
c: a b
d:a b
3
3
b
b
1
2
4
a b c
ac
a d
5
d b
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Conflict-Directed Clause-Learning (CDCL)
 Complete SAT solvers that extend DPLL with clauses
inferred via special forms of resolution inference
 Traces on unsatisfiable formulas yield resolution
refutations.
 See tomorrow’s tutorial by Sam Buss to see their
generality
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Proof systems based on other
representations of clauses
 Algebraic:
 Clause 𝑪 = (𝒙  ¬𝒚  𝒛) becomes equation
 𝒇𝑪 ≡ 𝟏 − 𝒙 𝒚 𝟏 − 𝒛 = 𝟎 plus 𝒙𝟐 − 𝒙 = 𝟎 or …
 𝒇𝑪 ≡ 𝒙′ 𝒚𝒛′ = 𝟎 plus equations 𝒙 + 𝒙’ − 𝟏 = 𝟎 etc.
 Derive 𝟏 = 𝟎
 0-1-inequalities:
 Clause 𝑪 = (𝒙 ¬𝒚  𝒛) becomes inequalities
 𝒙 + 𝟏 − 𝒚 + 𝒛 ≥ 𝟏 plus 𝒙 ≤ 𝟏 and 𝒙 ≥ 𝟎 etc.
 Derive 𝟎 ≥ 𝟏
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Hilbert’s Nullstellensatz
System of polynomials
𝒇𝟏(𝒙𝟏, … , 𝒙𝒏) = 𝟎, … , 𝒇𝒎(𝒙𝟏, … , 𝒙𝒏) = 𝟎
over field 𝑲 has no solution in any extension field of 𝑲

there exist polynomials 𝒈𝟏(𝒙𝟏, … , 𝒙𝒏), … , 𝒈𝒎(𝒙𝟏, … , 𝒙𝒏)
in 𝑲[𝒙𝟏, … , 𝒙𝒏] s.t. 𝒏𝒊=𝟏 𝒇𝒊 𝒈𝒊 ≡ 𝟏
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Static: Nullstellensatz proof system
 Clause 𝑪 = (𝒙𝟏 𝒙𝟐  𝒙𝟑) becomes equation
𝒇𝑪 ≡ (𝟏 − 𝒙𝟏)𝒙𝟐(𝟏 − 𝒙𝟑) = 𝟎
 Add equations 𝒇𝒊 ≡ 𝒙𝟐𝒊 − 𝒙𝒊 = 𝟎 for each variable
 Guarantees that only solutions are in {𝟎, 𝟏}𝒏
 A refutation is polynomials 𝒈𝑪 and 𝒈𝒊 proving
unsatisfiability: i.e. such that
𝑪 𝒇𝑪 𝒈𝒄 + 𝒊 𝒇𝒊 𝒈𝒊 ≡ 𝟏
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Polynomial Calculus with Resolution (PCR)
 Axiom/inference system similar to static Nullstellensatz except:
 Variables 𝒙𝒊 and 𝒙𝒊 ′ with 𝒙𝒊 + 𝒙′𝒊 = 𝟏, 𝒙𝟐𝒊 = 𝒙𝒊 , 𝒙′𝒊 𝟐 = 𝒙′𝒊
 Clause 𝑪 = (𝒙¬𝒚  𝒛) becomes 𝒇𝑪 = 𝒙′ 𝒚𝒛′ = 𝟎
 Given polynomials 𝑹 and 𝑺 can infer
 𝒂𝑹 + 𝒃𝑺 for any 𝒂, 𝒃 in 𝑲
 𝒙𝒊𝑹
 Derive constant polynomial 𝟏
 Degree = max degree of any polynomial in the proof
 Size = # of monomials in proof
 Can find proof of degree 𝒅 in time 𝒏𝐎
basis-like algorithm (linear algebra)
𝒅
using Groebner
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PCR p-simulates resolution
Resolution
PCR
(abcd) (abcr)
(bcdr)
Given 𝒂’𝒃’𝒄’𝒅 and 𝒂𝒃’𝒄’𝒓
derive 𝒂’𝒃’𝒄’𝒅 𝒓 + (𝒂𝒃’𝒄’𝒓)𝒅
= 𝒂’ + 𝒂 𝒃’𝒄’𝒅𝒓
= 𝒃’𝒄’𝒅𝒓
37
Proof systems based on other
representations of clauses
 Algebraic:
 Clause 𝑪 = (𝒙  ¬𝒚  𝒛) becomes equation
 𝒇𝑪 ≡ 𝟏 − 𝒙 𝒚 𝟏 − 𝒛 = 𝟎 plus 𝒙𝟐 −𝒙 = 𝟎 or …
 𝒇𝑪 ≡ 𝒙′ 𝒚𝒛′ = 𝟎 plus equations 𝒙 + 𝒙’ − 𝟏 = 𝟎 etc.
 Derive 𝟏 = 𝟎
 0-1-inequalities:
 Clause 𝑪 = (𝒙 ¬𝒚  𝒛) becomes inequality
 𝒙 + 𝟏 − 𝒚 + 𝒛 ≥ 𝟏 plus 𝒙 ≤ 𝟏 and 𝒙 ≥ 𝟎 etc.
 Derive 𝟎 ≥ 𝟏
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Cutting Planes [Gomory 59, Chvatal 73]:
 addition:
a1x1 + ... + anxn  A
b1x1 + ... + bnxn  B
(a1+b1)x1+...+(an+bn)xn  A+B
 multiplication by positive integer:
a1x1 + ... + anxn  A
ca1x1 + ... + canxn  cA
 Division by positive integer:
ca1x1 + ... + canxn  B
a1x1 + ... + anxn  B/c
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Cutting planes geometry
40
Cutting planes geometry
41
Cutting Planes p-simulates resolution
Resolution
Cutting
Planes
(abcd) (abcr)
(bcdr)
a + b + c + (1-d)  1
(1-a) + b + c + (1-r)  1
(1-d)  0
(1-r)  0
2b + 2c + 2(1-d) + 2(1-r)  1 Addition
b + c + (1-d) + (1-r)  1
Division
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Semi-algebraic proof systems
 Positivstellensatz
 Start with same polynomial equalities 𝒇𝒋 = 𝟎 as
Nullstellensatz over ℝ
 Write 𝒈𝒋 and 𝒉𝒌 such that
𝒋 𝒇𝒋 𝒈𝒋 = 𝟏 +
𝟐
𝒉
𝒌 𝒌
 OR, start with linear inequalities 𝑳𝒋 ≥ 𝟎, derive 𝟎 ≥ 𝟏
 Use higher-degree consequences including use of 𝒙𝟐 = 𝒙.
 Possibly use 𝒑𝟐 (𝒙) ≥ 𝟎 for polynomials 𝒑(𝒙). (SDP)
 Sherali-Adams static, no SDP
 Lasserre/Sum-of-Squares static, SDP
 Lovasz-Schrijver restricted dynamic, variants with SDP
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ZFC
Some Proof System Relationships
P/poly-Frege
Frege
TC0-Frege
AC0-Frege
Positivstellensatz
Cutting Planes
Lasserre/SOS
Res(k)
Polynomial Calculus/PCR
Resolution
Nullstellensatz
DPLL
Truth Tables
44
Why all these proof systems?
 Proof systems formalize different types of reasoning
 Why even include the weaker systems within a given type of
reasoning?
 many weaker proof systems have better associated proof
search strategies, e.g. DPLL, Polynomial Calculus.
 Natural correspondence with circuit complexity classes
 analyze systems working upwards in proof strength to gain
insight for techniques
 Some proof systems correspond to our best algorithms for
NP-hard optimization problems
 e.g. semi-definite programming and Lasserre/SOS
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Lower Bound Methods
46
Width vs size in resolution proofs
Defn: If 𝑭 is a set of clauses
 𝑹𝒆𝒔 (𝑭) = length of shortest resolution refutation of 𝑭
= ∞ if 𝑭 is satisfiable
 𝒘(𝑭) = length of longest clause in 𝑭
 𝒘𝒊𝒅𝒕𝒉(𝑭) = 𝐦𝐢𝐧𝑷 {𝒘(𝑷): 𝑷 is a resolution refutation of 𝑭}
Thm [Ben-Sasson,Wigderson]: Every resolution refutation of 𝑭 of
size 𝑺 can be converted to one of width 𝟐𝒏 ln 𝑺 + 𝒘(𝑭)
Corollary: 𝑹𝒆𝒔 𝑭 ≥ 𝒆𝒙𝒑(
𝒘𝒊𝒅𝒕𝒉 𝑭 – 𝒘 𝑭 𝟐
)
𝟐𝒏
47
Width vs size in resolution proofs
Thm [Ben-Sasson,Wigderson]: Every resolution refutation of 𝑭 of
size 𝑺 can be converted to one of width 𝟐𝒏 ln 𝑺 + 𝒘(𝑭)
Pf: Let 𝑾 = 𝟐𝒏 ln 𝑺. A clause 𝑪 is large iff 𝒘 𝑪 ≥ 𝑾.
By induction on 𝒏 and 𝒌: If (𝟏 − 𝑾/𝟐𝒏)𝒌 𝑺 ≤ 𝟏 then any 𝑭
with ≤ 𝑺 large clauses has refutation of width  𝒌 + 𝒘(𝑭)
 Choose literal 𝒙 most frequently occurring in large clauses
and set it to 𝟏, satisfying  𝑾𝑺/𝟐𝒏 large clauses
 Result is a proof of 𝑭𝒙𝟏 with  𝑺 𝟐(𝟏 − 𝑾/𝟐𝒏) large clauses
𝑾
−
𝑾 𝑺proof
 By
 𝒌𝒆−−ln
𝟏 +𝑺𝒘(𝑭)
𝟐𝒏 𝑺 =
𝒙𝟏 has
𝟏induction
− 𝑾 ∕ 𝑭𝟐𝒏
≤ 𝒆of width
𝑺=1
 So can derive 𝒙 from 𝑭 in width 𝒌 + 𝒘(𝑭)
 Also by induction, proof of 𝑭𝒙𝟎 of width 𝒌 + 𝒘(𝑭)
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Width vs size in resolution proofs
Thm [Ben-Sasson,Wigderson]: Every resolution refutation of 𝑭 of
size 𝑺 can be converted to one of width 𝟐𝒏 ln 𝑺 + 𝒘(𝑭)
Pf: Let 𝑾 = 𝟐𝒏 ln 𝑺. A clause 𝑪 is large iff 𝒘 𝑪 ≥ 𝑾.
By induction on 𝒏 and 𝒌: If (𝟏 − 𝑾/𝟐𝒏)𝒌 𝑺 ≤ 𝟏 then any 𝑭
with ≤ 𝑺 large clauses has refutation of width  𝒌 + 𝒘(𝑭)
 So can derive 𝒙 from 𝑭 in width 𝒌 + 𝒘(𝑭)
 Also by induction, refutation of 𝑭𝒙𝟎 of width 𝒌 + 𝒘(𝑭)
Width 𝒌 + 𝒘(𝑭) refutation of 𝑭:
 Derive 𝒙 from 𝑭
 Resolve 𝒙 with all clauses of 𝑭 to yield 𝑭𝒙𝟎
 Refute 𝑭𝒙𝟎
49
Width vs size in tree-resolution proofs
Defn: If 𝑭 is a set of clauses
 𝑹𝒆𝒔𝑡𝑟𝑒𝑒 (𝑭) = length of shortest tree-resolution
refutation of 𝑭
= ∞ if 𝑭 is satisfiable
 𝒘(𝑭) = length of longest clause in 𝑭
 𝒘𝒊𝒅𝒕𝒉 (𝑭) = 𝐦𝐢𝐧𝑷 {𝒘(𝑷): 𝑷 is a resolution refutation of 𝑭}
Thm [Ben-Sasson,Wigderson]: Every DPLL/tree-resolution
refutation of 𝑭 of size 𝑺 can be converted to one of width
log2 𝑺 + 𝒘(𝑭)
Corollary: 𝑹𝒆𝒔𝑡𝑟𝑒𝑒 (𝑭) ≥ 𝟐𝒘𝒊𝒅𝒕𝒉(𝑭) – 𝒘(𝑭)
50
Proof: Width of Tree-like Resolution
⊥
𝒙
𝒙
 S/2
S
51
Width of Tree-like Resolution
⊥
𝒙
𝑭 proves 𝒙 in
size at most S/2
𝒙
 S/2
S
52
Width of Tree-like Resolution
⊥
𝒙
𝑭 proves 𝒙 in
size at most S/2
𝑭 proves 𝒙 in
size less than S
𝒙
 S/2
S
53
Width of Tree-like Resolution
⊥
𝒙
Fx1 proves ⊥ in
size at most 𝑺/𝟐
Fx0 proves ⊥ in
size less than 𝑺
𝒙
 S/2
S
54
Width of Tree-like Resolution
⊥
𝒙
Fx1 proves ⊥ in
size at most 𝑺/𝟐
𝒙
 S/2
S
By induction Fx1 proves ⊥ in width at most
log2(𝑺/𝟐) + 𝒘(𝑭) = log2𝑺 + 𝒘(𝑭) − 𝟏
55
Width of Tree-like Resolution
⊥
𝒙
Fx1 proves ⊥ in
size at most 𝑺/𝟐
𝒙
𝒘
S
By induction Fx1 proves ⊥ in width at most
log2(𝑺/𝟐) + 𝒘(𝑭) = log2𝑺 + 𝒘(𝑭) − 𝟏
𝑭 proves in 𝒙 in width at most 𝒘 = log𝟐𝑺 + 𝒘(𝑭)
56
Width of Tree-like Resolution
⊥
𝒙
𝒙
𝒘
Fx0 proves ⊥ in
size < S
S
By induction Fx0 proves ⊥ in width
at most log2 𝑺 + 𝒘(𝑭)
57
Conclusion: Width of Tree-like Resolution
New Refutation:
1. Derive 𝒙 from 𝑭 in width 𝒘
2. Resolve 𝒙 with clauses
of 𝑭 containing 𝒙 to derive 𝑭𝒙𝟎
3. Prove ⊥ in width 𝒘 from 𝑭𝒙𝟎
𝒘 = log𝟐𝑺 + 𝒘(𝑭)
⊥
𝒘
𝒙
𝒙
𝒘
𝒙
𝒙
𝒘 𝒘 𝒘
58
Using width-size relationships
 Cor: [Ben-SassonWigderson]:
 𝑹𝒆𝒔
𝒘𝒊𝒅𝒕𝒉 𝑭 – 𝒘 𝑭 𝟐
𝑭 ≥ 𝒆𝒙𝒑
𝟐𝒏
(𝑭) ≥ 𝟐𝒘𝒊𝒅𝒕𝒉(𝑭) – 𝒘(𝑭)
 𝑹𝒆𝒔𝑡𝑟𝑒𝑒
 Cor: [BP 96]: If there is a resolution refutation of size
𝑺 then can find one in time/size 𝒏𝑶( 𝒏 ln 𝑺 )
 Derive all clauses of width at most 𝑾 = 𝟐𝒏 ln 𝑺.
 Cor: If there is a tree-resolution refutation of size 𝑺
then can find one in time/size 𝒏𝑶(𝐥𝐨𝐠 𝑺) = 𝑺𝑶(𝐥𝐨𝐠 𝒏) .
59
Notes
 Relationship between width and size is roughly
optimal for general resolution
 [Bonet, et al 99] Ordering tautologies with constant
input clause size and polynomial-size proofs that
require width 𝛀( 𝒏)
 DPLL/tree-resolution can require exponentially larger
proofs than general resolution [BEGJ 98],[BW 98].
 Polynomial versus 𝟐Ω(𝒏/log 𝒏) size
 Uses graph pebbling and width-based lower bound
60
Bounding width: boundary expansion
 𝑭 - a set of clauses
 𝜹𝑭 - boundary of 𝑭 = set of variables appearing
in exactly one clause of 𝑭
 𝒔(𝑭) - minimum size non-empty subset
of 𝑯 ⊆ 𝑭 with 𝜹𝑯 = ∅
 𝒆(𝑭) - boundary expansion of 𝑭
= min 𝜹𝑯 : 𝑯 ⊆ 𝑭,
𝒔(𝑭)
𝟑
≤ 𝑯 ≤
𝟐𝒔(𝑭)
𝟑
61
Width vs boundary expansion
 Lemma [ChvatalSzemeredi] 𝒘𝒊𝒅𝒕𝒉(𝑭) ≥ 𝒆(𝑭).
≥ 𝒔(𝑭)
⊥
62
Width vs boundary expansion
 Lemma [ChvatalSzemeredi] 𝒘𝒊𝒅𝒕𝒉(𝑭) ≥ 𝒆(𝑭).
≥ 𝒔(𝑭)
Start at ⊥
Go to parent with
more sources until
≤ 𝟐𝒔(𝑭)/𝟑 sources
⊥
63
Width vs boundary expansion
 Lemma [ChvatalSzemeredi] 𝒘𝒊𝒅𝒕𝒉(𝑭) ≥ 𝒆(𝑭).
≥ 𝒔(𝑭)
Start at ⊥
Go to parent with
more sources until
≤ 𝟐𝒔(𝑭)/𝟑 sources
⊥
64
Width vs boundary expansion
 Lemma [ChvatalSzemeredi] 𝒘𝒊𝒅𝒕𝒉(𝑭) ≥ 𝒆(𝑭).
≥ 𝒔(𝑭)
Start at ⊥
Go to parent with
more sources until
≤ 𝟐𝒔(𝑭)/𝟑 sources
⊥
65
Width vs boundary expansion
 Lemma [ChvatalSzemeredi] 𝒘𝒊𝒅𝒕𝒉(𝑭) ≥ 𝒆(𝑭).
𝒔(𝑭)/𝟑 to 𝟐𝒔(𝑭)/𝟑
𝑯
contains 𝛅𝑯
literals
𝛅𝑯 ≥ 𝒆(𝑭)
⊥
66
Boundary expansion implies large size
Corollary [Ben-SassonWigderson]:
 𝑹𝒆𝒔 𝑭 ≥ 𝒆𝒙𝒑
𝒆 𝑭 –𝒘 𝑭 𝟐
𝟐𝒏
 𝑹𝒆𝒔𝑡𝑟𝑒𝑒(𝑭) ≥ 𝟐𝒆(𝑭) – 𝒘(𝑭)
 So…to prove strong 𝟐𝛀 𝒏 resolution lower
bound for 𝑭 with small clauses
 prove that 𝒆(𝑭) is 𝛀 𝒏
67
PCR degree bounds imply size bounds
Thm [Clegg, Edmonds, Impagliazzo] : Every PCR proof of
𝑭 of size 𝑺 can be converted to one of degree
𝒅 ≤ 𝟐𝒏 ln 𝑺 + 𝒘(𝑭)
 [CEI] showed this for polynomial calculus before
BenSasson-Wigderson’s argument for resolution.
 Suffices to prove lower bounds on PCR degree to get
strong lower bounds.
68
Hard examples
69
Counting
 Pigeonhole principle 𝑷𝑯𝑷𝒎
𝒏
 No 1-1 function from 𝒎 to 𝒏 for 𝒎 > 𝒏
 If 𝒎 = 𝒏 + 𝟏 write as 𝑷𝑯𝑷𝒏
70
Pigeonhole propositional formulas
Variables
Complete bipartite graph of
variables 𝑷𝒊𝒋 representing 𝒇(𝒊) = 𝒋
Clauses
𝒇 is total: (𝑷𝒊𝟏 𝑷𝒊𝟐 … 𝑷𝒊𝒏) for 𝒊 = 𝟏, … , 𝒎
𝒇 is 1-1: (𝑷𝒊𝒋  𝑷𝒌𝒋) for 𝟏  𝒊  𝒌  𝒎, 𝒋 = 𝟏, … , 𝒏
𝑷𝑯𝑷(𝑮): for graph 𝑮 contained in 𝒎 × [𝒏]
𝒇 is total: 𝒋: 𝒊,𝒋 ∈𝑬(𝑮) 𝑷𝒊𝒋 for 𝒊 ∈ 𝒎
𝒇 is 1-1: 𝑷𝒊𝒋  𝑷𝒌𝒋 for 𝒊, 𝒋 ≠ 𝒌, 𝒋 ∈ 𝑬(𝑮), 𝒋 ∈ 𝒏
71
Resolution and 𝑷𝑯𝑷𝒏
 Thm [Haken 84, BP 96] Any resolution proof of 𝑷𝑯𝑷𝒏 requires size
at least 𝟐𝒏/𝟐𝟎 .
 Follows from more direct methods
 Can’t use width-size bound directly since 𝒘(𝑷𝑯𝑷𝒏) is large
(total clauses have size 𝒏).
 Instead use constant-degree bipartite graph 𝑮 on 𝒏 + 𝟏 × 𝒏
 Prove boundary expansion lower bound for 𝑷𝑯𝑷(𝑮)
 ≡ expansion lower bound on 𝑮
 Implies 𝟐𝛀
𝒏
lower bound for 𝑷𝑯𝑷𝒏
72
More counting
 Counting 𝒎𝒐𝒅 𝟐
𝑪𝒐𝒖𝒏𝒕𝟐𝒏+𝟏
𝟐
 no matching on an odd size set
 variable for each edge
 Counting 𝒎𝒐𝒅 𝒓
 no perfect 𝒓-partition if 𝒓 doesn’t divide 𝒏
𝑪𝒐𝒖𝒏𝒕𝒏𝒓 for 𝒏 ≢ 𝟎 (𝒎𝒐𝒅 𝒓)
73
Counting and algebraic/inequality proofs
 Cutting-planes can count:
 Efficient refutations of 𝑷𝑯𝑷𝒏 and 𝑪𝒐𝒖𝒏𝒕𝒏𝒓
 Algebraic proofs over field of characteristic 𝒑 can only
count mod 𝒑
 PCR proofs require 𝛀 𝒏 degree/exponential size
to refute 𝑷𝑯𝑷𝒏 and 𝑪𝒐𝒖𝒏𝒕𝒏𝒓 for most 𝒏 and 𝑟.
74
Tseitin formulas - odd-charged graphs
 Given a low degree graph 𝑮(𝑽, 𝑬) with 0-1 charge
𝝌 𝒗 on each node 𝒗 s.t. total charge is odd
 One variable x𝒆 per edge 𝒆𝑬
 Clauses saying parity of edges touching 𝒗 is 𝝌 𝒗
 Needs 𝟐𝒅𝒆𝒈 𝒗 −𝟏 clauses
 If degree is large, add extension variables to compute
parity at each vertex
0
 Unsatisfiable
1
1
0
 Sum of degrees is even
0
1
75
Expander graphs
Defn: Let 𝑮 = (𝑽, 𝑬) be a graph. Let 𝑬(𝑺, 𝑺) ⊆ 𝑬 be those edges
with one endpoint in 𝑺 and one outside 𝑺.
Graph 𝑮 has expansion 𝜺 if |𝑬 𝑺, 𝑺 | ≥ 𝜺|𝑺| for all subsets 𝑺 of
size at most 𝒏/𝟐 vertices.
Fact: Random constant-degree regular graphs almost always
have constant expansion 𝜺 > 𝟎.
 Explicit constructions also with many applications in complexity
 Originally considered for regular resolution lower bounds [Galil]
76
Tseitin formulas using mod 2 reasoning
 Assume wlog # vertices 𝒏 is odd and all charges are 𝟏.
 Input parity equations are just
𝒗∈𝒆 𝒙𝒆 = 𝟏 mod 𝟐 for each 𝒗 ∈ 𝑽.
 Can add equations mod 2 to get 𝟎 = 𝟏.
 Adding equations for all vertices in 𝑺 ⊆ 𝑽 yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
77
Tseitin formulas and mod 2 equations
Adding equations for all vertices in 𝑺 ⊆ 𝑽
yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
78
Tseitin formulas and mod 2 equations
Adding equations for all vertices in 𝑺 ⊆ 𝑽
yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
79
Tseitin formulas and mod 2 equations
Adding equations for all vertices in 𝑺 ⊆ 𝑽
yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
80
Tseitin formulas and mod 2 equations
81
Tseitin formulas and mod 2 equations
Adding equations for all vertices in 𝑺 ⊆ 𝑽
yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
82
Tseitin formulas using mod 2 reasoning
 Assume wlog # vertices 𝒏 is odd and all charges are 𝟏.
 Input parity equations are just
𝒗∈𝒆 𝒙𝒆 = 𝟏 mod 𝟐 for each 𝒗 ∈ 𝑽.
 Can add equations mod 2 to get 𝟎 = 𝟏.
 Adding equations for all vertices in 𝑺 ⊆ 𝑽 yields
𝒗∈𝑬(𝑺,𝑺) 𝒙𝒆 = |𝑺| mod 𝟐
 Graph 𝑮 with expansion 𝜺 needs equation with at
least 𝜺𝒏/𝟐 variables to derive a contradiction.
We relate this to degree in PCR…
83
Parity equations and PCR
 Given equations of form
 𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 = 𝟎 (mod 𝟐)
 Represent in the “Fourier basis” over {𝟏, −𝟏}
 Polynomial equation 𝒚𝒊𝟐 − 𝟏 = 𝟎 for each variable
 𝒚𝒊 = 𝟏 − 𝟐𝒙𝒊 = (−𝟏)𝒙𝒊
 Polynomial equation 𝒚𝟏 𝒚𝟐 𝒚𝟑 − 𝟏 = 𝟎
 would be 𝒚𝟏 𝒚𝟐 𝒚𝟑 + 𝟏 = 𝟎 if RHS were 𝟏
 Thm: Since transformation is linear and invertible it
preserves degrees of PCR proofs if field’s characteristic
is not 2.
84
Tseitin formulas in Fourier basis
Variables are in {𝟏, −𝟏}
 𝒚𝟐𝒆 = 𝟏 for every 𝒆 ∈ 𝑬
Parity of edges touching 𝒗 equals charge at 𝒗

𝝌(𝒗) for every 𝒗 ∈ 𝑽
𝒚
=
(−𝟏)
𝒗∈𝒆 𝒆
Degree of input polynomials equals degree of graph
Thm: There is a constant-degree graph 𝑮 s.t. a Tseitin formula for
𝑮 with all charges 𝟏 and odd # of vertices requires
 degree 𝛀(𝒏) to refute in Nullstellensatz [Grigoriev]
 degree 𝛀(𝒏) to refute in PCR [BussGrigorievImpagliazzoPitassi]
85
Proof idea: binomial equations
Every inferred polynomial from 𝑻𝒔𝒆𝒊𝒕𝒊𝒏(𝑮) over Fourier
basis has two terms with ±𝟏 coefficients
 Every monomial corresponds to a parity of a subset of edges
 So … each equivalence corresponds to a parity equation:
 e.g. 𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 = 𝟏 + 𝒙𝟐 + 𝒙𝟒 mod 𝟐
 Equivalently… 𝒙𝟏 + 𝒙𝟑 + 𝒙𝟒 = 𝟏 mod 𝟐
 # of variables is at most 𝟐 × degree of PCR proof.
 Degree of the PCR proof of 𝑻𝒔𝒆𝒊𝒕𝒊𝒏(𝑮)
𝟏
≥ min # variables in mod 2 equations proof of 𝑻𝒔𝒆𝒊𝒕𝒊𝒏(𝑮)
≥
𝟐
𝟏
𝜺𝒏/𝟐 =
𝟐
𝜺𝒏/𝟒
86
Random k-CNF formulas
 𝒎 = 𝒓𝒏 clauses chosen independently at random
𝒏
𝒌
from all 𝟐
clauses of size 𝒌.
𝒌
 Exponentially hard for PCR
 Idea [BenSasson-Impagliazzo]: If characteristic isn’t 2 then
lower bound holds even if each clause is strengthened to a
parity
 [Alekhnovich-Razborov]: Same bound holds for
characteristic 2
87
DPLL on random 3-CNF
Proof complexity
shows 𝟐𝛀(𝒏 /𝒓) time
# of DPLL
backtracks
is required for
unsatisfiable formulas
for 𝒓 > 𝒓∗𝟑
[B,Karp,Saks,Pitassi 98]
[Ben-Sasson 02]
1
0
4.267
ratio of clauses to variables
𝒓
88
ZFC
Much more…
P/poly-Frege
Frege
TC0-Frege
AC0-Frege
Positivstellensatz
Cutting Planes
Lasserre/SOS
Res(k)
Polynomial Calculus/PCR
Resolution
Nullstellensatz
DPLL
Truth Tables
89
Thanks for listening!
Questions…?
90