Abstract
Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-style calculus for DQBF, thus opening future avenues into CDCL-based DQBF solving.
Highlights
Proof complexity investigates the resources for proving logical theorems, focussing foremost on the minimal size of proofs needed in a particular calculus
Given what we know about the semantics of dependency QBFs (DQBF), we pose the following question: What is the impact of the existence of type C and D DQBFs on the transfer of solving techniques from quantified Boolean formulas (QBF)? We argue that the impact is visible in theoretical models of solving
Merge Resolution (M-Res) works with Herbrand-form DQBFs, whereas the system in [13] was defined for Skolem-form DQBFs
Summary
Proof complexity investigates the resources for proving logical theorems, focussing foremost on the minimal size of proofs needed in a particular calculus. This is in contrast to all previous existing QBF calculi in the literature, where strategies are algorithmically constructed from proofs This applies to the approaches taken in [23,54] for LD-Q-Res and in [15] for reductionless LD-Q-Res. and the choice of our representation as merge maps matters: as [15,54] both represent (partial) strategies as trees, the constructed strategies may grow exponentially in the proof size, losing the property of efficient strategy extraction desired for practice. In M-Res we have explicit representations of strategies and can allow resolution steps as long as the strategies in both parent clauses are isomorphic to each other, a property that we can check efficiently for merge maps This last mentioned advantage of allowing resolution steps in M-Res forbidden in (reductionless) LD-Q-Res manifests in shorter proofs. By design our DQBF system has efficient strategy extraction
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