Proof Compression and NP Versus PSPACE II

Abstract

We upgrade [3] to a complete proof of the conjecture NP = PSPACE that is known as one of the fundamental open problems in the mathematical theory of computational complexity; this proof is based on [2]. Since minimal propositional logic is known to be PSPACE complete, while PSPACE to include NP, it suffices to show that every valid purely implicational formula ρ has a proof whose weight (= total number of symbols) and time complexity of the provability involved are both polynomial in the weight of ρ. As in [3], we use proof theoretic approach. Recall that in [3] we considered any valid ρ in question that had (by the definition of validity) a "short" tree-like proof π in the Hudelmaier-style cutfree sequent calculus for minimal logic. The "shortness" means that the height of π and the total weight of different formulas occurring in it are both polynomial in the weight of ρ. However, the size (= total number of nodes), and hence also the weight, of π could be exponential in that of ρ. To overcome this trouble we embedded π into Prawitz's proof system of natural deductions containing single formulas, instead of sequents. As in π, the height and the total weight of different formulas of the resulting tree-like natural deduction ∂1 were polynomial, although the size of ∂1 still could be exponential, in the weight of ρ. In our next, crucial move, ∂1 was deterministically compressed into a "small", although multipremise, dag-like deduction ∂ whose horizontal levels contained only mutually different formulas, which made the whole weight polynomial in that of ρ. However, ∂ required a more complicated verification of the underlying provability of ρ. In this paper we present a nondeterministic compression of ∂ into a desired standard dag-like deduction ∂0 that deterministically proves ρ in time and space polynomial in the weight of ρ. Together with [3] this completes the proof of NP = PSPACE.
 Natural deductions are essential for our proof. Tree-to-dag horizontal compression of π merging equal sequents, instead of formulas, is (possible but) not sufficient, since the total number of different sequents in π might be exponential in the weight of ρ − even assuming that all formulas occurring in sequents are subformulas of ρ. On the other hand, we need Hudelmaier's cutfree sequent calculus in order to control both the height and total weight of different formulas of the initial tree-like proof π, since standard Prawitz's normalization although providing natural deductions with the subformula property does not preserve polynomial heights. It is not clear yet if we can omit references to π even in the proof of the weaker result NP = coNP.

Yet another argument in favour of NP=CoNP

“…In [5] and [4], we show the existence of short certificates for every M ⊃ valid formula via compression of Natural Deduction proofs into Directed Acyclic Graphs (DAGs). We are aware that NP = PSPACE implies that NP = CoNP.…”

“…= CoNP in terms of proof systems. The background theory and terminology come from [9], [6], [11], [10], [5] and [4]. In section 2, we briefly review the related work.…”

Proof Compression and NP Versus PSPACE II: Addendum

“…Recall that in [2,3] we proved that intuitionistically valid purely implicational formulas ρ have dag-like ND proofs ∂ whose weights (= the total numbers of symbols) are polynomial in the weights |ρ| of ρ. ∂ were defined by a suitable two-fold horizontal compression of the appropriate tree-like ND ∂ 1 obtained by standard conversion of basic tree-like HSC proofs π existing by the validity of ρ.…”

“…Moreover, it is readily seen that the latter conclusion holds true for any tree-like ND ∂ with the polynomial upper bounds on the height and total weight of distinct formulas used. We just arrived at the following Theorem 1.3, where NM → is standard purely implicational ND for minimal logic (see also Appendix and [3] for more details).…”

“…

In [3] we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC) [5] with the horizontal compressing in the corresponding minimal Prawitzstyle natural deduction (ND) [6]. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC.

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