computing the density of states of boolean formulas stefano ermon, carla gomes, and bart selman...

Computing the Density of States of Boolean FormulasStefano Ermon,

Carla Gomes, and Bart Selman

Cornell University, September 2010

From 100 variables, 200 constraints (early 90’s)to over 1,000,000 vars. and 5,000,000 clausesin 20 years.

Applications: Hardware and Software Verification, Planning, Scheduling, Optimal Control, Protocol Design, Routing, Multi-agent systems, E-Commerce (E-auctions and electronic trading agents), etc.

Motivation:Significant progress in SAT

SAT: Given a Boolean formula Φ in CNF, Φ=C1ΛC2 Λ…ΛCmdoes Φ have a satisfying assignment?

Extending SAT technology Model counting problem (number of distinct

satisfying assignments): probabilistic inference problems multi-agent / adversarial reasoning (bounded)

[Roth ‘96, Littman et. al. ‘01, Sang et. al. ‘04, Darwiche ‘05,

Domingos ‘06]

MAX-SAT and Weighted MAX-SAT: find a truth assignment that maximizes the number of satisfied clauses or the sum of their weights

beyond decision (NP) [Hansen at al. ’90] hard and soft constraints [Heras et al. ’08, Cohen et al. ’06]

How can we combine both challenges?

Density of states Given a Boolean formula Φ in CNF,

Φ=C1ΛC2 Λ…ΛCm with m clauses

the density of states is a function

that gives the number of truth assignments that violate exactly i clauses, for i =0,..,m

n(0) = number of assignments that violate 0 clauses (models) n(1) = number of assignments that violate exactly 1 clause

],..,0[: mn

Density of states: a challenging problem Generalizes SAT

Decision problem: Φ is satisfiable if and only if n(0)>0

Generalizes MAX-SAT MAX-SAT is the minimum i such that n(i)>0

Generalizes #SAT Number of models = n(0)

Statistical physics

More generally, the density of states (DOS) gives the number of microstates with energy E Microstates = truth assignments Energy = number of violated clauses Ground states = maximally satisfying assignments

Compact, very informative characterization of a physical system Macroscopic thermodynamic quantities (free energy,

internal energy,..) Partition function, phase transitions,..

Motivation

DOS provides a finer characterization of the structure of a combinatorial search space

Statistical physics and CSPs: Insights on problem structure, hardness, new

algorithms, Survey Propagation [Montanari et. al. ‘07, Monasson et. al. ‘96, Mézard ’02, Parisi ‘02]

By defining different energy functions, it can be naturally used for probabilistic style inference (e.g. Markov Logic, [Domingos ‘06] )

Talk Outline

Prior work A novel sampling strategy: MCMC-FlatSat Empirical Validation

Small formulas with ground truth Synthetic formulas Random 3-SAT Large structured instances Model counting

Conclusions

Density of states: prior work

Exact Method: Enumeration (exponential) Approximate

Uniform Sampling [Belaidouni et. al. ‘02] Sample density from K random truth assignments Impractical, unlikely to hit rare assignments (e.g. solutions)

Metropolis Sampling [Rose et. al. ‘96] In theory, the density can be extracted from the Boltzmann

distribution Impractical, difficult choice of the temperature and slow

mixing times [Wei et. al. ‘04 ]

How can we improve the sampling strategy?

The flat histogram idea

Idea: Set up a Markov Chain that visits all energy levels equally often [F. Wang and Landau ’02, J. Wang

et. al. ’99, De Oliveira et. al. ’96] e.g. an equal amount of time at the set of truth assignments with 0 unsat clauses, 1 unsat clause, ...

How? Flip a variable, accept new state σ’ with probability

Always accepts “rarer” states (when n(E’)<n(E))

)(

)(,1minEn

Enp

Example

1

11

11

RED ↔ 0 unsat clauses, n(0)=1GREEN ↔ 1 unsat clauses, n(1)=3BLUE ↔ 2 unsat clauses, n(2)=10

Goal: visit all energy levels (colors) equally often

Example

3/103/10

3/10

3/10



Example

1

11



Example

1

1 1

11



Example

1

11

11



Example

1

1

1

1

1



Example

1/10

1

1/3

1/10

1/10



Example

…and finally…

Example RED ↔ 0 unsat clauses, n(0)=1GREEN ↔ 1 unsat clauses, n(1)=3BLUE ↔ 2 unsat clauses, n(2)=10

Flat histogram: Intuition Detailed balance holds with those

transition probabilities

Properly biased towards “rare” states But there are few “rare” states

The total time spent in each type of state is the same (flat visit histogram).

)(

)(,1minEn

Enp

(Red, Blue, Green)

Note: in contrast, Simulated Annealing concentrates sampling around low energy states (more greedy!)

Flat histogram

But… how can we even run the Markov Chain?

Acceptance probability:

The density n() is unknown and is precisely what we want to compute!

)(

)(,1minEn

Enp

Adaptive sampling

Start with an initial guess g (our estimate of the true density n)

Random walk: Use g for guidance (acceptance probability) Chain will initially not sample uniformly across energy

levels Each step, adjust g using a modification factor F Keep track of the visit histogram H

When we see a flat H, we must have the right density!

The modification factor, F

F controls the tradeoff between convergence rate and accuracy

Use large modification factors F at the beginning to get rough estimates fast convergence

Keep reducing F to get finer estimates Analogous to an annealing process

MCMC-FlatSat

Inner Loop

Initialization

Inner Loop : adaptive sampling until the visit histogram is flat (g becomes our new guess for n)

H

96 99105

0

50

100

0 (R ed) 1(G reen)

2 (B lue)

MCMC-FlatSat

OuterLoop

Reduce modification factor and repeat inner loop until g≈n

Outline of empirical validation

Empirical validation on combinatorial problems Convergence Efficiency (number of samples vs search space size) Accuracy

We study: Small formulas with ground truth Large synthetic formulas Random 3-SAT Large structured instances Model counting

Formulas with known ground truth

Instances from MAXSAT2007 competition (Ramsey, Spin Glass, Max Clique)

Direct enumeration is possible (n<=28), so we can compare our estimate with ground truth

Metrics: KL divergence:

Relative error per point

EKL Eg

En

Z

EngnD

)(

)(log)(

)||(

Spin glass instance, 27 variables, 162 clauses Needs ~ 8 *106 flips << search space size

(227≈1.3 *108)

Energy (# unsat clauses)

Log-

dens

ity

Rel

. err

or (

%)

Energy

3.1 % maximum relative error

n – ground truthg – estimateX

Instance variables clausesKL-

divergence Max rel. error Entropy

ramk3n7.ra0 21 70 3.9 E-05 2.39 % 2.45

ramk3n8.ra0 28 126 1.1 E-05 5.1 % 3.93

johnson8-2-4.clq 28 420 4.5 E-05 5.5 % 2.90

T3pm3-5555.spn 27 162 1.3 E-05 3.15 % 3.34

Ramsey Max Clique


Log-

dens

ity


Log-

dens

ity


ng X

Synthetic formulas

Ground truth for larger formulas? Construct synthetic formulas

Formulas for which we derive a closed form solution for the density

Result on the composition (logical conjunction) of independent formulas (do not share variables)

The density of F is the convolution of the density of Φ with itself l times

)(...)()(),...,,( 2121 ll xxxxxxF

Instance variables clauses KL-divergence Max relative error (%) Entropy

UnifConv50_100 50 100 1.19 E-05 3.05 % 7.3

PigHoleConv410 200 750 1.26 E-07 2.2 % 33.7

Convolution of uniform densities

Convolution of pigeon hole formulas


Log-

dens

ity


Log-

dens

ity

Needs ~ 2*107 flips << state space size (250≈1015)


ng X

Random k-SAT formulas

Well known phase transitions for the satisfiability property in terms of the ratio α (Φ is satisfiable ↔ nΦ(0)>0 )

Analytic result on the average density

given a truth assignment, the probability of having a clause that is violated is 1/2k for random k-SAT

nim

k

i

ki

minE 2

2

11

2

1)]([

]0)0([ nP

n=50 variables (average over 1000 instances) Needs ~ 108 flips << state space size (250≈1015)

Average DensitiesLo

g-de

nsity

Ratio clauses to var. α

Large structured instances

No ground truth known Consistency checks

Number of models, when exact model counting is feasible (# models=n(0))

Method of the moments: Sample K assignments at random, compute their

energies (unsat clauses) Compute sample moments (e.g. average energy,

2nd order moment of energy, ..) Compare with moments obtained using the

estimated density g

Large structured instances

Instance variables clauses g(0) # models Ms(1) M(1) Ms(2) M(2)

brock400 2.clq 40 1188 0 0 297.0 297.0 88365.9 88372.3

Spinglass5.pm 125 750 0 0 187.4 187.4 35249.2 35247.1

MANN a27.clq 42 1690 0 0 422.4 422.4 178709 178703

bw large.a 459 4675 1 1 995.2 995.3 996349 996634

Logistic instance Planning problem:

n=459 variables m=4675 clauses

Huge search space (2459 truth assignments), but MCMC-FlatSat returns within hours

Remarkable precision: Finds the only existing model g(0)=n(0)=1! The mode of the estimated distribution is e300 times larger than the number of

models g(0) counts the needles and the haystack!

(the moments method indicates g is accurate)


Log-

dens

ity

Model counting Comparison with state-of-the-art model counters

SampleCount [Gomes et. al. ‘06] SampleMiniSAT [Gogate et. al. ‘07]

MCMCFlatSat is very accurate Timings are competitive when ratio clauses to variables is

not too large DOS provides guidance Information on what is not a model Overhead because it provides more information

Model counting comparison

Instance variables clausesExact # models

SampleCount SampleMiniSAT MCMC-FlatSat

Models Time

(s) Models Time

(s) Models Time

(s)

2bitmax 252 766 2.10×1029 >2.40×1028 29 2.08×1029 345 1.96×1029 1863

wff-3-3.5 150 525 1.40×1014 >1.60×1013 145 1.60×1013 240 1.34×1014 393

wff-3.1.5 100 150 1.80×1021 >1.00×1020 240 1.58×1021 128 1.83×1021 21

wff-4-5.0 100 500 >8.00×1015 120 1.09×1017 191 8.64×1016 189

ls8-norm 301 1603 5.40×1011 >3.10×1010 1140 2.22×1011 168 5.93×1011 2693

Conclusions Computing the density of states is a hard problem

(encompasses SAT, MAX-SAT, #SAT) MCMCFlatSat: sampling strategy adapted from physics for

combinatorial spaces that adaptively explores the space while collecting statistics Extremely accurate, very efficient (few samples) Provides a compact, rich description of the search space. New

insights about structure and local search Very general method: any property can be used for search

space partitioning. Many applications to counting and inference problems

Extra slides

Future work

SAT-specific improvements Energy saturation Energy barriers (and related normalization issues) Walksat heuristics (with Metropolis-Hastings updates)

Direct application to inference in Markov Logic Formal proof of convergence / counterexamples Application to other counting problems in

combinatorial spaces

Runtime for random 3-SAT

Search space 2^50 Flips 10^8 ≈ 2^26

Related work on random 3-SAT

Lots of work on the i=0 (i.e. SAT/UNSAT) case [Gent et. al. ‘94]

Previous experimental work for i>0 [Zhang ‘01] Different definition: “no more than i unsat clauses”

versus “exactly i unsat clauses” Location on the phase transitions Apparently, same location

Analytic results [Achlioptas et al. ‘05] We can see two phase transitions: assignments

are unlikely to violate a large number of clauses

Other phase transition

50 variables, 1000 instances

Other structured instances

Histogram flatness (formal)

Flatness condition of the visit histogram H is a necessary condition for convergence If g is equal to the true density n, the detailed balance

is satisfied by

upon convergence, the steady state probability is proportional to the reciprocal of the density of the corresponding energy level

pppp )()(

))((

1)(

Enp

Histogram flatness (formal)

Flatness condition of the visit histogram H is a necessary condition for convergence

If g=n, the visit histogram H will be flati.e. energy levels visited equally often

Problem: the density n() is unknown and is precisely what we want to compute!

cEn

cpEH

EEEE

)(|)(| )(

)()(

Satifiability (SAT): Given a formula in propositional calculus,

does it have a model, i.e., is there an assignment to its

variables making it true?

(a b c) AND (b c) AND (a c)

SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971)

Propositional Satisfiability (SAT)

Prior work - Metropolis

Approximate [Rose et. al., ‘96 ] Metropolis sampling at temperature T Energy = number of violated clauses Measure the empirical probability P(E) of energy

levels, rescale to get the density n(E)

Slow mixing times [Wei et. al., ‘04 ] Difficult choice of the temperature

)exp()()( TEEnEP

Theoretical Convergence

Stated as a conjecture in Lee,Okabe and Landau, Convergence and

Refinement of the Wang-Landau Algorithm, Computer Physics Communications, 2006

Proof of convergence inAtchade and Liu,The Wang-Landau Algorithm for Monte Carlo computation in general state spaces, Statistica Sinica, 2009

Closed forms

k-SAT, m clauses such that each variable appears exactly in one clause, the DOS is

where is the fraction of assignments of the variables in a single clause not satisfying it (e.g. in 3-SAT, only 1 over 8 assignments does not satisfy a clause)

nEm

k

E

kE

mEn 2

2

11

2

1)(

k21

Implementation details

Use of log-densities New assignment generated by flipping a variable

uniformly at random Flatness condition (within 10% of the max) Normalization

The density is obtained only up to a constant factor We use the normalization constraint

F0=1.5, reduce by Flatness is checked every 1000 moves

E

nEg 2)(

FF

Random k-SAT formulas

k-SAT: each clause has at most k literals Random k-SAT: formulas Φ generated with

n variables m clauses produced independently by

randomly choosing a set of k variables from n available negating each one with probability 0.5

α is the ratio clauses to variables (m/n)

Let be the probability measure associated with this generative model for Φ

()P

Composition

Logical conjunction of formulas Φ that do not share variables

The density of F is the convolution of the density of Φ with itself l times

Analogous to the probability density of the sum of l independent random variables

)(...)()(),...,,( 2121 ll xxxxxxF

Closed forms

Starting with a formula Φ with uniform density (e.g. )

we construct

Sum of n s-sided dices has closed form probability distribution:

)()(),( 21212121 xxxxxxxx

),(...),(),(),...,,( 1212121 lll xxxxxxxxxF

computing the density of states of boolean formulas stefano ermon, carla gomes, and bart selman...

Documents

number of assignments

number of truth assignments

truth assignments energy

n0 slide

density of states dos

clause slide

number of microstates

number of models