K-consistency and SAT
Download
1 / 13

K-consistency and SAT Teague Lasser - PowerPoint PPT Presentation


  • 96 Views
  • Uploaded on

K-consistency and SAT Teague Lasser. Petke, J. and Jeavons, P. Local Consistency and SAT-Solvers CP 2010 pp.398-413. Introduction. The relationship between constraint satisfaction problems and boolean satisfiability problems has been an area of active research in the last decade

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' K-consistency and SAT Teague Lasser' - xandra-hamilton


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

K-consistency and SAT

Teague Lasser

Petke, J. and Jeavons, P.

Local Consistency and SAT-Solvers

CP 2010 pp.398-413


Introduction
Introduction

  • The relationship between constraint satisfaction problems and boolean satisfiability problems has been an area of active research in the last decade

  • CP solvers attempt to learn new constraints but the belief of the community has been that attempting to enforce stronger consistency (k-consistency) than generalized arc consistency is inefficient

  • SAT solvers which rely primarily on resolution to produce refutations have shown remarkable


K consistency
k-Consistency

  • K-consistency extends arc consistency and path consistency to arbitrarily high levels of consistency

  • A nearly-optimal algorithm KS for k-consistency for any k proposed in 1989 suggests that if k-consistency is used as a method of relaxing the problem its time complexity is

    O(n2(a + 1)n)

    where n is the number of variables and a is a number that relates the number of labelings by a constraint on the domain1

[1] Cooper, M.

An Optimal k-Consistency Algorithm

Artificial Intelligence 41 pp.89-95


Converting a csp to a sat problem
Converting a CSP to a SAT problem

  • Take a CSP of the form P = {V, D, C}

  • Introduce a set of boolean variables of the form xvd for each v∈V with domain element d∈Dv

  • Along the each independent domain, ensure that only one variable can be true for the clause to be satisfied. ∨d∈DvXvd , ¬xvi∨¬xvj for all i,j∈Dv

  • Encode constraints as clauses of 2 or more boolean variables for each partial assignment that does not satisfy the constraint

    ∨v∈S¬xvf(v)


Resolution
Resolution

  • We can solve clauses by inference across some clauses of the form

  • C1∨ x and C2∨ ¬x to produce a new clause (the resolvent) of the form C1∨ C2

  • If we have a collection of clauses of the form Ci∨¬xi for

  • i =1, 2,...,r, where each xi is a Boolean variable, and a purely positive clause

  • x1∨ x2∨···∨ xr, then we can deduce the clause C1∨ C2∨···∨ Cr.

  • They dubbed this form of inference positive-hyper-resolution and the resultant

  • clause C1∨ C2∨···∨ Cr the positive-hyper-resolvent.


K consistency and positive hyper resolution
K-Consistency and Positive-Hyper-Resolution

Theorem 1. The k-consistency closure of a CSP instance P is empty if and only if its direct encoding as a set of clauses has a positive-hyper-resolution refutation of width at most k.

Lemma 1.1. Let P be a CSP instance, and let Φ be its direct encoding as a set of clauses. If Φ has no positive-hyper-resolution refutation of width k or less, then the k-consistency closure of P is non-empty.

Lemma 1.2. Let P be a CSP instance, and let Φ be its direct encoding as a set of clauses. If the k-consistency closure of P is non-empty, then Φ has no positive-hyper-resolution refutation of width k or less.


Proof discussion
Proof discussion

∨d∈DvXvd , ¬xvi∨¬xvj for all i, j∈Dv

∨v∈S¬xvf(v)

C1VC2Vx1 ¬x1V¬x2 x2VC3VC4

C1VC2vC3VC4


  • Theorem 2. If a set of non-empty clauses Δ over n Boolean variables has a positive-hyper-resolution refutation of width k and length m, where all derived clauses contain only negative literals, then the expected number of restarts required by a standard randomised SAT-solver to discover that Δ is unsatisfiable is less than mnk(n on k).


  • Theorem 3. If a set of non-empty clauses Δ over n Boolean variables has a positive-hyper-resolution refutation of width k and length m, where all derived clauses contain only negative literals, then the expected number of restarts required by a standard randomised SAT-solver using the Decision learning scheme to discover that Δ is unsatisfiable is less than m(n on k).


  • Theorem 4. If the k-consistency closure of a CSP instance P is empty, then the expected number of restarts required by a standard randomised SAT-solver using the Decision learning scheme to discover that the direct encoding of P is unsatisfiable is O(n2kd2k), where n is the number of variables in P and d is the maximum domain size.


A problem with no solution
A problem with no solution

  • n = ((d-1)*w + 2)*w over w groups d = {0..d-1} tree width = 2w - 1



Acknowledgements
Acknowledgements

  • Thank you to Justyna Petke for allowing me to use her modified SAT solver

  • Thank you to Peter for allowing me to use his laptop


ad