1 Paul Beame University of Washington Phase Transitions in Proof Complexity and Satisfiability...
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Transcript of 1 Paul Beame University of Washington Phase Transitions in Proof Complexity and Satisfiability...
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Paul Beame University of Washington
Phase Transitions in Proof Complexity and Satisfiability Search
Dimitris Achlioptas Michael Molloy Microsoft Research U. Toronto
with
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Satisfiability
F (x1 x2 x4) (x1 x3) (x3 x2) (x4 x3)
satisfying assignment for F: x1, x2, x3, x4
Given F does such an assignment exist?
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Satisfiability Algorithms
• Incomplete Algorithms will (likely) find a satisfying assignment but
will simply give up if one is not found
• Complete Algorithms will either find a satisfying assignment or
determine that no such assignment exists
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Satisfiability Algorithms
• Incomplete Algorithms Local search
GSAT [Selman,Levesque,Mitchell 92] Walksat [Kautz,Selman 96]
Belief PropagationSP [Braunstein, Mezard, Zecchina 02]
• Complete Algorithms Backtracking search
DPLL [Davis,Putnam 60] [Davis,Logeman,Loveland 62]
DPLL + “clause learning” GRASP, SATO, zchaff
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Simplification and Satisfaction
F (x1 x2 x4) (x1 x3) (x3 x2) (x4 x3)
satisfying assignment for F: x1, x2, x3, x4
• Simplifying F after setting literal x3 to true
F (x1 x2 x4) (x1 x3) (x3 x2) (x4 x3)
F|x3 (x1 x2 x4) (x2) (x4)
• F is satisfied if all clauses disappear under simplification given the assignment
1-clauses
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Backtracking search/DPLL
DPLL(F) while F contains a 1-clause l’
F F|l’ if F has no clauses output ‘satisfiable’ halt if F has an empty clause
backtrackelse select a literal l = some x or x
DPLL(F|l)
if backtrack then DPLL(F|l)
Residual formula
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Some standard select choices for DPLL algorithms
• UC: Unit Clause/Ordered DLL Choose variables in a fixed order Always set True first
• UCwm: Unit Clause with majority Choose variables in a fixed order Apply a majority vote among 3-clauses for
assigning each value
• GUC: Generalized Unit Clause Choose a variable v in a shortest clause C Set v to satisfy C
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Random k-CNF formulas
• Distribution Fk,n(r) Randomly choose rn clauses over n
variables independently, each of size k Each size k clause is equally likely
• Threshold value rk*• r rk*, almost certainly satisfiable
• r rk*, almost certainly unsatisfiable
• Hardest problems near threshold
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DPLL on random 3-CNF*
0
1
probability satisfiable
4.267
ratio of clauses to variables
# of DPLLbacktracks
* n = 50 variables
Proof complexity
shows 2(n/r) time is required for
unsatisfiable formulas for r r3*
[B,Karp,Saks,Pitassi 98][Ben-Sasson 02]
What about satisfiableformulas below threshold?
r[Mitchell,Selman,Levesque 92]
Exponential lower bounds for 3-CNF formulas below ratio 4.267
r3UC = 3.81
r3UCwm = 3.83
r3GUC
= 4.01
Theorem Let A {UC, UCwm, GUC}. Let
w.h.p. algorithm A takes exponential time on a random FF3,n
(r) for r r3A
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Exponential lower bounds for satisfiable formulas below the k-CNF threshold
Theorem There exist lk2k/k and uk2ks.t. for every k 4 and for FFk,n
(r) with lk r uk w.h.p.• F is satisfiable• UC takes exponential time on F
Note These formulas have huge numbers of satisfying assignments (more than 2 (1-) n out of a possible 2n) but still are hard
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Ideas
Part I:Use differential equations to analyze trajectory of
algorithm as a function of the clause-variable ratio for r larger than lk
Use resolution proof complexity to show that some residual formula along this trajectory requires large DPLL running time
Part II: Show that formulas up to ratio uk are satisfiable
[Achlioptas, Peres 03] uk=2k ln 2 – (k+4)/2
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Algorithmic behavior using simple select choices
• On input FFk,n (r) before the first backtrack
occurs, the residual formula F’ is distributed as F2Fk where
FjFj,n’ (rj) for j=2,,k only has clauses of size k
Fj are mutually independent
• Values of rj almost surely follow algorithm-dependent trajectories given by differential equations
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Proof Complexity
• Study of the number of symbols required for proofs of unsatisfiability (or tautology) in propositional logic
• Does not address algorithmic issue How would you find short proofs if they existed?
• Existence of short proofs for every unsatisfiable formula is equivalent to NP = co-NP (and is implied by P=NP) Generally believed that such proofs don’t exist
• Active research area with rich theory and many open questions
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Resolution
• Start with clauses of CNF formula F
• Resolution rule Given (A x), (B x) can derive (A
B)
• The empty clause is derivable F is unsatisfiable
• Proof size = # of clauses used
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Resolution and DPLL
• Running DPLL with any select rule on an unsatisfiable formula F generates a Resolution refutation of F # of clauses running time
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Backtracking search/DPLL
DPLL(F) while F contains a 1-clause l’
F F|l’ if F has no clauses output ‘satisfiable’ halt if F has an empty clause
backtrackelse select a literal l = some x or x
DPLL(F|l)
if backtrack then DPLL(F|l)
Residual formula
Long-running DPLL Executions
Residual formula at is unsatisfiable
Algorithm’sproof of unsatisfiability is exponentially long
Every resolution
Residual formula at each node is a mix of 2- and 3-clauses
2n
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Satisfiability for mixed random formulas: proven properties
1
4.501
SAT
UNSAT
3.522/3
?
??
?
?
?
?
?
?
?
?
?
2.28
3-clause ratio
2-c
laus
e r
atio
[Achlioptas et al 96]
[Kaporis et al 03]
[Dubois 01]
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Resolution proof complexity of mixed random formulas
Theorem A random CNF formula FF2,n (r2) is
Satisfiable w.h.p. if r2<1 Unsatisfiable w.h.p if r2>1 and has linear size resolution
proofs [Chvatal-Reed 91], [Goerdt 91], [De La Vega 91]
Theorem For any constant r30, w.h.p. GF3,n (r3)
requires an exponential-size resolution proof of unsatisfiability [Chvatal,Szemeredi 88]
Theorem For any constants r21 and r3 0, w.h.p. for FF2,n
(r2) and GF3,n (r3) the combined formula
FG requires an exponential-size resolution proof of unsatisfiability
Easy
Hard
Easy Hard = Hard
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Sharp Threshold in Resolution Proof Complexity
• Define distribution Hn(r) on CNF formulas of the form H=FG where GF3,n
(r3) for some r32.28 and
FF2,n (r).
• Then for HHn(r) w.h.p. H is unsatisfiable For r 1, H has O(n) size resolution proofs For r 1, H requires 2(n) size resolution proofs
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Trajectory on 3-CNF
1UC Algorithm Trajectory
2-c
laus
e r
atio
4.51
Provably UNSAT& Hard
3.52 4.267
ProvablySAT & Easy
3-clause ratio
3.81
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UC trajectory for k 4
• Start with 2.752kn/k k-clauses
• Wait until 3n/(k-1) variables remain
• With high probability: The 2-clauses remained satisfiable throughout The residual formula overall is unsatisfiable Its resolution complexity is exponential
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Directions
• What price completeness?
• Closing gap for unsatisfiability of mixed formulas would yield an algorithm-dependent phase transition Below rA algorithm runs in linear time
Above rA algorithm requires exponential time
• Backtracking algorithms for other random problems with phase transitions? e.g. k-colorability on random graphs G(n,r/n)
• Unsatisfiable phase exp(cn/rk) [B, Culberson, Mitchell, Moore 03]