On Davis-Putnam reductions for minimally …cs.swan.ac.uk/~csoliver/Artikel/2012_SAT_talk.pdfOn Davis-Putnam reductions for minimally unsatis able clause-sets Oliver Kullmann1 Xishun

On Davis-Putnam reductions forminimally unsatisfiable clause-sets

Oliver Kullmann1 Xishun Zhao2

1Computer Science, Swansea University, UK

2Institute of Logic and Cognition, Sun Yat-sen University, China

SAT 2012, June 20, 2012

O Kullmann (Swansea University) Combinatorics of DP and MU Trento 2012 1 / 24

http://cs.swan.ac.uk/~csoliver

http://logic.sysu.edu.cn/faculty/zhaoxishun/en/

http://www.swan.ac.uk/compsci/

http://logic.sysu.edu.cn/logic/english/Index.asp

http://sat2012.fbk.eu/

Introduction

Two sources: DP and MU

Under what circumstances is DP-reduction (single steps of theDP-reduction from Davis and Putnam [2]; also called “variableelimination”) confluent or confluent modulo isomorphism?

Further steps towards the classification of all minimally unsatisfiableclause-sets.


Introduction

Bibliographical remarks

Underlying paper is Kullmann and Zhao [9].

The (extended) underlying technical report is Kullmann and Zhao [10](containing also some technical corrections).


Introduction

Outline

1 Introduction

2 BackgroundDP reductions

3 Minimal unsatisfiability

4 ApplicationsThe classification of MU — understanding unsatisfiability

5 Singular DP-reductionThe main results

6 Underlying insightsDP-reductionsNeighbour exchanges

7 Conclusions


Background

Clause-sets

Literals are variables v and their complements v.

A clause C is a finite and clash-free set of literals, i.e., C ∩ C = ∅.A clause-set is a finite set of clauses.

The set of all clause-sets is CLS.

For example

F2 ={{v1, v2}, {v1, v2}, {v1, v2}, {v2, v1}

}is a clause-set (minimally unsatisfiable, deficiency 2).

Remark: Clause-sets are considered as (precise) combinatorial objects, asgeneralised hypergraphs.


Background

Applying partial assignments

The application of a partial assignment ϕ ∈ PASS to a clause-setF ∈ CLS is denoted by

ϕ ∗ F ∈ CLS.

Satisfied clauses are removed, then falsified literals.A special clause-set is > := ∅, a special clause is ⊥ := ∅.

F is satisfiable iff there is ϕ ∈ PASS with ϕ ∗ F = >.

> is satisfiable.

{⊥} is unsatisfiable.

More generally, every F with ⊥ ∈ F is unsatisfiable.

CLS = SAT ·∪USAT .


Background DP reductions

Resolution and DP-reduction

Clauses C ,D are resolvable if C ∩ D = {x}:

C �D := (C \ {x}) ∪ (D \ {x}).

DP-reduction (or “variable elimination”):

DPv(F) := {C ∈ F : v /∈ var(C )} ∪{C �D : C ,D ∈ F ∧ C ∩ D = {v}

}.

DPv (F ) is semantically the existential quantification of F by v :

DPv (F )←→ ∃ v : F .


Background DP reductions

Singular variables

A variable v is singular for F if

in one sign it occurs only once,

while in the other sign it occurs at least once.

Singular DP-reduction (“sDP-reduction”) is F ; DPv (F )for a singular variable.

sDP-reduction decreases the number of clauses at least by one.


Minimal unsatisfiability

TOC: MU and its structures

MU, SMU

deficiency

MU(1)

non-singularity

MU(2)

See Kleine Buning and Kullmann [5].



MU and SMU

Minimal unsatisfiability: no clause can be removed.

MU := {F ∈ USAT | ∀C ∈ F : F \ {C} ∈ SAT }.

Saturated minimal unsatisfiability: no literal can be added.

SMU := {F ∈MU | ∀C ∈ F ∀C ′ ⊃ C : (F \ {C}) ∪ {C ′} ∈ SAT }.

Lemma

F ∈ SMU iff for all v ∈ var(F ) and ε ∈ {0, 1} holds 〈v → ε〉 ∗ F ∈MU .



Deficiency

The basic complexity parameter of F ∈MU is the deficiency:

δ(F) := c(F )− n(F )

c(F ) := |F |n(F ) := |var(F )|.

The basic fact:F ∈MU =⇒ δ(F ) ≥ 1.

We use hereMU(k) := {F ∈MU : δ(F ) = k}.



MU(1)

Timeline:

Aharoni and Linial [1] (1986) SMU(1)

Davydov, Davydova, and Kleine Buning [3] (1998) MU(1)

Kullmann [6] (2000) tree model

a

bww

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c''

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1 {{a, b}, {a, b}, {a, c}, {a, c}} ∈ SMU(1). Singular: b, c .2 {{a, b}, {b}, {a, c}, {a, c}} ∈ MU(1). Singular: all.3 {{a, b}, {b}, {a, c}, {c}} ∈ MMU(1) (“marginal”).



Eliminating trivialities

We consider sDP-reduction as eliminating “trivialities”.

So all of MU(1) boils down to {⊥}.Intuitively one can understand applying sDP as removing some“MU(1)-hunch”.

A clause-set F is called non-singular if it does not contain singularvariables.

The set of singular variables of F is vars(F) ⊆ var(F ).

So F is nonsingular iff vars(F ) = ∅.

MU ′ := {F ∈MU : vars(F ) = ∅}.



MU(2)

A breakthrough was achieved by Kleine Buning [4] (2000).The elements of MU ′(2) = SMU(2) are precisely the followingclause-sets for n ≥ 2:

x1 → x2, x2 → x3, . . . , xn−1 → xn, xn → x1

{x1, . . . , xn},{x1, . . . , xn}.

That is, one cycle, with opposed forced “directions” (n = 6):

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Applications The classification of MU — understanding unsatisfiability

Understanding UNSAT — understanding MU

For unsatisfiable F ∈ USAT

we want to “understand” its unsatisfiability.

We want to SEE it.

Each F ′ ⊆ F with F ′ ∈MU contains one explanation:

The additional clauses in F \ F ′ can make the contradiction of F ′

more easily accessible, but do not contribute “another reason”.

Different F ′ provide different explanations.

So we have to “explain” unsatisfiability for F ∈MU .

Explain MU ′(k) for k = 1, 2, 3, . . . .



The Classification Conjecture

Conjecture For every k ∈ N there are finitely many “patterns”, whichexplain MU ′(k) completely.

1 MU ′(1) = {⊥}2 MU ′(2) = cycles of length n ≥ 2 plus “at least one variable is false”

plus “at least one variable is true”.

3 MU ′(3): in preparation.



Two problems

For F ∈MU let

sDP(F) := {F ′ ∈MU ′ : FsDP−−→∗ F ′}.

First problem: We want to understand F ∈MU(k). For that weconsider sDP(F ). Note sDP(F ) ⊂MU(k).

Easiest is |sDP(F )| = 1 — when does this hold?

Second problem: A transition FsDP−−→∗ F ′ ∈MU ′ removes “trivialities”

— what if we have to understand these trivialities?


Singular DP-reduction The main results

The “singularity index”

Theorem

For F ∈MU and F ′,F ′′ ∈ sDP(F ) we have n(F ′) = n(F ′′).

This allows us to define the singularity index si(F) ∈ N0 assi(F ) := n(F )− n(F ′) for some F ′ ∈ sDP(F ).

Corollary

If F ∈MU(2), then for F ′,F ′′ ∈ sDP(F ) we have F ′ ∼= F ′′.

Here F ′ ∼= F ′′ denotes isomorphism.



Confluence

Theorem

If F ∈ SMU , then |sDP(F )| = 1.



Confluence modulo isomorphism

Theorem

If for F ∈MU we have sDP(F ) ⊆ SMU , then for F ′,F ′′ ∈ sDP(F ) wehave F ′ ∼= F ′′.

Corollary

If F ∈MU(2), then for F ′,F ′′ ∈ sDP(F ) we have F ′ ∼= F ′′.


Underlying insights DP-reductions

Commutativity

In Kullmann and Luckhardt [7, 8] it is shown:

Lemma

For F ∈ CLS, variables v1, . . . , vk and a permutation π ∈ Sk we have that

DPv1,...,vk (F ), DPvπ(1),...,vπ(k)(F )

are equal modulo subsumption.


Underlying insights Neighbour exchanges

Controlled permutations

The key is to exchange neighbouring sDP-reductions such that weagain obtain an sDP-reduction.

Distinguish between 1-singular variables (occurring in both signs onlyonce) and non-1-singular variables.


Conclusions

Conclusions

Contributions:

For minimally unsatisfiable clause-sets the number of sDP-reductionsuntil reaching nonsingularity does not depend on the choice ofreductions.

Confluence of sDP-reduction for saturated minimally unsatisfiableclause-sets.

Confluence modulo isomorphism of sDP-reduction in case all maximalsDP-reductions yield saturated minimally unsatisfiable clause-sets.

Obtaining unique normalforms (up to isomorphism) for MU(2) viasDP-reduction.

Next steps:

Fuller understanding of sDP-reduction.

Generalisation to all of CLS.


Conclusions

End

(references on the remaining slides).


Conclusions

Bibliography I

[1] Ron Aharoni and Nathan Linial. Minimal non-two-colorablehypergraphs and minimal unsatisfiable formulas. Journal ofCombinatorial Theory, A 43:196–204, 1986.

[2] Martin Davis and Hilary Putnam. A computing procedure forquantification theory. Journal of the ACM, 7:201–215, 1960.

[3] Gennady Davydov, Inna Davydova, and Hans Kleine Buning. Anefficient algorithm for the minimal unsatisfiability problem for asubclass of CNF. Annals of Mathematics and Artificial Intelligence,23:229–245, 1998.

[4] Hans Kleine Buning. On subclasses of minimal unsatisfiable formulas.Discrete Applied Mathematics, 107:83–98, 2000.


Conclusions

Bibliography II

[5] Hans Kleine Buning and Oliver Kullmann. Minimal unsatisfiabilityand autarkies. In Armin Biere, Marijn J.H. Heule, Hans van Maaren,and Toby Walsh, editors, Handbook of Satisfiability, volume 185 ofFrontiers in Artificial Intelligence and Applications, chapter 11, pages339–401. IOS Press, February 2009. ISBN 978-1-58603-929-5. doi:10.3233/978-1-58603-929-5-339.

[6] Oliver Kullmann. An application of matroid theory to the SATproblem. In Fifteenth Annual IEEE Conference on ComputationalComplexity, pages 116–124. IEEE Computer Society, July 2000. Seealso TR00-018, Electronic Colloquium on Computational Complexity(ECCC), March 2000.


Conclusions

Bibliography III

[7] Oliver Kullmann and Horst Luckhardt. Deciding propositionaltautologies: Algorithms and their complexity. Preprint, 82 pages; theps-file can be obtained athttp://cs.swan.ac.uk/~csoliver/Artikel/tg.ps, January1997.

[8] Oliver Kullmann and Horst Luckhardt. Algorithms for SAT/TAUTdecision based on various measures. Preprint, 71 pages; the ps-filecan be obtained fromhttp://cs.swan.ac.uk/~csoliver/Artikel/TAUT.ps, February1999.

[9] Oliver Kullmann and Xishun Zhao. On Davis-Putnam reductions forminimally unsatisfiable clause-sets. In Alessandro Cimatti andRoberto Sebastiani, editors, Theory and Applications of SatisfiabilityTesting - SAT 2012, volume LNCS 7317 of Lecture Notes inComputer Science, pages 270–283. Springer, 2012.


http://cs.swan.ac.uk/~csoliver/Artikel/tg.ps

http://cs.swan.ac.uk/~csoliver/Artikel/TAUT.ps

Conclusions

Bibliography IV

[10] Oliver Kullmann and Xishun Zhao. On Davis-Putnam reductions forminimally unsatisfiable clause-sets. Technical ReportarXiv:1202.2600v3 [cs.DM], arXiv, May 2012.


Documents

On Davis-Putnam reductions for minimally …cs.swan.ac.uk/~csoliver/Artikel/2012_SAT_talk.pdfOn Davis-Putnam reductions for minimally unsatis able clause-sets Oliver Kullmann1 Xishun