searching for autarkies to trim unsatisfiable clause sets mark h. liffiton and karem a. sakallah...

Searching for Autarkiesto

Trim Unsatisfiable Clause Sets

Mark H. Liffiton and Karem A. Sakallah

University of Michigan

{liffiton, karem}@eecs.umich.edu

May 12, 2008

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(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)

c: TRUEd: TRUEe: FALSE

Autarkies

Autarky: Self-sufficiency, Independence.

A partial assignment satisfying all clauses containing one of the assigned variables.

In this talk: “Autarky” = the satisfied clauses.

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(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)(a)(¬a b)(¬a ¬b)(¬b c)(d e)(¬d ¬e)

Autarkies and Infeasibility

Autarkies are disjoint from any clauses involved with the infeasibility of an instance.

(Resolution proofs, minimal unsatisfiable subsets, etc.)

[O. Kullmann. Investigations on autark assignments. Discrete Applied Mathematics, 107(1-3):99–137, 2000.]


AutarkyMinimal Unsatisfiable Subset (MUS);

Leaves of any resolution refutation.

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Outline

Introduction

Previous work

Encoding MaxAutarky to SAT Optimization Comparison to previous work

Autarkies in industrial instances

Trimming autarkies

Conclusion

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Previous Work

Early work used autarkies within SAT algorithms (Monien and Speckenmeyer, Van Gelder)

Dealt mainly with "conditional autarkies“ Not developed for finding maximal autarkies

This work is motivated mainly by Kullmann's work on autarkies

Showed that clauses in a formula are partitioned into 1) the maximal autarky and2) clauses in any resolution proof

Developed an algorithm for MaxAutarky based on this concept

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1. : UNSAT

Previous Work: Finding Autarkies

[O. Kullmann. On the use of autarkies for satisfiability decision. (SAT-2001), pages 231–253.]

Kullmann’s method for MaxAutarkyGiven a formula F:

1. If F is SAT, return the satisfying assignment.

2. Form a resolution refutation, T, of F.

3. Remove variables in T from F.

4. Goto 1.


(a)

(¬a b)

(¬a ¬b)

(¬b)

(b) Т

{a,b}

(c)(d e)(¬d ¬e)(c)(d e)(¬d ¬e)

c=TRUE d=TRUE e=FALSE

2. 3. 1. SAT

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An Encoding to SAT

Deciding whether a formula has an autarky is NP-complete, thus it can be reduced to SAT.

A SAT encoding turns MaxAutarky into a SAT optimization problem.

Our MaxAutarky algorithm “Sifter”: Instrument CNF with extra variables that

enable/disable variables and clauses. Variables and clauses enabled in a satisfying

assignment indicate an autark assignment. Maximize enabled clauses to find maximal

autarky.

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An Encoding to SAT

1. Replace literals with literal substitutes:

2. Add clause-selector variables:

3. Link literal substitutes to variable-selectors:

4. Link clause-selectors with variable-selectors:


(a1)(a0 b1)(a0 b0)(b0 c1)(d1 e1)(d0 e0)(a1)(a0 b1)(a0 b0)(b0 c1)(d1 e1)(d0 e0)

(¬y1 a1)(¬y2 a0 b1)(¬y3 a0 b0) . . .(¬y1 a1)(¬y2 a0 b1)(¬y3 a0 b0) . . .

(¬y1 a1)(¬y2 a0 b1)(¬y3 a0 b0) . . .(a1 = a a+) (a0 = ¬a a+) (b1 = b b+) (b0 = ¬b b+) . . .



(a+ y1)(a+ y2)(a+ y3) . . . (e+ y5)(e+ y6)


(a+ y1)(a+ y2)(a+ y3) . . . (e+ y5)(e+ y6)

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An Encoding to SAT, Optimization

Objective: Maximize ∑ yi

Solved incrementally using a sliding objective Implemented with AtMost constraints

AtMost({l1,l2,…,ln},k) = ∑ assign(li) ≤ k

Employs a single incremental search,exploiting common SAT techniques.


(a+ y1)(a+ y2)(a+ y3) . . . (e+ y5)(e+ y6)


(a+ y1)(a+ y2)(a+ y3) . . . (e+ y5)(e+ y6)

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Removing Pure Literals

Pure literals: Variables appearing in one polarity (pos./neg.) throughout a formula.

Pure literal = [partial] autark assignment

First steps for MaxAutarky:1. Scan for and remove clauses w/ pure literals

2. Repeat until no pure literals remain


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Comparison: Experimental

“Scraper” (Kullmann’s MaxAutarky) vs “Sifter”

Both use same pure literal code (C++)

Scraper Perl script Calls zchaff and zverify_df (zchaff.64bit.2007.3.12)

Measured CPU time (excludes I/O time)

Sifter C++ Built on MiniSAT v.1.12b

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Comparison: Experimental

0.0001

0.001

0.01

0.1

1

10

100

1000

0.0001 0.001 0.01 0.1 1 10 100 1000

Scraper (seconds)

Sift

er (

seco

nds)

Benz

Miter

Dimacs

nPipe

BMC:Barrel

BMC:Longmult

BMC:Queueinvar

600 sec.timeout

600 sec.timeout

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Autarkies in Industrial InstancesFamily Variables Clauses |autarky|

min max min max min max

Benz 1,513 1,891 4,013 9,957 2,097 7,025

Miter 1,266 17,303 1,027 34,238 1 1,831

Dimacs 389 7,767 1,115 20,812 0 0

nPipe 861 15,469 6,695 394,739 0 0

BMC:Barrel 50 8,903 159 36,606 0 0

BMC:Longmult 437 7,807 1,206 24,351 2 2

BMC:Queueinvar 116 2,435 399 20,671 0 0

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Comparison: Qualitative

Both are built around standard SAT solvers, picking up advances as they are made

The encoding used in Sifter is a simple base for future work (e.g., local search)

Sifter is an anytime algorithm

Sifter can be easily modified to find approximations (modify the optimization technique)

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Trimming Autarkies as Preprocessing

Remove independent portions independent before analyzing infeasibility

Only possibly useful in cases where complexity of MaxAutarky is less than that of the analysis.

No: Single US, Single MUS Yes: Smallest MUS, All MUSes

Tested on “Benz” instances (manageable size, substantial autarkies)

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Trimming Autarkies for SMUS

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000

Without trimming autarkies (seconds)

Au

tark

ies

trim

me

d (

seco

nd

s)

600 sec.timeout

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Trimming Autarkies for CAMUS (phase 1)

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000

Without trimming autarkies (seconds)

Au

tark

ies

trim

me

d (

seco

nd

s)

600 sec.timeout

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Conclusion

Novel encoding of MaxAut to SAT

Experimental evaluation of Autarky algorithms

Investigating autarkies for several industrial benchmarks

Investigation of trimming autarkies for infeasibility analysis

Thank you

Mark Liffiton

[email protected]

www.eecs.umich.edu/~liffiton/

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CAMUS

Compute All Minimal Unsatisfiable Subsets

1. Finds all MCSes of a constraint system

2. Computes minimal hitting sets of the MCSes

Constraints

Find MCSes MCSesCompute Minimal

Hitting Sets

All MUSes

CAMUS

searching for autarkies to trim unsatisfiable clause sets mark h. liffiton and karem a. sakallah...

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