the computational complexity of satisfiability lance fortnow nec laboratories america
TRANSCRIPT
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The Computational Complexity of Satisfiability
Lance FortnowNEC Laboratories America
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Boolean Formula
u v w x: variables take on TRUE or FALSE NOT u u OR v u AND v
u v w u w x v w x
uu vu v
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Assignment
u TRUEv FALSEw FALSEx TRUE
u v w u w x v w x
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Satisfying Assignment
u TRUEv FALSEw TRUEx TRUE
u v w u w x v w x
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Satisfiability
A formula is satisfiable if it has a satisfying assignment.
SAT is the set of formula with satisfying assignments.
SAT is in the class NP, the set of problems with easily verifiable witnesses.
u v w u w x v w x
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NP-Completeness of SAT
In 1971, Cook and Levin showed that SAT is NP-complete.
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NP-Completeness of SAT
In 1971, Cook and Levin showed that SAT is NP-complete. Every set A in NP reduces to SAT.
A
SAT
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NP-Completeness of SAT
In 1971, Cook and Levin showed that SAT is NP-complete. Every set A in NP reduces to SAT.
A
SATf
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NP-Completeness of SAT
True even for SAT in 3-CNF form.
A
SATf
u v w u w x v w x
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NP-Complete Problems SAT has same complexity as
Map Coloring Traveling Salesman Job Scheduling Integer Programming Clique …
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Questions about SAT How much time and memory do we need
to determine satisfiability? Can one prove that a formula is
not satisfiable? Are two SAT questions better
than one? Is SAT the same as every other NP-
complete set? Can we solve SAT quickly on other models
of computation?
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How Much Time and Memory Do We Need to Determine Satisfiability?
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Solving SAT
TIME
SPACElog nn
n
2n
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Solving SAT Search all of the
assignments. Best known for
general formulas.TIME
SPACElog nn
n
2n
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Solving SAT Can solve 2-CNF
formula quickly.
TIME
SPACElog nn
n
2n
2-CNF
u w u v u v
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Solving SAT
TIME
SPACElog nn
n
2n
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Solving SAT Schöning (1999)3-CNF satisfiabilitysolvable in time (4/3)nT
IME
SPACElog nn
n
2n
1.33n 3-CNF
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Schöning’s Algorithm Pick an assignment a at random. Repeat 3n times:
If a is satisfying then HALT Pick an unsatisfied clause. Pick a random variable x in that clause. Flip the truth value of a(x).
Pick a new a and try again.
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Solving SAT Is SAT computable
in polynomial-time?
Equivalent toP = NP question.
Clay Math Institute Millennium Prize
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
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Solving SAT Can we solve SAT
in linear time?
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
?
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Solving SAT Does SAT have
a linear-time algorithm? Unknown.T
IME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
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Solving SAT Does SAT have
a linear-time algorithm? Unknown.
Does SAT have a log-space algorithm?
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP?
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Solving SAT Does SAT have
a linear-time algorithm? Unknown.
Does SAT have a log-space algorithm? Unknown.
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
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Solving SAT Does SAT have
an algorithm that uses linear time and logarithmic space?
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
?
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Solving SAT Does SAT have
an algorithm that uses linear time and logarithmic space? No! [Fortnow ’99]
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
X
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Idea of Separation Assume SAT can be solved in linear
time and logarithmic space. Show certain alternating automata
can be simulated in log-space. Nepomnjaščiĭ (1970) shows such
machines can simulate super-logarithmic space.
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Solving SAT Improved by
Lipton-Viglas and Fortnow-van Melkebeek.
Impossible intime na and polylogarithmic space for any a less than the Golden Ratio.
TIME
SPACElog nn
n
2n
1.33n 3-CNF
nc P = NP
n1.618
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Solving SAT Fortnow and van
Melkebeek ’00 More General Time-
Space TradeoffsTIME
SPACElog nn
2n
1.33n 3-CNF
nc P = NP
n1.618
n
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Solving SAT Fortnow and van
Melkebeek ’00 More General Time-
Space Tradeoffs Current State of
Knowledge for Worst Case
TIME
SPACElog nn
2n
1.33n 3-CNF
nc P = NP
n1.618
n
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Solving SAT Fortnow and van
Melkebeek ’00 More General Time-
Space Tradeoffs Current State of
Knowledge for Worst Case
Other Work on Random Instances
TIME
SPACElog nn
2n
1.33n 3-CNF
nc P = NP
n1.618
n
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Can One Prove That a Formula is not Satisfiable?
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SAT as Proof Verification
u v u v
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SAT as Proof Verification
u v u v
is satisfiable
u = True; v = True
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SAT as Proof Verification
u u v u v
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SAT as Proof Verification
u u v u v
is satisfiable
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SAT as Proof Verification
u u v u v
is satisfiable
Cannot producesatisfying assignment
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Verifying Unsatisfiability
u u v u v
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Verifying Unsatisfiability
u u v u v
u = true; v = true
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Verifying Unsatisfiability
u v u v
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Verifying Unsatisfiability
u v u v
u = true; v = false
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Verifying Unsatisfiability
Not possible unless NP = co-NP
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Interactive Proof System
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Interactive Proof System
HTTHHHTH
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Interactive Proof System
HTTHHHTH010101000110
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Interactive Proof System
HTTHHHTH010101000110THTHHTHHTTH001111001010
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Interactive Proof System
HTTHHHTH010101000110THTHHTHHTTH
THTTHHHHTTHHH001111001010
100100011110101
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Interactive Proof System
HTTHHHTH010101000110THTHHTHHTTH
THTTHHHHTTHHH001111001010
100100011110101
Developed in 1985 by Babaiand Goldwasser-Micali-Rackoff
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Interactive Proof System
HTTHHHTH010101000110THTHHTHHTTH
THTTHHHHTTHHH001111001010
100100011110101
Lund-Fortnow-Karloff-Nisan 1990: There is an interactive proof system for showing a formula not satisfiable.
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Interactive Proof for co-SAT
u u v u v
(1 ) (1 ) (1 )u u v u v
For any u in {0,1} and v in {0,1} value is zero.
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Interactive Proof for co-SAT
(1 ) (1 ) (1 )u u v u v
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Interactive Proof for co-SAT
1 1
0 0
(1 ) (1 ) (1 )u v
u u v u v
Value is zero.
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Interactive Proof for co-SAT
1
0
(1 ) (1 ) (1 )v
u u v u v
3 23 2u u u
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Interactive Proof for co-SAT
1
0
(1 ) (1 ) (1 )v
u u v u v
1
3 2
0
3 2 0u
u u u
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Interactive Proof for co-SAT
1
0
(1 ) (1 ) (1 )v
u u v u v
3 23 2u u u
Picks u at random, say u = 17.
3 23 2 4080u u u
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Interactive Proof for co-SAT
1
0
(1 ) (1 ) (1 )v
u u v u v
u = 17
4080
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Interactive Proof for co-SAT
1
0
17 (1 17) (1 17) (1 )v
v v
u = 17
4080
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Interactive Proof for co-SAT
17 (1 17) (1 17) (1 )v v
217 17 4080v v
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Interactive Proof for co-SAT
17 (1 17) (1 17) (1 )v v
12
0
17 17 4080 4080v
v v
u = 174080
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Interactive Proof for co-SAT
17 (1 17) (1 17) (1 )v v
217 17 4080v v
u = 17v = 63570
Pick random v, say v=6.
217 17 4080 3570v v
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Interactive Proof for co-SAT
u = 17v = 63570
(1 ) (1 ) (1 )u u v u v
Plug in 17 for u and 6 for v.Evaluates to 3570.
A PERFECT MATCH!
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Interactive Proof for co-SAT If formula was satisfiable
then any evil prover would fail with high probability.
Uses fact that polynomials are low-degree.
Two low-degree polynomials cannot agree on many places.
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Extensions Shamir 1990
Interactive Proof System for every PSPACE language.
GMW/BCC 1990 SAT has interactive proof
that does not reveal any information about the satisfying assignment.
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Probabilistically Checkable Proof Systems
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Probabilistically Checkable Proof Systems
Queries bitsof the proof
Defined by Fortnow-Rompel-Sipser 1988
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Probabilistically Checkable Proof Systems
Queries bitsof the proof
Babai-Fortnow-Lund 1990 PCP = NEXP
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Probabilistically Checkable Proof Systems
Queries bitsof the proof
Babai-Fortnow-Levin-Szegedy 1991 Roughly linear-size proof of SAT verifiable
with small number of queries.
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Probabilistically Checkable Proof Systems
Queries bitsof the proof
ALMSS 1991 Proofs of SAT using constant queries and
logarithmic number of random coins.
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Probabilistically Checkable Proof Systems
Queries bitsof the proof
ALMSS 1991 Many applications for showing hardness of
approximation for optimization problems.
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Hard to Approximate Clique Size Traveling Salesman Max-Sat Shortest Vector in Lattice Graph Coloring Independent Set …
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Are Two SAT Questions Better Than One?
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Questions to SAT
Does the number of queries matter? Focus on what happens if two
queries to SAT can be simulated by a single SAT query.
Oracle willing to honestly answera limited number of SAT questions.
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Are Two Queries Better Than One? Series of results by
Kadin 1988 Wagner 1988 Chang-Kadin 1990 Amir-Beigel-Gasarch 1990 Beigel-Chang-Ogihara 1993 Buhrman-Fortnow 1998 Fortnow-Pavan-Sengupta 2002
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If One Query as Powerful as Two Queries …
Polynomial-Time hierarchy collapses to Symmetric Polynomial-Time.
Any polynomial number of adaptive SAT queries, can be simulated by a single SAT query.
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Alternation
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Alternation
Model inventedby CKS 1981. Unbounded
Alternation = PSPACE
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Alternation
Model inventedby CKS 1981. Constant
Alternation =PolynomialHierarchy
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Symmetric P
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Symmetric P
Defined by Russelland Sundaram 1996
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If One Query as Powerful as Two Queries …
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If One Query as Powerful as Two Queries …
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Hard-Easy Strings If one query as powerful as two then
for every unsatisfiable , either There is a nondeterministic proof that
is not satisfiable, or One can use as advice to solve
satisfiability for all formulas of the same length.
Proofs use applications of this fact.
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Is SAT the Same as Every Other NP-Complete Set?
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NP-Completeness of SAT
A
SAT
* *
f
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Isomorphisms of SAT
A
SAT
* *
f
A set A is isomorphic to SAT if A reduces to SAT via a 1-1, onto, easily computable and invertible reduction.
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Are all NP-complete sets the same as SAT?
A
SAT
* *
f
Berman and Hartmanis 1978 All of the known NP-complete sets are
isomorphic.
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Are all NP-complete sets the same as SAT?
A
SAT
* *
f
Berman and Hartmanis 1978 Conjecture: All of the NP-complete sets
are isomorphic.
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Are all NP-complete sets the same as SAT?
A
SAT
* *
f
If conjecture is true… All NP-complete sets, like SAT, must
have an exponential number of strings at every length.
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What if SAT reduces to a small set? Mahaney’s Theorem (1978)
For many-one reduction then P=NP. Ogihara and Watanabe (1991)
For reductions that ask a constant number of queries still P=NP.
Karp-Lipton(1980)/Sengupta(2001) For arbitrary reductions, polynomial
hierarchy collapses to Symmetric-P.
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Are all NP-complete sets the same as SAT?
A
SAT
* *
f
Still Open Look at relativized worlds
Universes that show us limitations of most proof techniques.
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Are all NP-complete sets the same as SAT?
A
SAT
* *
f
Fenner-Fortnow-Kurtz 1992 A relativized world where the
isomorphism conjecture holds.
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Can We Solve SAT Quickly on Other Models of Computation?
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Solving SAT on Other Models of Computation
RANDOM QUANTUMDNA
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Can we solve SAT Quickly with Random Coins?
Would imply collapse of the polynomial-time hierarchy.
Reasonable assumptions imply randomness computation not any stronger than deterministic computation. IW ’97: If EXP does not have
subexponential-size circuits then we can derandomize.
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Can we solve SAT Quickly with DNA Computing?
Adleman has solved TSP on 20 cities with DNA manipulation.
Problem: Exponential Growth
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Exponential Growth
20 Cities
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Exponential Growth
75 Cities
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Can we solve SAT Quickly with DNA Computing?
Adleman has solved TSP on 20 cities with DNA manipulation.
Problem: Exponential Growth Adleman
The less pleasing part is that we learned enough about our methods to conclude that they would not allow us to outperform electronic computers.
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Can we solve SAT Quickly on a Quantum Computer?
Basic element is qubit that is in a superposition of zero and one.
N qubits can be entangled to form 2N quantum states.
States can have negative amplitudes that can cancel each other out.
Transformations are limited to a unitary manner.
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Can we solve SAT Quickly on a Quantum Computer?
Shor 1994 Factoring can be solved
quickly on a quantum computer.
Grover 1996 Search a database of size N
using N1/2 queries. Yields quadratic improvement
for general satisfiability. Best possible in a black-box
model.
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Can we solve SAT Quickly on a Quantum Computer?
Fortnow-Rogers Relativized world where
quantum computing is no easier than classical, yetPNP and the polynomial hierarchy does not collapse.
Physical Difficulties Maintain Entanglement Handle Errors High Precision
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Other Research Lower Bounds for proving non-
satisfiabilility in weak logical models. Circuit complexity approaches to
lower bounds for satisfiability. Solving SAT on “Typical” instances. Many other structural questions
about satisfiability.
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Conclusions The satisfiability question captures
nondeterministic computation and much of the interest in computational complexity.
We have made much progress on these fronts but many questions remain.
Prove PNP!