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This study merges dependency quantified Boolean formulas (DQBF) and QBF dependency schemes by reinterpreting schemes as DQBF mappings. It reveals

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Area of Science:

  • Automated Reasoning
  • Computational Logic
  • Theoretical Computer Science

Background:

  • Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes are distinct yet related extensions of QBF.
  • Both formalisms replace linear quantifier order with partial orders, enabling more expressive reasoning.
  • Existing literature treats these concepts separately, hindering a unified understanding of their proof systems.

Purpose of the Study:

  • To unify the treatment of DQBF and QBF dependency schemes.
  • To reinterpret dependency schemes as mappings from QBF to DQBF.
  • To gain new insights into the soundness of proof systems for QBF with dependency schemes.

Main Methods:

  • Reinterpreting dependency schemes as mappings into DQBF.
  • Investigating DQBF proof systems.
  • Applying the reinterpretation to analyze soundness properties ('full exhibition') in universal reduction and universal expansion paradigms.

Main Results:

  • Full exhibition is sufficient but not necessary for soundness in universal reduction systems.
  • Full exhibition exactly characterizes soundness in universal expansion systems.
  • The reflexive resolution path dependency scheme is demonstrated to be fully exhibited.

Conclusions:

  • The proposed reinterpretation provides a unified framework for DQBF and QBF dependency schemes.
  • The property of 'full exhibition' plays a crucial role in characterizing soundness across different proof system paradigms.
  • A long-standing conjecture by Slivovsky regarding the reflexive resolution path dependency scheme is proven.