Should Decisions in QCDCL Follow Prefix Order?

Authors:
raw_author_name: Benjamin Böhm
raw_affiliation_string: Friedrich Schiller University Jena, Jena, Germany
raw_author_name: Tomáš Peitl
raw_affiliation_string: TU Wien, Vienna, Austria
raw_author_name: Olaf Beyersdorff
raw_affiliation_string: Friedrich Schiller University Jena, Jena, Germany

Abstract: Abstract Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions. We investigate an alternative model for QCDCL solving where decisions can be made in arbitrary order. The resulting system $$\textsf{QCDCL}^\textsf {{A\tiny {\MakeUppercase {ny}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>QCDCL</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>NY</mml:mi> </mml:mrow> </mml:msup> </mml:math> is still sound and terminating, but does not necessarily allow to always learn asserting clauses or cubes. To address this potential drawback, we additionally introduce two subsystems that guarantee to always learn asserting clauses ( $$\textsf{QCDCL}^\textsf {{U\tiny {\MakeUppercase {ni}}-A\tiny {\MakeUppercase {ny}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>QCDCL</mml:mi> <mml:mrow> <mml:mi>U</mml:mi> <mml:mi>NI</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mi>NY</mml:mi> </mml:mrow> </mml:msup> </mml:math> ) and asserting cubes ( $$\textsf{QCDCL}^\textsf {{E\tiny {\MakeUppercase {xi}}-A\tiny {\MakeUppercase {ny}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>QCDCL</mml:mi> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>XI</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mi>NY</mml:mi> </mml:mrow> </mml:msup> </mml:math> ), respectively. We model all four approaches by formal proof systems and show that $$\textsf{QCDCL}^\textsf {{U\tiny {\MakeUppercase {ni}}-A\tiny {\MakeUppercase {ny}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>QCDCL</mml:mi> <mml:mrow> <mml:mi>U</mml:mi> <mml:mi>NI</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mi>NY</mml:mi> </mml:mrow> </mml:msup> </mml:math> is exponentially better than $$\mathsf{{QCDCL}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>QCDCL</mml:mi> </mml:math> on false formulas, whereas $$\textsf{QCDCL}^\textsf {{E\tiny {\MakeUppercase {xi}}-A\tiny {\MakeUppercase {ny}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>QCDCL</mml:mi> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>XI</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> <mml:mi>NY</mml:mi> </mml:mrow> </mml:msup> </mml:math> is exponentially better than $$\mathsf{{QCDCL}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>QCDCL</mml:mi> </mml:math> on true QBFs. Technically, this involves constructing specific QBF families and showing lower and upper bounds in the respective proof systems. We complement our theoretical study with some initial experiments that confirm our theoretical findings.