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We present a proof of the conjecture NP = PSPACE by showing that arbitrary tautologies of Johansson's minimal propositional logic admit "small" polynomial-size dag-like natural deductions in Prawitz's system for minimal propositional logic. These "small" deductions arise from standard "large" tree-like inputs by horizontal dag-like compression that is obtained by merging distinct nodes labeled with identical formulas occurring in horizontal sections of deductions involved. The underlyingarXiv:1609.09562v1 fatcat:27ljismue5gfdn3ukdb7tvvqyy
more »... ric" idea: if the height, h( ∂) , and the total number of distinct formulas, ϕ( ∂) , of a given tree-like deduction ∂ of a minimal tautology ρ are both polynomial in the length of ρ, | ρ|, then the size of the horizontal dag-like compression is at most h( ∂) ×ϕ( ∂) , and hence polynomial in | ρ|. The attached proof is due to the first author, but it was the second author who proposed an initial idea to attack a weaker conjecture NP= coNP by reductions in diverse natural deduction formalisms for propositional logic. That idea included interactive use of minimal, intuitionistic and classical formalisms, so its practical implementation was too involved. The attached proof of NP=PSPACE runs inside the natural deduction interpretation of Hudelmaier's cutfree sequent calculus for minimal logic.
We upgrade  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 . 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 weightdoi:10.18778/0138-0680.2020.16 fatcat:kgl5di3jdnd5tlboqackju2jeu
more »... of ρ. As in , we use proof theoretic approach. Recall that in  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  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.
In  we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC)  with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction (ND) . In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea (due to the second author) is to omit full minimal logic and compress only \naive" normal tree-like ND refutations of thedoi:10.18778/0138-0680.2022.01 fatcat:d734nbldjfb3baul2svn773b7e
more »... tence of Hamiltonian cycles in given non-Hamiltonian graphs, since the Hamiltonian graph problem in NP-complete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE .
Traditional proof theory of Propositional Logic deals with proofs whose size can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The use of proof-graphs, instead of trees or lists, for representing proofs is getting popular among proof-theoreticians. Proof-graphs serve as a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs.doi:10.1016/j.entcs.2016.06.012 fatcat:f4wwpgry4fc5rmowu6ew2a2hlu
more »... Fpl-graphs were initially developed for minimal implicational logic representing proofs through references rather than copy. Thus, formulas and sub-deductions preserved in the graph structure, can be shared deleting unnecessary sub-deductions resulting in the reduced proof. In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting the number of maximal formulas in the original derivation. In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph and the case of the hidden maximum formula that usually occurs in the tree-form derivation is already represented in the fpl-graph.
It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of thisdoi:10.4204/eptcs.144.2 fatcat:stbdnq2enneoliegqxe3dokhey
more »... m is that formulas can be shared in the reduced proof. In the present paper we give a precise definition of proof-graphs for the minimal implicational logic, together with a normalization procedure for these proof-graphs. In contrast to standard tree-like formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proof-graph representations.
Gordeev, Lev Gordeev, Lew ............ See Gordeev, Lev Gordillo Ardila, Jorge Enrique .... * 11001 Gordon, Cameron McA. $7012,57036 Gordon, E. ...
National Union Catalog
Tolstol, Lev , Lew Ni laevich, grat. voina | mir. PG3365. V65G6 NjP MH InU NNC VtMiM NIC L. N. Tolstogo. 59-26265 t Gordeichuk, Nikolai Maksi sch see Hordiichuk, Mykola Maksymovych. ... TS1490.G65 °59-38829 t Gordeev, V A Yerpoficrso m oGcayaAusanne antoMaTMYeCcKAX TKAWKUX ctanxos, Mocxsa, W3,-s0 mayyno-texn. aut-ps:) PCOCP, 1960. 182 p._ illus. 28cm. ...
SMPTE Motion Imaging Journal
GORDEEV, M.G. GORDON, V.N. GOROKHOV, A.P. GRECHISHNIKOV, V. GRIBOV, G. GRINEVETSKY, G. INSHIN, E.|. IVANOV, V.I. IVANOV, V.P. IVANOVSKY, V.Z. KHAIMOV, V.1. KHLEBNIKOV, I.D. KHOLIN, |.A.l. ... LEW, J.M. REDOVIAN SANTA CLARA: T.F. BURCHILL, M. CHULANI, R.G. GREEN, J.C. KUO, M.D. LANDA, R.P. MANFREDO, C.A. PRESSLER, H.K. REGNIER JR, G. SADOWSKI, J. SHIKE, D.K. TAO, J.P. WADDELL, G.L. ...
Journal of the Electrochemical Society
Gordeev, G. V., “Ionization by collision at n-p junctions,” Fiz. Tverdogo Tela, 1, 851 (1959). Handbuch der Physik, ... ., “Internal field emission and lew temperature thermionic emission into vacuum,” Proc. IRE, 48, 1644 (1960) . Goetzberger, A., and W. Shockley, “Metal precipitates in silicon p-n junctions,” J. ...
SMPTE Motion Imaging Journal
LEW, R. A. MIZER, S. QUINN, R. J. STEWART, M. |. ZIEGLER SANTA CLARA: N. R. BAKER, F. W. BARLING, R. J. BEST, A. T. BOURKE, D. T. CHANG, A. EAGLE, M. A. ELSTON, R. L. GOODWIN, H. GRAY, C. A. ... FROLOV, ~ GORDEEV, M. G. GORDON, V. N. GOROKHOV, . GRECHISHNIKOV, G. GRINEVETSKY, IVANOV, V. |. IVANOV, V. P. IVANOVSKY, KHLEBNIKOV, |. D. KHOLIN, C. Z. KOCHUASHVILI, . KRIVOVIAZ, L. G. LISHIN, E. G. ...
SMPTE Motion Imaging Journal
Gordeev, M. G. Gordon, V. N. Gorokhov, A. P. Grechishnikov, G. Grinevetsky, V. I. Ivanov, E. I. Ivanov, V. P. Ivanovsky, V. I. Khlebnikov, I. D. Kholin, C. Z. Kochuashvili, V. A. Kontorovitch, V. F. ... Lew, S. S. Mirzad, R. A. Mizer, R. J. Stewart Sanger: R. C. Vaughn Santa Clara: F. W. Barling, A. J. Benedetti, R. J. Best, D. T. Chang, A. Chopra, G. W. Cintas, C. Fields, H. Gray, C. A. Hardman, H. ...
Gordeev, Istoriâ kazačestva, Veče, Moskva 2006, s. 8, 13. 36 R. ... pochodzić m.in. od rosyjskich zbiegów (władimir Broniewski 18 , siergiej sołowjow 19 , wasilij kluczewski i siergiej Płatonow 20 , Aleksandr szennikow 21 , nikołaj Mininkow 22 ), ple- mion chazarskich (Lew ...doi:10.18778/1505-9057.61.01 fatcat:anpxkgalbfhmbdcopxc23jywga
We would like very much to thank professor Lew Gordeev for the work we have done together and the inspiration to follow this alternative approach. ...arXiv:2101.00003v1 fatcat:p3na5hk4grcevpoesvjx7stli4
(with Gordeev, E. N.) Qualitative investigation of trajectory problems. (Russian. English summary) Kibernetika (Kiev) 1986, no. 5, 82-89, 105, 135. ... Lew) 88g:41028 §41A60 (42A38, 44A15, 65R10) (with Gabutti, Bruno) The numerical performance of Tricomi’s formula for inverting the Laplace transform. Numer. Math. 51 (1987), no. 3, 369-380. (P. M. ...
Negative amplitude responses to SLR at the mouth of the Delaware Bay are additionally worked out by a regression of annual M 2 changes (ΔH) against sea level changes (Δs) at station Lewes (1957 Lewes ... ., Gordeev et al., 1977; Ray, 1998 ) but non-trivial in practice; a complete treatment in forward models requires either global convolution integrals (Farrell, 1973; Stepanov & Hughes, 2004) or spherical ...doi:10.1029/2018jc013959 fatcat:3gmqwcwdunbppfhadqpfxz4hya
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