QRAT Polynomially Simulates Merge Resolution

07/20/2021 ∙ by Sravanthi Chede, et al. ∙ Indian Institute of Technology Ropar 0

Merge Resolution (MRes [Beyersdorff et al. J. Autom. Reason.'2021] ) is a refutational proof system for quantified Boolean formulas (QBF). Each line of MRes consists of clauses with only existential literals, together with information of countermodels stored as merge maps. As a result, MRes has strategy extraction by design. The QRAT [Heule et al. J. Autom. Reason.'2017] proof system was designed to capture QBF preprocessing. QRAT can simulate both the expansion-based proof system ∀Exp+Res and CDCL-based QBF proof system LD-Q-Res. A family of false QBFs called SquaredEquality formulas were introduced in [Beyersdorff et al. J. Autom. Reason.'2021] and shown to be easy for MRes but need exponential size proofs in Q-Res, QU-Res, CP+∀red, ∀Exp+Res, IR-calc and reductionless LD-Q-Res. As a result none of these systems can simulate MRes. In this paper, we show a short QRAT refutation of the SquaredEquality formulas. We further show that QRAT strictly p-simulates MRes. Besides highlighting the power of QRAT system, this work also presents the first simulation result for MRes.

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1 Introduction

Quantified Boolean formulas (QBF) extend propositional logic with quantifications, there exists () and for all (). QBF proof complexity deals with understanding the limitations and strength of various QBF solving approaches. In the literature, there exists mainly two solving approaches i.e. Conflict-Driven-Clause-Learning (CDCL) and expansion-based solving. Several QBF proof systems have been developed to capture these solving approaches. Q-resolution (Q-Res[KBKF95] is the base of CDCL-based approach. It is further extended to QU-resolution (QU-Res[qures_simul] and Long-Distance-resolution (LD-Q-Res[ldqres_paper]. On the other hand, proof system Exp+Res [JM15] is the base of expansion-based solving. It is further extended to powerful proof systems IR-calc [BCJ14] and IRM-calc [BCJ14]. The simulation orders of these proof systems are well studied in the literature [BCJ15, Figure 1].

Recently, a new proof system Merge resolution (MRes[mres_paper] has been developed. It follows a different QBF-solving approach. In MRes, winning strategies for the universal player are explicitly represented within the proof in the form of deterministic branching programs, known as merge maps [mres_paper]. MRes builds partial strategies at each line of the proof such that the strategy at the last line (corresponding to the empty clause) forms the complete countermodel for the input QBF. As a result, MRes admits strategy extraction by design. While performing resolution steps, MRes merges the partial strategies of the two hypotheses carefully if their corresponding merge maps are isomorphic or consistent. Note that whether two merge maps are isomorphic or consistent can be checked efficiently. This allows those resolution steps to be performed in MRes which would have been blocked in LD-Q-Res.

To be precise, in LD-Q-Res universal variables could appear in both polarities in the hypotheses and get merged in the resolvent provided appears in the right of the pivot variable in the quantifier prefix. MRes relaxed this restriction by allowing resolution steps even if is on the left of the pivot variable provided the merge maps of in both the hypotheses are isomorphic. This makes MRes powerful as compared to reductionless LD-Q-Res [BjornerJK15, PeitlSS19a]. In fact there exists a family of false QBFs SquaredEquality formulas (Definition LABEL:def:squared) with short refutations in MRes [mres_paper] but require exponential size refutations in Q-Res, QU-Res, CP+red [qcp18], Exp+Res, IR-calc [BeyersdorffB20, BeyersdorffBH19] and reductionless LD-Q-Res [mres_paper]. Therefore, none of these proof systems can simulate MRes.

Quantified Resolution Asymmetric Tautologies (QRAT) proof system is introduced in [qrat_paper] to capture the preprocessing steps performed by several QBF-solvers. It has been shown in [qrat_paper] that QRAT can efficiently simulate all the existing preprocessing steps used by present-day QBF solvers. Recently, it has been shown that QRAT can simulate both the expansion-based proof system Exp+Res [ecalculus_simul] and CDCL-based proof system LD-Q-Res [ldqres_simul]. Since QRAT allows resolution steps with universal variables as pivot, it simulates QU-Res as well [ldqres_simul]. It is also known that QRAT is strictly stronger than Exp+Res, LD-Q-Res and QU-Res [ldqres_simul, Figure 2].

In this short paper, we extend the importance of QRAT among QBF proof systems by showing that QRAT even polynomially simulates MRes. We also show that refuting the SquaredEquality formulas in QRAT is easy. Thus the semantic structure of these formulas which makes it harder to refute in all other proof systems is not a restriction for QRAT. We explain these contributions in the following subsection.

1.1 Our contributions

Short QRAT refutation of SquaredEquality formulas: SquaredEquality formulas, a variant of equality formulas [BeyersdorffBH19], have been defined in [mres_paper] to show that MRes is strictly stronger than reductionless LD-Q-Res [BjornerJK15, PeitlSS19a]. The original equality formulas which are hard for Q-Res but easy for LD-Q-Res have been extended in a way that prohibits the resolution step in reductionless LD-Q-Res but not in MRes.
In this paper, we show that the SquaredEquality formulas have a short refutation in QRAT (Theorem LABEL:short_proof). Also, since the original equality formulas are easy for LD-Q-Res and QRAT can simulate LD-Q-Res, the formulas are easy for QRAT as well.
Note that, all other known families of false QBFs used to establish the incomparability results among QBF proof systems are easy for QRAT: KBKF [KBKF95, Theorem 3.2] formulas are easy for QU-Res [qures_simul, Example 5.5] hence they are easy for QRAT. Similarly, QPARITY [BCJ15] formulas are easy for Exp+Res [BCJ15, Lemma 15] and hence easy for QRAT. Variants of these formulas were used to show the incomparability results among proof systems known to be simulated by QRAT. Hence these formulas are also easy for QRAT. Thus the presented short QRAT refutation of the SquaredEquality formulas makes this formulas also easy for QRAT.

QRAT polynomially simulates MRes: It has been shown that MRes can simulate reductionless LD-Q-Res [mres_paper]. However, none of the proof systems Q-Res, QU-Res, CP+red, Exp+Res, IR-calc and reductionless LD-Q-Res are capable of simulating MRes. The difficulty for these proof systems lies in simulating the axiom steps of MRes. To be precise, MRes gets rid of all the universal variables from the input clauses just by maintaining the partial strategies for them. On the other hand, the above mentioned proof systems have different and restricted rules for handling the universal variables. For example, Q-Res and QU-Res use universal reduction(UR) rule, which allows dropping a universal variable only if it is not blocked. Similarly, expansion-based proof systems like Exp+Res and IR-calc handle the universal variables by introducing the annotated existential variables only.
In this paper, we show how QRAT handles this hurdle and polynomially simulates MRes. We show this by proving that the downloaded clauses in MRes proofs are all Asymmetric Tautology (AT) (Definition LABEL:asymmetric-tautology) with respect to the input QBF (Lemma LABEL:existential_clause). Therefore, they can easily be added in QRAT. Since the resolution step can be easily simulated by QRAT (Observation LABEL:obs:qrat-simulate-res), the remaining resolution steps in MRes refutation can also be simulated (Theorem LABEL:simul).

Emphasizing the importance of QRAT among QBF proof systems: QRAT has been shown to simulate varieties of QBF solving approaches. That is on one hand, QRAT can simulate the expansion-based system Exp+Res and on the other, it can simulate the powerful CDCL-based system LD-Q-Res. Since MRes is based on an entirely different QBF-solving approach; by showing that QRAT can polynomially simulate MRes, the paper extends the importance of QRAT system.

QRAT is a possible candidate for the universal checking format which can verify all existing QBF-solving techniques [chew21]. Our simulation result is a small step in this direction. (The other possible candidate is the extended Frege for QBFs, denoted as, eFrege+red [chew21, Conjecture 1]). For the simulation order and incomparabilities involving QRAT and several QBF proof systems, refer Figure LABEL:fig1.