Short Proofs for Some Symmetric Quantified Boolean Formulas

04/04/2018 ∙ by Manuel Kauers, et al. ∙ Johannes Kepler University Linz 0

We exploit symmetries to give short proofs for two prominent formula families of QBF proof complexity. On the one hand, we employ symmetry breakers. On the other hand, we enrich the (relatively weak) QBF resolution calculus Q-Res with the symmetry rule and obtain separations to powerful QBF calculi.

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1 Introduction

A Quantified Boolean Formula (QBF) is a formula of the form , where is a propositional formula, say in the variables , and is a quantifier prefix with . From QBF proof complexity, it is well-known that the shortest proof of certain QBFs may have exponential size in a resolution-based calculus DBLP:books/daglib/0075409 ; DBLP:conf/stacs/BeyersdorffCJ15 . We consider here two families of QBFs (cf. Section 2) which play a prominent role in QBF proof complexity for separating various calculi. We make the observation that short proofs can be obtained if we take into account the symmetries of the formulas. In Section 3, we do so by using symmetry breakers. In Section 4, we enrich the oldest variant of the resolution calculus for QBF, Q-Res DBLP:journals/iandc/BuningKF95 , by a symmetry rule, generalizing an idea reported in DBLP:journals/acta/Krishnamurthy85 ; DBLP:journals/dam/Urquhart99 for SAT. In both cases, it turns out that the proof sizes for both families of formulas shrinks from exponential to linear. As consequences, we obtain separation results between Q-Res with the symmetry rule and powerful proof systems like IR-calc DBLP:conf/stacs/BeyersdorffCJ15 and LQU DBLP:conf/sat/BalabanovWJ14 (cf. Section 5).

Let us recall some basic facts and fix some notation. We only consider QBFs where is in conjunctive normal form (CNF), i.e., is a conjunction of clauses, each clause being a disjunction of literals, each literal being a variable or a negated variable, i.e., if is a variable, and are literals. We also view clauses as sets of literals. The prefix imposes an order on its variables: if . The Q-Res calculus DBLP:journals/iandc/BuningKF95 applies the following rules on a QBF :

  1. Any clause of can be derived.

  2. From the already derived clauses and with existentially quantified variable and such that is not a tautology, the clause can be derived.

  3. Let be an already derived clause where is a universal literal, and all existential literals are such that . Then the clause can be derived.

In the following, we do not mention the application of the axiom rule A explicitly. We write , and for the application of R and U. A refutation of a QBF is the consecutive application of the resolution rule R and the universal reduction rule U until the empty clause is derived. Q-Res is sound and complete.

Finally, let us recall the notion of (syntactic) symmetries for QBFs. A bijective map from the set of literals to itself is called admissible for a prefix if for all and for all , we have only if and belong to the same quantifier block, i.e., . An admissible function is called a symmetry for a QBF with in CNF if applying to all literals in maps to itself (possibly up to reordering clauses and literals).

2 Formula Families

We consider the following two families of formulas.

Definition 0 (DBLP:books/daglib/0075409 )

For , the formula is defined by the prefix

and the following clauses:

  • for :


    .

  • ,

  • for :

    and .

For every , the formula is false, and it is known DBLP:books/daglib/0075409 that any Q-Res refutation needs a number of steps which is at least exponential in .

Definition 0 (DBLP:conf/stacs/BeyersdorffCJ15 )

For with , the formula is defined by the prefix

and the following clauses:




  • for :




  • and

  • for , are obtained from by replacing by .

is a variant of the family DBLP:conf/stacs/BeyersdorffCJ15 which encodes , where stands for exclusive or. Obviously all these formulas are false. Refuting needs an exponential number of steps in the calculus Q-Res, but not in the stronger calculus LQU. We use instead of because for this family, also LQU needs exponentially many steps DBLP:conf/stacs/BeyersdorffCJ15 . This will be used in Section 5.

3 Symmetry Breakers

Let be a set of symmetries for a QBF . A symmetry breaker is a certain Boolean formula such that when is true, so is . Writing , it was shown in audemard2007efficient ; DBLP:journals/corr/abs-1802-03993 that

is a symmetry breaker.

For the formulas (Def. 1), we have for every the symmetry which exchanges the variables , the literals , and the literals . Therefore,

is a symmetry breaker for .

Proposition 0

For , write as , and let be the symmetry breaker from above. Then has a refutation proof with no more than steps.

The proof proceeds as follows.

  • ,.

  • for , do

    ,.

    ,.

    Then .

  • ,.

  • for , do

    ,.

    Then .

  • for , do

    .

    is the empty clause.

For the formulas , the argument is similar. In this case, we have the symmetries and

for every . There are some further symmetries which we will not need. The symmetries give rise to the symmetry breaker

for .

Proposition 0

For with , write as , and let be the symmetry breaker from above. Then has a refutation proof with no more than steps.

The proof proceeds as follows.

  • ,.

  • ,.

  • for , do

    ,.

  • for , do

    ,.

  • ,.

  • empty clause.

4 The Symmetry Rule

As an alternative to using symmetry breakers, we can enrich the calculus Q-Res as introduced in Section 1 to the calculus Q-Res+S by adding the following rule, which allows us to exploit symmetries of the input formula within the proof.

  1. From an already derived clause and a symmetry of , the clause can be derived.

Several variants of this rule have been proposed for SAT in DBLP:journals/acta/Krishnamurthy85 ; DBLP:journals/dam/Urquhart99 , but to our knowledge it has not yet been considered in the context of QBF. However, it is easy to see that the rule also works for QBF.

Proposition 0

Let be a QBF, and suppose that is a clause which can be derived from using the rules S, R, U. Then it can also be derived using only the rules R, U.

Proof. Suppose otherwise. Then there are clauses which can be derived with S, R, U but not with R, U alone. Let be such a clause, and consider a derivation of with a minimal number of applications of S. The rule S is used at least once during the derivation. Consider its earliest application, suppose this application derives from the clause . If we can show that can also be derived using only R and U, then we can eliminate this first application of S in the derivation of  and obtain a contradiction to the assumed minimality.

To show that can be derived using only R and U, observe first that was derived only using R and U. For an admissible function , we have for every variable . Therefore, if a clause can be derived by R from two clauses and , we can derive by R from and . Furthermore, an admissible function cannot permute literals across quantifier blocks, which implies that if can be derived by U from , then can be derived by U from . Finally, when is a symmetry of and is a clause of , then also is a clause of . By combining these three observations, it follows that applying to all clauses appearing in the derivation of yields a derivation of . This completes the proof. ∎

According to the previous proposition, with S we cannot derive any clause that we cannot also derive without. Therefore, soundness of Q-Res+S follows from soundness of Q-Res. Next, we illustrate that Q-Res+S allows for shorter proofs than Q-Res. For the application of S, we write ,.

Proposition 0

For every , the formula can be refuted by no more than applications of S, R, U.

We proceed as follows by using the symmetries of the form for .

  • set .

  • for , do

    ,.

  • set .

  • for , do

    .
    ,.
    ,.
    ,.

  • .

  • ,.

  • ,.

  • ,empty clause.

Proposition 0

For every with , the formula can be refuted by no more than applications of S, R, U.

Recall from Section 4 that has the symmetries and for .

  • ,.

  • for , do

    ,.

  • ,.

  • .

  • for , do

    ,.
    ,.

  • ,.

  • ,.

  • ,empty clause.

5 Consequences

From recent results, it is known that plain Q-Res is rather weak (for a fine-grained comparison of QBF proof systems see DBLP:conf/stacs/BeyersdorffCJ15 ). Both, the expansion-based proof system IR-calc and the CDCL-based proof system LQU are strictly stronger than Q-Res. The addition of the symmetry rule changes the situation. While the formulas are hard for LQU and the formulas are hard for IR-calc, we have shown that both are easy for Q-Res+S. Now one may ask if Q-Res+S is strictly stronger than IR-calc or LQU. The answer is clearly “no”. For , the application of the symmetry rule can be hindered by introducing universally quantified variables which are placed between and in the prefix. Further, each clause changes to . For this modified formula, LQU can still find a short proof, but Q-Res+S can only apply R and U, hence it falls back to Q-Res which does not exhibit short proofs for . In a similar way, can be modified such that these formulas remain simple for IR-calc, but become hard for Q-Res+S.

Proposition 0

Q-Res+S and IR-calc are incomparable, and so are Q-Res+S and LQU.

For the future, the effects of adding S to more powerful proof systems than Q-Res remain to be investigated.

Acknowledgements. Parts of this work were supported by the Austrian Science Fund (FWF) under grant numbers NFN S11408-N23 (RiSE), Y464-N18, and SFB F5004.

References

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