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Combinatorica 14(4), 417–433 (1994)\nSanjeev Arora & Boaz Barak (2009). Computational complexity: a modern approach Cambridge University Press.\nAlbert Atserias & Neil Thapen: The ordering principle in a fragment of approximate counting. ACM Transactions on Computational Logic 15(4), 29 (2014)\nEli Ben-Sasson & Avi Wigderson: Short proofs are narrow—resolution made simple. Journal of the ACM 48(2), 149–169 (2001)\nBuss, Samuel, Impagliazzo, Russell, Krajíček, Jan, Pudlák, Pavel, Razborov, Alexander, Sgall, Jiri: Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. Computational Complexity 6(3), 256–298 (1996)\nBuss, Samuel, Kołodziejczyk, Leszek A., Thapen, Neil: Fragments of approximate counting. Journal of Symbolic Logic 79(2), 496–525 (2014)\nBuss, Samuel, Kołodziejczyk, Leszek A., Zdanowski, Konrad: Collapsing modular counting in bounded arithmetic and constant depth propositional proofs. Transactions of the American Mathematical Society 367(11), 7517–7563 (2015)\nMario Chiari & Jan Krajíček: Witnessing functions in bounded arithmetic and search problems. Journal of Symbolic Logic 63(3), 1095–1115 (1998)\nVašek Chvátal & Endre Szemerédi: Many hard examples for resolution. Journal of the ACM 35(4), 759–768 (1988)\nStephen Cook & Robert Reckhow: The relative efficiency of propositional proof systems. Journal of Symbolic Logic 44(1), 36–50 (1979)\nRussell Impagliazzo & Jan Krajíček: A note on conservativity relations among bounded arithmetic theories. Mathematical Logic Quarterly 48(3), 375–377 (2002)\nRussell Impagliazzo, Toniann Pitassi & Alasdair Urquhart (1994). Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of LICS'94, 220–228\nJeřábek, Emil: On independence of variants of the weak pigeonhole principle. Journal of Logic and Computation 17(3), 587–604 (2007)\nBala Kalyanasundaram & Georg Schintger: The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics 5(4), 545–557 (1992)\nKrajíček, Jan: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. Journal of Symbolic Logic 62(2), 457–486 (1997)\nKrajíček, Jan: On the weak pigeonhole principle. Fundamenta Mathematicae 170(1–2), 123–140 (2001)\nJan Krajíček (2017). A feasible interpolation for random resolution. Logical Methods in Computer Science 13(1).\nJan Krajíček (2018). Randomized feasible interpolation and monotone circuits with a local oracle. Journal of Mathematical Logic 18(2).\nKrajíček, Jan, Pudlák, Pavel, Woods, Alan: An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures & Algorithms 7(1), 15–39 (1995)\nKrajíček, Jan, Skelley, Alan, Thapen, Neil: NP search problems in low fragments of bounded arithmetic. Journal of Symbolic Logic 72(2), 649–672 (2007)\nRan Raz & Avi Wigderson: Monotone circuits for matching require linear depth. Journal of the ACM 39(3), 736–744 (1992)\nAlexander Razborov (2015). Pseudorandom generators hard for k-DNF resolution and polynomial calculus resolution. Annals of Mathematics 415–472\nAlan Skelley & Neil Thapen: The provably total search problems of bounded arithmetic. Proceedings of the London Mathematical Society 103(1), 106–138 (2011)\nThapen, Neil: A tradeoff between length and width in resolution. Theory of Computing 12, 5 (2016)\nAlasdair Urquhart & Xudong Fu: Simplified lower bounds for propositional proofs. Notre Dame Journal of Formal Logic 37(4), 534–544 (1996)",{"EN":142},"We study the random resolution refutation system defined in Buss et al. (J Symb Logic 79(2):496–525, 2014). This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if \n                  \n                    \n                  \n                  $${{\\bf P} \\neq {\\bf NP}}$$\n                  \n                    \n                  \n                , then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time. We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in Buss et al. (2014). We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant-depth Frege.",{"EN":144},"Random resolution refutations",{"VOID":146},"10.1007\u002Fs00037-019-00182-7","PUBLICATION","VERIFIED","Auto Verify","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00037-019-00182-7",[152,168],{"id":153,"sortIndex":19,"researcher":18,"roles":154,"affiliations":156,"properties":165},"6fc56f22-f477-459e-b120-cc9a2c0d71ee",[155],"AUTHOR",[157],{"id":18,"sortIndex":19,"affiliation":158,"properties":18},{"id":159,"createTime":160,"updateTime":160,"relativeEntities":161,"slug":18,"properties":162,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"44b00b6b-b5c3-45ee-9fe5-fcaee92f66d7","2024-01-16T23:01:34.711+00:00",[],{"title":163},{"VI":164},"Institute of Mathematics, Czech Academy of Sciences, Praha 1, Czech Republic",{"title":166},{"VI":167},"Pavel Pudlák",{"id":169,"sortIndex":123,"researcher":18,"roles":170,"affiliations":171,"properties":177},"67a3d995-6c1a-43b6-9a70-89b3161e6ec3",[155],[172],{"id":18,"sortIndex":19,"affiliation":173,"properties":18},{"id":159,"createTime":160,"updateTime":160,"relativeEntities":174,"slug":18,"properties":175,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},[],{"title":176},{"VI":164},{"title":178},{"VI":179},"Neil Thapen","ARTICLE",{"url":150,"publisher":182,"properties":209},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":183,"slug":10,"properties":184,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":187,"manageAffiliations":188,"indexDatabases":189,"url":18,"thumbnailPath":18,"statistic":204,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":185,"title":186},{"VOID":13},{"VOID":15},[],[],[190,197],{"id":100,"indexDatabase":191,"url":113,"indexYears":114,"academicFieldIds":196,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":192,"label":193,"description":194,"key":110,"publicationTags":195,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":198,"url":95,"indexYears":18,"academicFieldIds":203,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":199,"label":200,"description":201,"key":91,"publicationTags":202,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":205,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":206,"totalCitation":19,"totalCitationByYear":207,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":208,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":210,"pages":212},{"VOID":211},"28",{"VOID":213},"185-239","2019-04-22",2019,false,{"id":218,"createTime":219,"updateTime":220,"relativeEntities":221,"slug":222,"properties":223,"entityType":147,"verifyStatus":148,"verifyTime":220,"verifyNote":149,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":123,"primaryUrl":230,"fullTextUrl":18,"authors":231,"publicationType":180,"publisherRelationship":288,"citationCount":18,"citationInfo":18,"publishDate":321,"publishYear":322,"citationAnalyzeStatus":17,"lastCitationAnalyze":18,"indexDatabases":18,"openAccess":18,"references":18,"isForceReanalyzing":216},"e1e48296-d10d-4faa-8670-0ea1e282a66c","2024-01-10T23:44:36.420+00:00","2024-12-10T23:50:45.373+00:00",[],"On-interactive-proofs-with-a-laconic-prover",{"abstract":224,"title":226,"doi":228},{"EN":225},"We continue the investigation of interactive proofs with\nbounded communication, as initiated by Goldreich & Håstad (1998).\nLet L be a language that has an interactive proof in which the prover\nsends few (say b) bits to the verifier. We prove that the complement\n$\\bar L$ has a constant-round interactive proof of complexity that depends\nonly exponentially on b. This provides the first evidence that for NP-complete\nlanguages, we cannot expect interactive provers to be much\nmore “laconic” than the standard NP proof. When the proof system is\nfurther restricted (e.g., when b = 1, or when we have perfect completeness),\nwe get significantly better upper bounds on the complexity of $\\bar L$.",{"EN":227},"On interactive proofs with a laconic prover",{"VOID":229},"10.1007\u002Fs00037-002-0169-0","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00037-002-0169-0",[232,252,269],{"id":233,"sortIndex":234,"researcher":18,"roles":235,"affiliations":236,"properties":249},"41077b28-a670-4294-ae9c-19a0a3e343dd",2,[155],[237],{"id":238,"sortIndex":19,"affiliation":239,"properties":246},"d795eca4-467a-47f7-89a1-ea651b256fb1",{"id":240,"createTime":241,"updateTime":241,"relativeEntities":242,"slug":18,"properties":243,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"dbe7a303-d35f-4d8e-8474-0af814718b19","2023-12-31T12:19:06.034+00:00",[],{"title":244},{"VI":245},"Institute for Advanced Study, Princeton, U.S.A",{"title":247},{"VI":248},"Institute for Advanced Study, Princeton, USA",{"title":250},{"VI":251},"Avi Wigderson",{"id":253,"sortIndex":19,"researcher":18,"roles":254,"affiliations":255,"properties":266},"f7faae33-0c78-4fb8-9770-70244e4a64a0",[155],[256],{"id":18,"sortIndex":19,"affiliation":257,"properties":18},{"id":258,"createTime":259,"updateTime":260,"relativeEntities":261,"slug":262,"properties":263,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"9b6efb55-74e2-4ab2-8240-63e88610e28c","2024-01-13T19:27:56.712+00:00","2024-10-11T19:08:45.418+00:00",[],"Department-of-Computer-Science-and-Applied-Mathematics-Weizmann-Institute-of-Science-Rehovot-Israel",{"title":264},{"VI":265},"Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel",{"title":267},{"VI":268},"Oded Goldreich",{"id":270,"sortIndex":123,"researcher":18,"roles":271,"affiliations":272,"properties":285},"7380f5f5-cace-445e-ad3b-50dbfd82d48b",[155],[273],{"id":274,"sortIndex":19,"affiliation":275,"properties":282},"b199f78c-fe36-4fa2-9dee-05d695d032fd",{"id":276,"createTime":277,"updateTime":277,"relativeEntities":278,"slug":18,"properties":279,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"fe131f3a-eb40-485d-ae96-1581b482ac7c","2024-01-09T14:13:36.501+00:00",[],{"title":280},{"VI":281},"Division of Engineering and Applied Sciences, Harvard University, Cambridge, U.S.A.",{"title":283},{"VI":284},"Division of Engineering and Applied Sciences, Harvard University, Cambridge, USA",{"title":286},{"VI":287},"Salil Vadhan",{"url":230,"publisher":289,"properties":316},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":290,"slug":10,"properties":291,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":294,"manageAffiliations":295,"indexDatabases":296,"url":18,"thumbnailPath":18,"statistic":311,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":292,"title":293},{"VOID":13},{"VOID":15},[],[],[297,304],{"id":100,"indexDatabase":298,"url":113,"indexYears":114,"academicFieldIds":303,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":299,"label":300,"description":301,"key":110,"publicationTags":302,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":305,"url":95,"indexYears":18,"academicFieldIds":310,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":306,"label":307,"description":308,"key":91,"publicationTags":309,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":312,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":313,"totalCitation":19,"totalCitationByYear":314,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":315,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":317,"pages":319},{"VOID":318},"11",{"VOID":320},"1-53","2002-06-01",2002,{"id":324,"createTime":325,"updateTime":326,"relativeEntities":327,"slug":328,"properties":329,"entityType":147,"verifyStatus":148,"verifyTime":326,"verifyNote":149,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":338,"fullTextUrl":18,"authors":339,"publicationType":180,"publisherRelationship":405,"citationCount":18,"citationInfo":18,"publishDate":438,"publishYear":439,"citationAnalyzeStatus":17,"lastCitationAnalyze":18,"indexDatabases":18,"openAccess":18,"references":18,"isForceReanalyzing":216},"bfaf335d-638a-4319-a773-245b92a64300","2024-01-18T15:44:35.237+00:00","2025-02-03T23:46:14.925+00:00",[],"Two-tapes-versus-one-for-off-line-Turing-machines",{"references":330,"abstract":332,"title":334,"doi":336},{"VOID":331},"N. Alon andW. Maass, Meanders and their application in lower bound arguments.J. Comput. System Sci. 37 (1988), 118–129.\nL. Babai, P. Pudlák, V. Rödl, andE. Szemerédi, Lower bounds to the complexity of symmetric boolean functions.Theoret. Comput. Sci. 74 (1990), 313–323.\nM. Dietzfelbinger, The speed of copying on one-tape off-line turing machines.Inform. Process. Lett. 33 (1989), 83–89.\nM. Dietzfelbinger and W. Maass, The complexity of matrix transposition on one-tape off-line turing machines with output tape, 1986. To appear inTheoret. Comput. Sci.\nP. Duris, Z. Galil, W. J. Paul, and R. Reischuk, Two nonlinear bounds. InProc. Fifteenth Ann. ACM Symp. Theor. Comput., 1983, 127–132.\nF. C. Hennie, One-tape off-line turing machine computations.Inform. and Control 8 (1965), 553–578.\nJ. E. Hopcroft and J. D. Ullman,Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979.\nM. Li andP. M. B. Vitanyi, Tape versus quene and stacks: the lower bounds.Inform. and Comput. 78 (1988), 56–85.\nM. Li, L. Longpre, andP. M. B. Vitanyi, The power of the queue.Proc. of Structure in Complexity Theory, Lecture Notes in Computer Science 223 (1986), 219–233.\nR. J. Lipton andR. E. Tarjan, A separator theorem for planar graphs.SIAM J. Appl. Math. 36 (1979), 177–189.\nW. Maass, Combinatorial lower bound arguments for deterministic and nondeterministic turing machines.Trans. Amer. Math. Soc. 292 (1985), 675–693.\nW. Maass andG. Schnitger, An optimal lower bound for turing machines with one work tape and a two-way input tape.Proc. of Structure in Complexity Theory, Lecture Notes in Computer Science 223 (1986), 249–264.\nW. Maass, G. Schnitger, and E.Szemerédi, Two tapes are better than one for off-line turing machines. InProc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 94–100.\nW. J. Paul, N. Pippenger, E. Szemerédi, and W. Trotter, On determinism versus nondeterminism and related problems. InProc. 24th Ann. Symp. Found. Comput. Sci., 1983, 429–438.\nM. O. Rabin, Real time computation.Israel J. of Math. 1 (1963), 203–211.\nJ.E. Savage, The performance of multilective vlsi algorithms.J. Comput. System Sci. 29 (1984), 243–272.",{"EN":333},"We prove the first superlinear lower bound for a concrete, polynomial time recognizable decision problem on a Turing machine with one work tape and a two-way input tape (also called off-line 1-tape Turing machine). In particular, for off-line Turing machines we show that two tapes are better than one and that three pushdown stores are better than two (both in the deterministic and in the nondeterministic case).",{"EN":335},"Two tapes versus one for off-line Turing machines",{"VOID":337},"10.1007\u002FBF01275490","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01275490",[340,357,372,387],{"id":341,"sortIndex":234,"researcher":18,"roles":342,"affiliations":343,"properties":354},"9b78acaa-3ae5-4f5d-95c5-e895ea73884c",[155],[344],{"id":18,"sortIndex":19,"affiliation":345,"properties":18},{"id":346,"createTime":347,"updateTime":348,"relativeEntities":349,"slug":350,"properties":351,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"097dbd6b-a90f-40bb-93b0-cffdd5211dcd","2024-04-17T09:03:02.422+00:00","2025-06-11T22:26:04.719+00:00",[],"-Department-of-Computer-Science-Rutgers-University-New-Brunswick-USA-",{"title":352},{"EN":353},"[Department of Computer Science, Rutgers University, New Brunswick, USA]",{"title":355},{"VI":356},"Endre Szemerédi",{"id":358,"sortIndex":19,"researcher":18,"roles":359,"affiliations":360,"properties":369},"294a8ba0-42d4-4467-8d51-48e4fb220afe",[155],[361],{"id":18,"sortIndex":19,"affiliation":362,"properties":18},{"id":363,"createTime":364,"updateTime":364,"relativeEntities":365,"slug":18,"properties":366,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"c1cb8cdf-a52a-40ca-b59e-2f3aa005c44b","2024-01-09T22:06:20.856+00:00",[],{"title":367},{"VI":368},"Institute for Theoretical Computer Science, Technische Universität Graz, Graz, Austria",{"title":370},{"VI":371},"Wolfgang Maass",{"id":373,"sortIndex":123,"researcher":18,"roles":374,"affiliations":375,"properties":384},"05b5d229-a3e7-43b3-9931-8cad6da0459a",[155],[376],{"id":18,"sortIndex":19,"affiliation":377,"properties":18},{"id":378,"createTime":379,"updateTime":379,"relativeEntities":380,"slug":18,"properties":381,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"84aeddd9-aafb-4ee3-b0ec-160f407845b5","2024-01-18T15:44:35.269+00:00",[],{"title":382},{"VI":383},"FB 17, Mathematik-Informatik, Universität Paderborn, Paderborn, Germany",{"title":385},{"VI":386},"Georg Schnitger",{"id":388,"sortIndex":389,"researcher":18,"roles":390,"affiliations":391,"properties":402},"63ea0fdc-71d2-4f13-ba89-f040f0f8782b",3,[155],[392],{"id":18,"sortIndex":19,"affiliation":393,"properties":18},{"id":394,"createTime":395,"updateTime":396,"relativeEntities":397,"slug":398,"properties":399,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"691c731f-8008-411c-b3a3-558efbf28de3","2024-02-05T16:42:24.040+00:00","2024-10-15T06:34:37.872+00:00",[],"Department-of-Mathematics-Statistics-and-Computer-Science-University-of-Illinois-at-Chicago-Chicago-USA",{"title":400},{"VI":401},"Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA",{"title":403},{"VI":404},"György Turán",{"url":338,"publisher":406,"properties":433},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":407,"slug":10,"properties":408,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":411,"manageAffiliations":412,"indexDatabases":413,"url":18,"thumbnailPath":18,"statistic":428,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":409,"title":410},{"VOID":13},{"VOID":15},[],[],[414,421],{"id":100,"indexDatabase":415,"url":113,"indexYears":114,"academicFieldIds":420,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":416,"label":417,"description":418,"key":110,"publicationTags":419,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":422,"url":95,"indexYears":18,"academicFieldIds":427,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":423,"label":424,"description":425,"key":91,"publicationTags":426,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":429,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":430,"totalCitation":19,"totalCitationByYear":431,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":432,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":434,"pages":436},{"VOID":435},"3",{"VOID":437},"392-401","1993-12-01",1993,{"id":441,"createTime":442,"updateTime":443,"relativeEntities":444,"slug":445,"properties":446,"entityType":147,"verifyStatus":17,"verifyTime":443,"verifyNote":455,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":456,"fullTextUrl":18,"authors":457,"publicationType":180,"publisherRelationship":510,"citationCount":18,"citationInfo":18,"publishDate":543,"publishYear":544,"citationAnalyzeStatus":17,"lastCitationAnalyze":18,"indexDatabases":18,"openAccess":18,"references":18,"isForceReanalyzing":216},"474af352-17ee-443f-88ca-8a9b9cf4fd3b","2023-12-11T22:07:59.002+00:00","2024-12-14T23:44:01.515+00:00",[],"A-lower-bound-for-randomized-algebraic-decision-trees",{"references":447,"abstract":449,"title":451,"doi":453},{"VOID":448},"M. Ben-Or,Lower Bounds for Algebraic Computation Trees, Proc. 15th ACM STOC (1983), pp. 80–86.\nC.H. Bennett andJ. Gill,Relative to a Random Oracle A, P A ≠ NP A ≠ co-NP A with Probability 1, SIAM J. Comput.10 (1981), pp. 96–113.\nA. Björner, L. Lovász and A. Yao,Linear Decision Trees: Volume Estimates and Topological Bounds, Proc. 24th ACM STOC (1992), pp. 170–177.\nP. Bürgisser, M. Karpinski andT. Lickteig,On Randomized Algebraic Test Complexity. J. of Complexity9 (1993), pp. 231–251.\nD.P. Dobkin andR.J. Lipton,A Lower Bound of 1\u002F2n 2 on Linear Search Programms for the Knapsack Problem, J. Compt. Syst. Sci.16 (1978), pp. 413–417.\nH. Edelsbrunner,Algorithms in Computational Geometry, Springer, 1987.\nD. Grigoriev andM. Karpinski,Lower Bounds on Complexity of Testing Membership to a Polygon for Algebraic and Randomized Computation Trees, Technical Report TR-93-042, International Computer Science Institute, Berkeley, 1993.\nD. Grigoriev and M. Karpinski,Lower Bound for Randomized Linear Decision Tree Recognizing a Union of Hyperplanes in a Generic Position, Research Report No. 85114-CS, University of Bonn, 1994.\nD. Grigoriev, M. Karpinski, F. Meyer Auf Der Heide and R. Smolensky,A Lower Bound for Randomized Algebraic Decision Trees, Proc. ACM STOC (1996), pp. 612–619.\nD. Grigoriev, M. Karpinski andN. Vorobjov,Lower Bound on Testing Membership to a Polyhedron by Algebraic Decision Trees, Discrete Comput. Geom. 17 (1997), pp. 191–215.\nD. Grigoriev andN. Vorobjov,Solving Systems of Polynomial Inequalities in Subexponential Time, Journal of Symbolic Comp.5 (1988), pp. 37–64.\nS. Lang,Algebra, Addison-Wesley, New York, 1984.\nP. McMullen andG. Shephard,Convex Polytopes and the Upper Bound Conjecture, Cambridge University Press, Cambridge (1971).\nS. Meiser,Point Location in Arrangements of Hyperplanes, Information and Computation106 (1993), pp. 286–303.\nF. Meyer Auf Der Heide,A Polynomial Linear Search Algorithm for the n-Dimensional Knapsack Problem, J. ACM31 (1984), pp. 668–676.\nF. Meyer auf der Heide,Nondeterministic versus Probabilistic Linear Search Algorithms, Proc. IEEE FOCS (1985a), pp. 65–73.\nF. Meyer auf der Heide,Lower Bounds for Solving Linear Diophantic Equations on Random Access Machines, J. ACM32 (1985b), pp. 929–937.\nF. Meyer auf der Heide,Simulating Probabilistic by Deterministic Algebraic Computation Trees, Theoretical Computer Science41 (1985c), pp. 325–330.\nU. Manber andM. Tompa,Probabilistic, Nondeterministic and Alternating Decision Trees, J. ACM, Vol. 32 (1985), pp. 720–732.\nM. Snir,Lower Bounds for Probabilistic Linear Decision Trees, Theor. Comput. Sci., Vol. 38 (1985), pp. 69–82.\nJ. M. Steele andA. C. Yao,Lower Bounds for Algebraic Decision Trees, J. of Algorithms3 (1982), pp. 1–8.\nA. Tarski,A Decision Method for Elementary Algebra and Geometry, University of California Press, 1951.\nA. Yao,A Lower Bound to Finding Convex Hulls, J. ACM28 (1981), pp. 780–787.\nA. Yao,Algebraic Decision Trees and Euler Characteristics, Proc. 33rd IEEE FOCS (1992), pp. 268–277.\nA. Yao,Decision Tree Complexity and Betti Numbers, Proc. 26th ACM STOC (1994), pp. 615–624.",{"EN":450},"We prove the firstnontrivial (andsuperlinear) lower bounds on the depth ofrandomized algebraic decision trees (with two-sided error) for problems being finite unions of hyperplanes and intersections of halfspaces, solving a long standing open problem. As an application, among other things, we derive, for the first time, an Ω(n\n2)randomized lower bound for theKnapsack Problem, and an Ω(n logn)randomized lower bound for theElement Distinctness Problem which were previously known only for deterministic algebraic decision trees. It is worth noting that for the languages being finite unions of hyperplanes our proof method yields also a new elementary lower bound technique for deterministic algebraic decision trees without making use of Milnor's bound on Betti number of algebraic varieties.",{"EN":452},"A lower bound for randomized algebraic decision trees",{"VOID":454},"10.1007\u002FBF01270387","Author affiliation is blank","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF01270387",[458,465,480,495],{"id":459,"sortIndex":389,"researcher":18,"roles":460,"affiliations":461,"properties":462},"03b69249-c0c4-4cd4-b42f-7add74062c3f",[155],[],{"title":463},{"VI":464},"Roman Smolensky",{"id":466,"sortIndex":19,"researcher":18,"roles":467,"affiliations":468,"properties":477},"9b8bec0b-4533-4d2a-a552-62c965e171d6",[155],[469],{"id":18,"sortIndex":19,"affiliation":470,"properties":18},{"id":471,"createTime":472,"updateTime":472,"relativeEntities":473,"slug":18,"properties":474,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"9cd7e253-ae49-4950-9b79-f2da546585d8","2023-12-11T22:07:50.384+00:00",[],{"title":475},{"VI":476},"Dept. of Computer Science and Mathematics, Penn State University, University Park",{"title":478},{"VI":479},"Dima Grigoriev",{"id":481,"sortIndex":234,"researcher":18,"roles":482,"affiliations":483,"properties":492},"a2519119-3527-4b92-9bc0-b153465fbd02",[155],[484],{"id":18,"sortIndex":19,"affiliation":485,"properties":18},{"id":486,"createTime":487,"updateTime":487,"relativeEntities":488,"slug":18,"properties":489,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"39cba1ca-a3bc-43fd-b2cf-791d7b2fe3f8","2023-12-11T22:07:59.051+00:00",[],{"title":490},{"VI":491},"Heinz Nixdorf Institute and Computer Science Department, University of Paderborn, Paderborn",{"title":493},{"VI":494},"Friedhelm Meyer auf der Heide",{"id":496,"sortIndex":123,"researcher":18,"roles":497,"affiliations":498,"properties":507},"6b1853af-10b2-42ea-b0cc-67bd3af88aa2",[155],[499],{"id":18,"sortIndex":19,"affiliation":500,"properties":18},{"id":501,"createTime":502,"updateTime":502,"relativeEntities":503,"slug":18,"properties":504,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"90a16e82-b970-4ece-945d-590b44614084","2023-12-21T07:10:27.195+00:00",[],{"title":505},{"VI":506},"Dept. of Computer Science, University of Bonn, Bonn",{"title":508},{"VI":509},"Marek Karpinski",{"url":456,"publisher":511,"properties":538},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":512,"slug":10,"properties":513,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":516,"manageAffiliations":517,"indexDatabases":518,"url":18,"thumbnailPath":18,"statistic":533,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":514,"title":515},{"VOID":13},{"VOID":15},[],[],[519,526],{"id":100,"indexDatabase":520,"url":113,"indexYears":114,"academicFieldIds":525,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":521,"label":522,"description":523,"key":110,"publicationTags":524,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":527,"url":95,"indexYears":18,"academicFieldIds":532,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":528,"label":529,"description":530,"key":91,"publicationTags":531,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":534,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":535,"totalCitation":19,"totalCitationByYear":536,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":537,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":539,"pages":541},{"VOID":540},"6",{"VOID":542},"357-375","1996-12-01",1996,{"id":546,"createTime":547,"updateTime":548,"relativeEntities":549,"slug":550,"properties":551,"entityType":147,"verifyStatus":148,"verifyTime":548,"verifyNote":149,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":560,"fullTextUrl":18,"authors":561,"publicationType":180,"publisherRelationship":581,"citationCount":18,"citationInfo":18,"publishDate":614,"publishYear":615,"citationAnalyzeStatus":17,"lastCitationAnalyze":18,"indexDatabases":18,"openAccess":18,"references":18,"isForceReanalyzing":216},"46145640-e1c7-4e61-950b-64217aaf9f58","2023-12-06T12:24:09.394+00:00","2025-01-03T23:42:31.947+00:00",[],"Unifying-Known-Lower-Bounds-via-Geometric-Complexity-Theory",{"references":552,"abstract":554,"title":556,"doi":558},{"VOID":553},"Scott Aaronson & Andrew Drucker (2009). Impagliazzo’s worlds in arithmetic complexity. Talk presented at the Workshop on Complexity and Cryptography: Status of Impagliazzo’s Worlds, Center for Computational Intractability, Princeton, NJ, June 5, 2009. Slides available at http:\u002F\u002Fwww.scottaaronson.com\u002Ftalks\u002Farith.ppt.\nB. Adsul, Milind Sohoni & K. V. Subrahmanyam (2009). Quantum deformations of the restriction of \\({GL_{mn}(\\mathbb{C})}\\)-modules to \\({GL_m(\\mathbb{C}) \\times GL_n(\\mathbb{C})}\\). arXiv:0905.0094 [math.RT].\nManindra Agrawal (2005). Proving lower bounds via pseudo-random generators. In FSTTCS 2005: Foundations of software technology and theoretical computer science, volume 3821 of Lecture Notes in Computer Science, 92–105. Springer, Berlin.\nManindra Agrawal & V. Vinay (2008). Arithmetic Circuits: A Chasm at Depth Four. In FOCS ’08: 49th, Annual IEEE Symposium on Foundations of Computer Science 67–75. IEEE Computer Society.\nEric Allender (1999). 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Programming 5, 255–266.",{"EN":555},"We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan–Wigderson), the results of Razborov and Smolensky on AC\n                0[p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur–Strassen), the connected components technique (Ben-Or, Steele–Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev–Karpinski), lower bounds on permanent versus determinant (Mignon–Ressayre, Landsberg–Manivel–Ressayre), lower bounds on matrix multiplication (Bürgisser–Ikenmeyer, Landsberg–Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta–Kayal–Kamath–Saptharishi, Agrawal–Vinay, Koiran, Tavenas), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.",{"EN":557},"Unifying Known Lower Bounds via Geometric Complexity Theory",{"VOID":559},"10.1007\u002Fs00037-015-0103-x","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00037-015-0103-x",[562],{"id":563,"sortIndex":19,"researcher":18,"roles":564,"affiliations":565,"properties":578},"38e13407-253e-4040-a77a-c21c7d69a0cb",[155],[566],{"id":567,"sortIndex":19,"affiliation":568,"properties":575},"5fca5cd0-4c0b-4c75-9d7e-d8ea0d35a155",{"id":569,"createTime":570,"updateTime":570,"relativeEntities":571,"slug":18,"properties":572,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"280a6730-11f6-4dce-a9b4-91131eaec99d","2023-12-08T09:29:11.906+00:00",[],{"title":573},{"VI":574},"Santa Fe Institute, Santa Fe, U.S.A.",{"title":576},{"VI":577},"Santa Fe Institute, Santa Fe, USA",{"title":579},{"VI":580},"Joshua A. Grochow",{"url":560,"publisher":582,"properties":609},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":583,"slug":10,"properties":584,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":587,"manageAffiliations":588,"indexDatabases":589,"url":18,"thumbnailPath":18,"statistic":604,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":585,"title":586},{"VOID":13},{"VOID":15},[],[],[590,597],{"id":100,"indexDatabase":591,"url":113,"indexYears":114,"academicFieldIds":596,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":592,"label":593,"description":594,"key":110,"publicationTags":595,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":598,"url":95,"indexYears":18,"academicFieldIds":603,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":599,"label":600,"description":601,"key":91,"publicationTags":602,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":605,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":606,"totalCitation":19,"totalCitationByYear":607,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":608,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":610,"pages":612},{"VOID":611},"24",{"VOID":613},"393-475","2015-05-07",2015,{"id":617,"createTime":618,"updateTime":619,"relativeEntities":620,"slug":621,"properties":622,"entityType":147,"verifyStatus":148,"verifyTime":633,"verifyNote":149,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":634,"fullTextUrl":18,"authors":635,"publicationType":180,"publisherRelationship":758,"citationCount":18,"citationInfo":18,"publishDate":791,"publishYear":792,"citationAnalyzeStatus":793,"lastCitationAnalyze":794,"indexDatabases":18,"openAccess":18,"references":18,"isForceReanalyzing":216},"ec7e0bbf-8104-4e59-8089-5bd5561b41e3","2024-01-08T06:04:12.223+00:00","2025-10-26T23:40:05.839+00:00",[],"Quantum-generalizations-of-the-polynomial-hierarchy-with-applications-to-QMA-2-",{"references":623,"abstract":625,"title":627,"doi":629,"gsPaper":631},{"VOID":624},"S. 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Yamakami (2002). Quantum NP and a quantum hierarchy. In Foundations of Information Technology in the Era of Network and Mobile Computing (TCS 2002), R. Baeza-Yates, U. Montanari & N. Santoro, editors, 323–336. Springer, Boston, MA.",{"EN":626},"The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, \n                \n                  \n                \n                $$\\rm{QCPH}$$\n                \n              , uses classical proofs, and the second, \n                \n                  \n                \n                $$\\rm{QPH}$$\n                \n              , uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, \n                \n                  \n                \n                $$\\rm{Q\\Sigma_3}$$\n                \n              , into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for \n                \n                  \n                \n                $$\\rm{QMA(2)}$$\n                \n              , the variant of Quantum Merlin-Arthur (\n                \n                  \n                \n                $$\\rm{QMA}$$\n                \n              ) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if \n                \n                  \n                \n                $$\\rm{QCPH = QPH}$$\n                \n               (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in \n                \n                  \n                \n                $${\\rm P}^{{\\rm pp}^{{\\rm pp}}}$$\n                \n              ). Second, because \n                \n                  \n                \n                $$\\rm{QMA(2)}\\subseteq \\rm{Q\\Sigma_3}$$\n                \n              , \n                \n                  \n                \n                $$\\rm{QMA(2)}$$\n                \n               is strictly contained in NEXP unless \n                \n                  \n                \n                $$\\rm{QMA(2)}=\\rm{Q\\Sigma_3}$$\n                \n               (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').",{"EN":628},"Quantum generalizations of the polynomial hierarchy with applications to 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Allender, A note on the power of threshold circuts, inProceedings of the 30th IEEE Symposium on Foundations of Computer Science, 1989, 580–584.\nJ. Bruck, Harmonic analysis of polynomial threshold functions,SIAM Journal on Discrete Mathematics 3(2) (1990), 168–177.\nJ. Bruck and R. Smolensky, Polynomial threshold functions,AC 0 functions and spectral norms, inProceedings of the 31st IEEE Symposium on Foundations of Computer Science, 1990, 632–641.\nY. Freund, Boosting a weak learning algorithm by majority, inWorkshop on Computational Learning Theory, Morgan Kaufmann, 1990, 202–216.\nM. Goldmann and M. Karpinski, Constructing depthd+1 majority circuits that simulate depthd threshold circuits, manuscript, 1992.\nA. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán, Threshold circuits of bounded depth, inProceedings of 28 IEEE Symposium on Foundations of Computer Science, 1987, 99–110.\nJ. Håstad and M. Goldmann, On the power of small-depth threshold circuits, inProceedings of the 31st IEEE Symposium on Foundations of Computer Science, 1990, 610–618.\nJ. Hertz, R. Krogh, and A. Palmer,An Introduction to the Theory of Neural Computation, Addison-Wesley, 1991.\nM. Hofri,Probabilistic Analysis of Algorithms, Springer-Verlag, 1987.\nM. Krause, Geometric arguments yield better bounds for threshold circuits and distributed computing, inProceedings of the 6th Structure in Complexity Theory Conference, 1991, 314–322.\nM. Krause and S. Waack, Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fanin, inProceedings of the 32nd IEEE Symposium on Foundations of Computer Science, 1991, 777–782.\nW. Maass, G. Schnitger, and E. Sontag, On the computational power of sigmoid versus boolean threshold circuits, inProceedings of the 32nd IEEE Symposium on Foundations of Computer Science, 1991, 767–776.\nS. Muroga,Threshold logic and its applications, Wiley-Interscience, 1971.\nJ. Myhill andW. H. Kautz, On the size of weights required for linear-input switching functions,IRE Trans. on Electronic Computers EC10(2) (1961), 288–290.\nG. Owen,Game theory, Academic press, second edition, 1982.\nK.-Y. Siu andJ. Bruck, On the power of threshold circuits with small weights,SIAM Journal on Discrete Mathematics 4(3) (1991) 423–435.\nK.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems, manuscript, 1992.\nS. Toda, On the computational power ofPP and ⊗P, inProceedings of the 30th IEEE Symposium on Foundations of Computer Science, 1989, 514–519.\nA. Yao, Some complexity questions related to distributive computing, inProceedings of the 11th ACM Symposium on Theory of Computing, 1979, 209–213.\nA. Yao, Lower bounds by probabilistic arguments, inProceedings of the 24th IEEE Symposium on Foundations of Computer Science, 1983, 420–428.\nA. Yao, Circuits and local computation, inProceedings of the 21st ACM Symposium on Theory of Computing, 1989, 186–196.\nA. Yao, OnACC and threshold circuits, inProceedings of the 31st IEEE Symposium on Foundations of Computer Science, 1990, 619–627.",{"EN":805},"In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following:\n                  \n                    \n                    \n                  \n                  \n                    \n                    \n                  \n                  \n                    \n                    \n                  \n                \n",{"EN":807},"Majority gates vs. general weighted threshold 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Babai andS. Moran, Arthur-Merlin games: a randomized proof-system, and a hierarchy of complexity classes.J. Comput. System. Sci. 36 (1988), 254–276.\nJ. L. Balcázar, J. Díaz andJ. Gabarró,Structural Complexity I. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.\nE. Börger,Computability, Complexity, Logic. North-Holland, Amsterdam, New York, Oxford, Tokyo, 1989.\nB. von Braunmühl, Alternationshierarchien von Turingmaschinen mit kleinem Speicher. Informatik Berichte 83, Institut für Informatik, Universität Bonn, 1991.\nJ. H. Chang, O. H. Ibarra, B. Ravikumar, andL. Berman Some observations concerning Turing machines using small space.Inform. Process. Lett. 25 (1987), 1–9. Erratum,Inform. Process. Lett. 25 (1988) 53.\nP. van Emde Boas, Machine models and simulations. InHandbook of Theoretical Computer Science, Volume A, Algorithms and Complexity ed.J. van Leeuwen, 1–66. Elsevier, Amsterdam, New York, Oxford, Tokyo, 1990.\nR. Freivalds, On time complexity of deterministic and nondeterministic Turing machines.Latvijski Mathematičeskij Eshegodnik 23 (1979), 158–165. In Russian.\nM. R. Garey andD. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Fransisco, 1979.\nV. Geffert, Nondeterministic computations in sublogarithmic space and space constructibility. InProc. 17 ICALP, Lecture Notes in Computer Science 443, 1990 111–124.\nV. Geffert, Tally versions of the Savitch and the Immerman-Szelepcsényi theorems for sublogarithmic space.SIAM J. Comput. 22 (1993), 102–113.\nL. A. Hemachandra, The strong exponential hierarchy collapses. InProc. Nineteenth Ann. ACM Symp. Theor. Comput, 1987, 110–122.\nN. Immerman, NSPACE is closed under complement.SIAM J. Comput. 17 (1988), 935–938.\nA. Ito, K. Inoue, andI. Takanami, A note on alternating Turing machines using small space.The Transactions of the IEICE E 70 no. 10 (1987), 990–996.\nK. Iwama, ASPACE(o(log log)) is regular. Research report KSU\u002FICS Kyoto Sangyo University, Kyoto, 603, Japan, 1986. See alsoSIAM J. Comput 22 (1993), 136–146.\nM. Kutylowski, M. Liśkiewicz, andK. Loryś, Reversal complexity classes for alternating Turing machines.SIAM J. Comput. 19 (1990), 207–221.\nP. M. Lewis, R. E. Stearns, and J. Hartmanis, Memory bounds for recognition of context-free and context-sensitive languages. InIEEE. Conf. Switch. Circuit Theory and Logic Design, 1965, 191–202.\nM. Liśkiewicz and R. Reischuk, Separating the lower levels of the sublogarithmic space hierarchy. InProc. 10.STACS, Lecture Notes in Computer Science 665, 1993a, 16–28.\nM. Liśkiewicz and R. Reischuk, The sublogarithmic space hierarchy is infinite. Technical report, Institut für theoretische Informatik, TH Darmstadt, 1993b.\nM. Liśkiewicz, K. Loryś, andM. Piotrów On reversal bounded alternating Turing machines.Theoret. Comput. Sci. 54 (1987), 331–339.\nW. J. Paul,Komplexitätstheorie. Teubner Studienbücher, Informatik, Stuttgart, 1978.\nU. Schöning and K. W. Wagner, Collapsing oracle hierarchies, census functions and logarithmically many queries. InProc. 5th. STACS 88, Lecture Notes in Computer Science 294, 1988, 91–97.\nM. Sipser, Halting bounded computations.,Theoret. Comput. Sci. 10 (1980), 335–338.\nM. Sipser, Borel sets and circuit complexity. InProc. Fifteenth Ann. ACM Symp. Theor. Comput., 1983, 330–335.\nR. Szelepcsényi, The method of forced enumeration for nondeterministic automata.Acta Infor. 26 (1988), 279–284.\nK. W. Wagner, The alternation hierarchy for sublogarithmic space: an exciting race to STACS'93 (Editorial note). InProc. 10 STACS, Lecture Notes in Computer Science 665, 1993, 2–4.\nK. Wagner andG. Wechsung,Computational Complexity. D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1986.",{"EN":1081},"The alternation hierarchy for Turing machines with a space bound between loglog and log is infinite. That applies to all common concepts, especially a) to two-way machines with weak space-bounds, b) to two-way machines with strong space-bounds, and c) to one-way machines with weak space-bounds. In all of these cases the ∑\n                  k\n                -and II\n                  k\n                -classes are not comparable fork>-2. Furthermore the ∑\n                  k\n                -classes are not closed under intersection and the II\n                  k\n                -classes are not closed under union. Thus these classes are not closed under complementation. The hierarchy results also apply to classes determined by an alternation depth which is a function depending on the input rather than on a constant.",{"EN":1083},"The alternation hierarchy for sublogarithmic space is infinite",{"VOID":1085},"10.1007\u002FBF01271368","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01271368",[1088,1103,1115],{"id":1089,"sortIndex":123,"researcher":18,"roles":1090,"affiliations":1091,"properties":1100},"1eb70b31-9817-46c9-b9a4-3e3b3d39be5c",[155],[1092],{"id":18,"sortIndex":19,"affiliation":1093,"properties":18},{"id":1094,"createTime":1095,"updateTime":1095,"relativeEntities":1096,"slug":18,"properties":1097,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},"dafa94fd-cf2e-458b-8462-f940afad1fef","2023-12-27T11:20:41.373+00:00",[],{"title":1098},{"VI":1099},"Institut für Informatik I, Universität Bonn, Bonn, Germany",{"title":1101},{"VI":1102},"Romain Gengler",{"id":1104,"sortIndex":234,"researcher":18,"roles":1105,"affiliations":1106,"properties":1112},"a0a4b59b-7600-4096-849d-7553014e0a21",[155],[1107],{"id":18,"sortIndex":19,"affiliation":1108,"properties":18},{"id":1094,"createTime":1095,"updateTime":1095,"relativeEntities":1109,"slug":18,"properties":1110,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},[],{"title":1111},{"VI":1099},{"title":1113},{"VI":1114},"Robert Rettinger",{"id":1116,"sortIndex":19,"researcher":18,"roles":1117,"affiliations":1118,"properties":1124},"27fedb1a-8742-455c-99fc-4772a44f2c6b",[155],[1119],{"id":18,"sortIndex":19,"affiliation":1120,"properties":18},{"id":1094,"createTime":1095,"updateTime":1095,"relativeEntities":1121,"slug":18,"properties":1122,"entityType":63,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19},[],{"title":1123},{"VI":1099},{"title":1125},{"VI":1126},"Burchard von Braunmühl",{"url":1086,"publisher":1128,"properties":1155},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1129,"slug":10,"properties":1130,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"syncStatus":17,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":1133,"manageAffiliations":1134,"indexDatabases":1135,"url":18,"thumbnailPath":18,"statistic":1150,"gsStatistic":18,"type":127,"analyzePriority":18},[],{"issn":1131,"title":1132},{"VOID":13},{"VOID":15},[],[],[1136,1143],{"id":100,"indexDatabase":1137,"url":113,"indexYears":114,"academicFieldIds":1142,"indexDatabaseRanking":120},{"id":102,"createTime":103,"updateTime":104,"relativeEntities":1138,"label":1139,"description":1140,"key":110,"publicationTags":1141,"standard":18},[],{"EN":107,"VI":107},{"EN":107,"VI":109},[112],[116,117,118,119],{"id":80,"indexDatabase":1144,"url":95,"indexYears":18,"academicFieldIds":1149,"indexDatabaseRanking":18},{"id":82,"createTime":83,"updateTime":84,"relativeEntities":1145,"label":1146,"description":1147,"key":91,"publicationTags":1148,"standard":18},[],{"EN":87,"VI":87},{"VI":89,"EN":90},[93,94],[97,98],{"impactFactor":19,"impactFactorByYear":1151,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":123,"totalPublicationByYear":1152,"totalCitation":19,"totalCitationByYear":1153,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":1154,"hindexLast5Year":19,"hindex":19},{},{"2008":123},{},{},{"volume":1156,"pages":1157},{"VOID":435},{"VOID":1158},"207-230","1993-09-01"]