Co-NP problems on random inputs
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Co-NP problems on random inputs
Paul Beame
University of Washington
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Basic idea
NP is characterized by a simple property - having short certificates of membership
Show that co-NP doesn’t have this property would separate P from NP so probably
quite hard Lots of nice, useful baby steps towards
answering this question
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Certifying language membership
Certificate of satisfiability Satisfying truth assignment Always short, SAT NP
Certificate of unsatisfiability ????? transcript of failed search for satisfying truth
assignment Frege-Hilbert proofs, resolution Can they always be short? If so then NP=co-NP.
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Proof systems
A proof system for L is a polynomial time algorithm A s.t. for all inputs x x is in L iff there exists a certificate
P s.t. A accepts input (P,x)
Complexity of a proof system How big |P| has to be in terms of |x|
NP = {L: L has polynomial-size proofs}
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Propositional proof systems
A propositional proof system is a polynomial time algorithm A s.t. for all formulas F F is unsatisfiable iff there exists a certificate P s.t.
A accepts input (P,F)
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Sample propositional proof systems
Truth tablesAxiom/Inference systems, e.g.
modus ponens A, (A -> B) | B excluded middle | (A v ~A)
Tableaux/Model Elimination systems search through sub-formulas of input formula
that might be true simultaneously e.g. if ~(A -> B) is true A must be true and
B must be false
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Frege Systems
Finite # of axioms/inference rulesProof of unsatisfiability of F - sequence
F1, …, Fr of formulas s.t. F1 = F
each Fj is an axiom or follows from previous ones via an inference rule
Fr = trivial falsehood
All of equivalent complexity up to poly
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Resolution
Frege-like system using CNF clauses onlyStart with original input clauses of CNF FResolution rule
(A v x), (B v ~x) | (A v B)Goal: derive empty clause
Most-popular systems for practical theorem-proving
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Davis-Putnam (DLL) Procedure
Both a proof system a collection of algorithms for finding proofs
As a proof system a special case of resolution where the
pattern of inferences forms a tree.The most widely used family of
complete algorithms for satisfiability
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Simple Davis-Putnam Algorithm
Refute(F) While (F contains a clause of size 1)
set variable to make that clause truesimplify all clauses using this assignment
If F has no clauses thenoutput “F is satisfiable” and HALT
If F does not contain an empty clause thenChoose smallest-numbered unset variable x Run Refute( )Run Refute( )0xF
1xF splitting rule
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Hilbert’s Nullstellensatz
System of polynomials Q1(x1,…,xn)=0,…,Qm(x1,…,xn)=0 over field K has no solution in any extension field of K iff there exist polynomials P1(x1,…,xn),…,Pm(x1,…,xn) in K[x1,…,xn] s.t.
1
QP ii
m
1i
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Nullstellensatz proof system
Clause (x1 v ~x2 v x3) becomes equation (1-x1)x2(1-x3)=0
Add equations xi2-xi =0 for each
variable
Proof: polynomials P1,…, Pm+n proving unsatisfiability
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Polynomial Calculus
Similar to Nullstellensatz except: Begin with Q1,…,Qm+n as before Given polynomials R and S can infer
a R + b S for any a, b in Kxi R
Derive constant polynomial 1 Degree = maximum degree of polynomial
appearing in the proof Can find proof of degree d in time nO(d) using
Groebner basis-like algorithm
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Cutting Planes
Introduced to relate integer and linear programming: Clause (x1 v ~x2 v x3)
becomes inequality x1+1-x2+x3 1
Add xi 0 and 1-xi 0
Derive 0 1 using rules for adding inequalities and Division Rule:
acx+bcy d implies ax+by d/c
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Some Proof System Relationships
Truth Tables
Davis-Putnam Nullstellensatz
Polynomial Calculus
Resolution
Cutting Planes
Frege
AC0-Frege
ZFC
P/poly-Frege
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Random k-CNF formulas
Make m independent choices of one of the clauses of length k
= m/n is the clause-density of the formula
Distribution
k
nk2
kn,F
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Threshold behavior of random k-SAT
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Contrast with ...
Theorem [CS]: For every constant , random k-CNF formulas almost certainly require resolution proofs of size 2(n)
What is the dependence on ?
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Width of resolution proofs
If P is a resolution proof width(P) = length of longest clause in P
Theorem [BW]: Every Davis-Putnam (DLL) proof of size S can be converted to one of width log2S
Theorem [BW]: Every resolution proof of size S can be converted to one of width
)logSnO(
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Sub-critical Expansion
F - a set of clausess(F) - minimum size subset of F that is
unsatisfiable F - boundary of F - set of variables
appearing in exactly one clause of Fe(F) - sub-critical expansion of F =
max min { |G|:
G F, s/2 < |G| s} s s(F)
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Width and expansion
Lemma [CS] : If P is a resolution proof of F then width(P) e(F).
s(F)
s/2 to s
G
containsG
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Consequences
Corollaries: Any Davis-Putnam (DLL) proof of F
requires size at least 2e(F)
Any resolution proof of F requires size at least
n(F)2e2
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s(F) and e(F) for random formulas
If F is a random formula from then s(F) is (n/1/(k-2)) almost certainly
e(F) is (n/2/(k-2)+) almost certainly
Proved for Hypergraph expansion
kn,F
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Hypergraph Expansion
F - hypergraph F - boundary of F - set of degree 1
vertices of FsH(F) - minimum size subset of F that does
not have a System of Distinct Representatives
eH(F) - sub-critical expansion of F - max min { |G|: G F, s/2
< |G| s} s sH(F)
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System of Distinct Representatives
sH(F) s(F) so eH(F) e(F)
variables/nodes
clauses/edges
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Density and SDR’s
The density of a hypergraph is #(edges)/#(vertices)
Hall’s Theorem: A hypergraph F has a system of distinct representatives iff every subgraph has density at most 1.
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Density and Boundary
A k-uniform hypergraph of density bounded below 2/k, say 2/k-has average degree bounded below 2 constant fraction of nodes are in
the boundary
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Density of random formulas
Fix set S of vertices/variables of size r Probability p that a single edge/clause
lands in S is at most (r/n)k
Probability that S contains at least q edges is at most
q
1k-
1k-q
nre
qnpe
qp)n,B(Pr
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s(F) for random formulas
Apply for q=r+1 for all r up to s using union bound:
for s = O(n/1/(k-2))
s
kr
1r
2k-
2k2
1r
1k-
1krs
kr
1r
1k-
1ks
kr
o(1)n
reenr
nre
rne
nre
r
n
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e(F) for random formulas
Apply for q=2r/k for all r between s/2 and s using union bound:
for s = (n/2/(k-2))
s
s/2r
2r/k
k/2-1k-
k/21kk/21
2r/k
1k-
1krs
s/2r
2r/k
1k-
1ks
s/2r
o(1)n
re
nre
rne
nre
r
n
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Hypergraph Expansion and Polynomial Calculus
Theorem [BI]: The degree of any polynomial calculus or Nullstellensatz proof of unsatisfiability of F is at least eH(F)/2 if the characteristic is not 2.
Groebner basis algorithm bound is only nO(eH(F))
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k-CNF and parity equations
Clause (x1 v ~x2 v x3) is implied by x1+(x2+1)+x3 = 1 (mod 2) i.e. x1+x2+x3 = 0 (mod 2)
Derive contradiction 0 = 1 (mod 2) by adding collections of equations
# of variables in longest line is at least eH(F)
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Parity equations and polynomial calculus
Given equations of form x1+x2+x3 = 0 (mod 2)
Polynomial equation yi2-1=0 for each variable
yi = 2xi-1
Polynomial equation y1 y2 y3-1=0
would be y1 y2 y3+1=0 if RHS were 1 Imply the old Nullstellensatz equations if
char(K) is not 2
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Lower bounds
For random k-CNF chosen from almost certainly for any >0: Any Davis-Putnam proof requires size
Any resolution proof requires size
Any polynomial calculus proof requires degree
kn,F
2)2/(kn/2 Δ
2)4/(kn/2 Δ
2)2/(kn/Δ
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Upper Bound
Theorem [BKPS]: For F chosen from and above the threshold, the simple Davis-Putnam (DLL) algorithm almost certainly finds a refutation of size
and this is a tight bound...
kn,F
O(1)n/O n22)1/(kΔ
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Idea of proof
2-clause digraph (x v y)
Contradictory cycle: contains both x and xAfter setting O(n/1/(k-2)) variables,
> 1/2 the variables are almost certainly in contradictory cycles of the 2-clause digraph a few splitting steps will pick one almost certainly setting clauses of size 1 will finish things off
x
y
x
y
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Implications
Random k-CNF formulas are provably hard for the most common proof search procedures.
This hardness extends well beyond the phase transition. Even at clause ratio =n1/3, current
algorithms on random 3-CNF formulas have asymptotically the same running time as the best factoring algorithms.
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Random graph k-colourability
Random graph G(n,p) where each edge occurs independently with probability p Sharp threshold for whether or not
graph is k-colourable, e.g. p ~ 4.6/n for k=3
What about proofs that the graph is not k-colourable?
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Lower Bound
Theorem [BCM 99]: Non-k-colourability requires exponentially large resolution proofs
Basic proof idea: same outline as before notion of boundary of a sub-graph
set of vertices of degree < k s(G) smallest non-k-colourable sub-graph
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Challenges
Better bound for e(F) for random F Can it be (s(F)) ?
If so, the simple Davis-Putnam algorithm has asymptotically best possible exponent of any DP algorithm.
Extend lower bounds to other proof systems must be based on something other than expansion
since certain formulas with high expansion have small Cutting Planes proofs.
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Challenges
Conjecture: Random k-CNF formulas are hard for Frege proofs
Extend to other random co-NP problems Independent Set?
Best algorithms only get within factor of 2 of the largest independent set in a random graph
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Sources
[Cook, Reckhow 79] [Chvatal, Szemeredi 89] [Mitchell, Selman, Levesque 93] [Beame, Pitassi 97] [Beame, Karp, Pitassi, Saks 98] [Beame, Pitassi 98] [Ben-Sasson, Wigderson 99] [Ben-Sasson, Impagliazzo 99] [Beame, Culberson, Mitchell 99]
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Circuit Complexity
P/poly - polysize circuitsNC1 - polysize formulasCNF - polysize CNF formulasAC0 - constant-depth polysize circuits
using and/or/not
AC0[m] - also = 0 mod m tests
TC0 - threshold instead
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C-Frege Proofs
Given circuit complexity class C can define C-Frege proofs to be Frege-like proofs that manipulate circuits in C rather than formulas
Frege = NC1-FregeResolution = CNF-FregeExtended-Frege = P/poly-FregeAC0-FregeAC0[m]-FregeTC0-Frege