Abstract
In this work we investigate how to extract alternating time bounds from ‘focussed’ proof systems. Our main result is the obtention of fragments of mathsf {MALL} {mathsf {w} } (mathsf {MALL} with weakening) complete for each level of the polynomial hierarchy. In one direction we encode QBF satisfiability and in the other we encode focussed proof search, and we show that the composition of the two encodings preserves quantifier alternation, yielding the required result. By carefully composing with well-known embeddings of mathsf {MALL} {mathsf {w} } into mathsf {MALL} , we obtain a similar delineation of mathsf {MALL} formulas, again carving out fragments complete for each level of the polynomial hierarchy. This refines the well-known results that both mathsf {MALL} {mathsf {w} } and mathsf {MALL} are mathbf {PSPACE}-complete. A key insight is that we have to refine the usual presentation of focussing to account for deterministic computations in proof search, which correspond to invertible rules that do not branch. This is so that we may more faithfully associate phases of focussed proof search to their alternating time complexity. This presentation seems to uncover further dualities, at the level of proof search, than usual presentations, so could be of proof theoretic interest in its own right.
Highlights
Introduction and MotivationProof search is one of the most general ways of deciding formulas of expressive logics, both automatically and interactively
To demonstrate the accuracy of this method, we show that these classes are, complete for their respective levels, via encodings from true quantified Boolean formulas (QBFs) of appropriate quantifier complexity, cf. [4]
We present an encoding of true QBFs into MALLw. (We will later adapt this into an encoding into multiplicative additive linear logic (MALL) in Sect. 7.) The former were used for the original proof that MALL is PSPACE-complete [17,18], though our encoding differs from theirs and leads to a more refined result, cf
Summary
Proof search is one of the most general ways of deciding formulas of expressive logics, both automatically and interactively. Focussed systems elegantly delineate the phases of invertible and non-invertible inferences in proofs, allowing the natural obtention of alternating time bounds for a logic. They significantly constrain the number of local choices available, resulting in reduced nondeterminism during proof search, while remaining complete. In this work we retain the classical abstract notion of focussing, but split the usual invertible, or ‘asynchronous’, phase into a ‘deterministic’ phase, with non-branching invertible rules, and a ‘co-nondeterministic’ phase, with branching invertible rules In this way, when expressing proof search as an alternating predicate, a ∀ quantifier needs only be introduced in a co-nondeterministic phase.
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