TY - GEN A1 - Baartse, Martijn A1 - Barmpalias, George T1 - On the gap between trivial and nontrivial initial segment prefix-free complexity T2 - Theory of Computing Systems N2 - An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 n ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that (⋆) for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a $\varDelta^{0}_{3}$ unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be $\varDelta^{0}_{2}$ hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound $\varDelta^{0}_{4}$ that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499–1502, 2006). Our second result is that if f is $\varDelta^{0}_{2}$ then there exists a non-empty $\varPi^{0}_{1}$ class of reals X with nontrivial prefix-free complexity which satisfy (⋆). This implies that in this case there uncountably many nontrivial reals X satisfying (⋆) in various well known classes from computability theory and algorithmic randomness; for example low for Ω, non-low for Ω and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452–463, 2011). KW - Kolmogorov complexity – Initial segment prefix-free complexity – K-triviality – Low for Ω Y1 - 2012 UR - http://www.springerlink.com/content/h0k3581g7202p574/ SN - 1433-0490 ER - TY - GEN A1 - Meer, Klaus A1 - Baartse, Martijn T1 - The PCP theorem for NP over the reals T2 - Foundations of Computational Mathematics KW - Probabilistically checkable proofs Real and complex number computations PCP theorem Hilbert Nullstellensatz decision problem Y1 - 2015 UR - http://link.springer.com/article/10.1007/s10208-014-9188-x?sa_campaign=email/event/articleAuthor/onlineFirst U6 - https://doi.org/10.1007/s10208-014-9188-x SN - 1615-3375 VL - 15 IS - 3 SP - 651 EP - 680 ER - TY - GEN A1 - Baartse, Martijn A1 - Meer, Klaus T1 - An algebraic proof of the real number PCP theorem T2 - Journal of Complexity N2 - The PCP theorem has recently been shown to hold as well in the real number model of Blum, Shub, and Smale (Baartse and Meer, 2015). The proof given there structurally closely follows the proof of the original PCP theorem by Dinur (2007). In this paper we show that the theorem also can be derived using algebraic techniques similar to those employed by Arora et al. (Arora et al., 1998; Arora and Safra, 1998) in the first proof of the PCP theorem. This needs considerable additional efforts. Due to severe problems when using low degree algebraic polynomials over the reals as codewords for one of the verifiers to be constructed, we work with certain trigonometric polynomials. This entails the necessity to design new segmentation procedures in order to obtain well structured real verifiers appropriate for applying the classical technique of verifier composition. We believe that designing as well an algebraic proof for the real PCP theorem on one side leads to interesting questions in real number complexity theory and on the other sheds light on which ingredients are necessary in order to prove an important result as the PCP theorem in different computational models. KW - Probabilistically checkable proofs KW - Real number model of computation KW - PCP theorem for NP over the reals KW - Testing trigonometric polynomials KW - Algebraic proofs Y1 - 2017 UR - http://www.sciencedirect.com/science/article/pii/S0885064X1630111X U6 - https://doi.org/10.1016/j.jco.2016.11.006 SN - 0885-064X VL - 40 SP - 34 EP - 77 ER - TY - GEN A1 - Baartse, Martijn A1 - Meer, Klaus T1 - Interactive proofs and a Shamir-like result for real number computations T2 - Computational Complexity Y1 - 2018 U6 - https://doi.org/10.1007/s00037-018-0174-6 SN - 1016-3328 SN - 1420-8954 ER - TY - CHAP A1 - Baartse, Martijn A1 - Meer, Klaus ED - Faliszewski, Piotr ED - Muscholl, Anca ED - Niedermeier, Rolf T1 - Real Interactive Proofs for VPSPACE T2 - 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0030-drops-64300 SN - 978-3-95977-016-3 SP - 14.1 EP - 14.13 PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH CY - Wadern ER - TY - CHAP A1 - Meer, Klaus A1 - Baartse, Martijn ED - Beckmann, Arnold ED - Mitrana, Victor ED - Soskova, Mariya T1 - Some results on interactive proofs for real computations T2 - Evolving Computability 11th Conference on Computability in Europe, CiE 2015, Bucharest, Romania, June 29-July 3, 2015, Proceedings Y1 - 2015 UR - http://www.springer.com/in/book/9783319200279 SN - 978-3-319-20027-9 U6 - https://doi.org/10.1007/978-3-319-20028-6 SP - 107 EP - 116 PB - Springer International Publishing CY - Berlin [u.a.] ER - TY - CHAP A1 - Meer, Klaus A1 - Baartse, Martijn ED - Hirsch, Edward ED - Kuznetsov, Sergei O. ED - Pin, Jean-Éric ED - Vereshchagin, Nikolay T1 - Testing Low Degree Trigonometric Polynomials T2 - Computer Science - Theory and Applications 9th International Computer Science Symposium in Russia, CSR 2014, Moscow, Russia, June 7-11, 2014, Proceedings Y1 - 2014 UR - http://www.springer.com/de/book/9783319066851 SN - 978-3-319-06685-1 U6 - https://doi.org/10.1007/978-3-319-06686-8 SP - 77 EP - 96 PB - Springer International Publishing CY - Berlin [u.a.] ER - TY - CHAP A1 - Meer, Klaus A1 - Baartse, Martijn T1 - An algebraic proof of the real number PCP theorem T2 - Mathematical Foundations of Computer Science 2015 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part II Y1 - 2015 UR - http://www.springer.com/us/book/9783662480533 SN - 978-3-662-48053-3 U6 - https://doi.org/10.1007/978-3-662-48054-0 SP - 50 EP - 61 PB - Springer-Verlag CY - Berlin [u.a.] ER - TY - CHAP A1 - Meer, Klaus A1 - Baartse, Martijn ED - Montaña, José Luis ED - Pardo, Luis Miguel T1 - Topics in Real and Complex Number Complexity Theory T2 - Recent Advances in Real Complexity and Computation Y1 - 2013 UR - http://bookstore.ams.org/conm-604?c=1 SN - 978-0-8218-9150-6 U6 - https://doi.org/10.1090/conm/604 SP - 1 EP - 53 PB - American Mathematical Society CY - Providence, RI ER - TY - CHAP A1 - Meer, Klaus A1 - Baartse, Martijn T1 - The PCP theorem for NP over the reals T2 - 30th Symposium on Theoretical Aspects of Computer Science STACS , Kiel, 2013 Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0030-drops-39262 UR - http://drops.dagstuhl.de/opus/volltexte/2013/3926/ SP - 104 EP - 115 CY - Leibnitz ER -