메뉴 건너뛰기




Volumn 51, Issue 1, 2005, Pages 62-72

Effective fractal dimensions

Author keywords

Algorithmic information; Circuit size complexity; Computational complexity; Constructive dimension; Entropy; Gales; Hausdorff dimension; Kolmogorov complexity; Martingales; Packing dimension; Polynomial time degrees; Random sequences; Resource bounded dimension

Indexed keywords


EID: 11944273760     PISSN: 09425616     EISSN: None     Source Type: Journal    
DOI: 10.1002/malq.200310127     Document Type: Article
Times cited : (34)

References (66)
  • 1
    • 0003130451 scopus 로고    scopus 로고
    • Resource-bounded measure and randomness
    • Complexity, Logic and Recursion Theory (A. Sorbi, ed.), Marcel Dekker, New York
    • K. Ambos-Spies and E. Mayordomo, Resource-bounded measure and randomness. In: Complexity, Logic and Recursion Theory (A. Sorbi, ed.), pp. l-47, Lecture Notes in Pure and Applied Mathematics (Marcel Dekker, New York 1997).
    • (1997) Lecture Notes in Pure and Applied Mathematics
    • Ambos-Spies, K.1    Mayordomo, E.2
  • 5
    • 11944249588 scopus 로고    scopus 로고
    • Complete sets and structure in subrecursive classes
    • Lecture Notes in Logic
    • H. Buhrman and L. Torenvliet, Complete sets and structure in subrecursive classes. In: Proceedings of Logic Colloquium '96, Lecture Notes in Logic 12, 45-78 (1998).
    • (1998) Proceedings of Logic Colloquium '96 , vol.12 , pp. 45-78
    • Buhrman, H.1    Torenvliet, L.2
  • 6
    • 0342468141 scopus 로고    scopus 로고
    • A generalization of resource-bounded measure, with application to the BPP vs. EXP Problem
    • H. Buhrman, D. van Melkebeek, K. W. Regan, D. Sivakumar, and M. Strauss, A generalization of resource-bounded measure, with application to the BPP vs. EXP Problem. SIAM J. Computing 30, 576-601 (2001).
    • (2001) SIAM J. Computing , vol.30 , pp. 576-601
    • Buhrman, H.1    Van Melkebeek, D.2    Regan, K.W.3    Sivakumar, D.4    Strauss, M.5
  • 7
    • 0028712369 scopus 로고
    • On hausdorff and topological dimensions of the kolmogorov complexity of the real line
    • J. Cai and J. Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. J. Computer and Systems Sciences 49, 605-619 (1994).
    • (1994) J. Computer and Systems Sciences , vol.49 , pp. 605-619
    • Cai, J.1    Hartmanis, J.2
  • 8
    • 0016532771 scopus 로고
    • A theory of program size formally identical to information theory
    • G. J. Chaitin, A theory of program size formally identical to information theory. J. ACM 22, 329-340 (1975).
    • (1975) J. ACM , vol.22 , pp. 329-340
    • Chaitin, G.J.1
  • 9
    • 0000761947 scopus 로고
    • Incompleteness theorems for random reals
    • G. J. Chaitin, Incompleteness theorems for random reals. Advances in Applied Mathematics 8, 119-146 (1987).
    • (1987) Advances in Applied Mathematics , vol.8 , pp. 119-146
    • Chaitin, G.J.1
  • 12
    • 0742269491 scopus 로고    scopus 로고
    • On the autoreducibility of random sequences
    • T. Ebert, W. Merkle, and H. Vollmer, On the autoreducibility of random sequences. SIAM J. Computing 32, 1542-1569 (2003).
    • (2003) SIAM J. Computing , vol.32 , pp. 1542-1569
    • Ebert, T.1    Merkle, W.2    Vollmer, H.3
  • 15
    • 0344360944 scopus 로고    scopus 로고
    • Gales and supergales are equivalent for defining constructive Hausdorff dimension
    • Computing Research Repository
    • S. A. Fenner, Gales and supergales are equivalent for defining constructive Hausdorff dimension. Technical Report cs.CC/0208044, Computing Research Repository 2002.
    • (2002) Technical Report Cs.CC/0208044
    • Fenner, S.A.1
  • 17
    • 34250950477 scopus 로고
    • Dimension und äusseres Maß
    • F. Hausdorff, Dimension und äusseres Maß. Math. Annalen 79, 157-179 (1919).
    • (1919) Math. Annalen , vol.79 , pp. 157-179
    • Hausdorff, F.1
  • 21
    • 0037163961 scopus 로고    scopus 로고
    • MAX3SAT is exponentially hard to approximate if NP has positive dimension
    • J. M. Hitchcock, MAX3SAT is exponentially hard to approximate if NP has positive dimension. Theoret. Comput. Sci. 289, 861-869 (2002).
    • (2002) Theoret. Comput. Sci. , vol.289 , pp. 861-869
    • Hitchcock, J.M.1
  • 22
    • 0037811185 scopus 로고    scopus 로고
    • Fractal dimension and logarithmic loss unpredictability
    • J. M. Hitchcock, Fractal dimension and logarithmic loss unpredictability. Theoret. Comp. Sci. 304, 431-441 (2003).
    • (2003) Theoret. Comp. Sci. , vol.304 , pp. 431-441
    • Hitchcock, J.M.1
  • 23
    • 0037447594 scopus 로고    scopus 로고
    • Gales suffice for constructive dimension
    • J. M. Hitchcock, Gales suffice for constructive dimension. Information Processing Letters 86, 9-12 (2003).
    • (2003) Information Processing Letters , vol.86 , pp. 9-12
    • Hitchcock, J.M.1
  • 25
    • 35248853282 scopus 로고    scopus 로고
    • Scaled dimension and nonuniform complexity
    • (to appear). A preliminary version appeared in: Proceedings of the 30th International Colloquium on Automata, Languages, and Programming
    • J. M. Hitchcock, J. H. Lutz, and E. Mayordomo, Scaled dimension and nonuniform complexity. J. Computer and System Sciences (to appear). A preliminary version appeared in: Proceedings of the 30th International Colloquium on Automata, Languages, and Programming, pp. 278-290 (2003).
    • (2003) J. Computer and System Sciences , pp. 278-290
    • Hitchcock, J.M.1    Lutz, J.H.2    Mayordomo, E.3
  • 27
    • 0029287714 scopus 로고
    • The complexity and distribution of hard problems
    • D. W. Jucdes and J. H. Lutz, The complexity and distribution of hard problems. SIAM J. Computing 24, 279-295 (1995).
    • (1995) SIAM J. Computing , vol.24 , pp. 279-295
    • Jucdes, D.W.1    Lutz, J.H.2
  • 28
    • 0000508181 scopus 로고
    • On the notion of a random sequence
    • L. A. Levin, On the notion of a random sequence. Soviet Mathematics Doklady 14, 1413-1416 (1973).
    • (1973) Soviet Mathematics Doklady , vol.14 , pp. 1413-1416
    • Levin, L.A.1
  • 29
    • 0016078993 scopus 로고
    • Laws of information conservation (nongrowth) and aspects of the foundation of probability theory
    • L. A. Levin, Laws of information conservation (nongrowth) and aspects of the foundation of probability theory. Problems of Information Transmission 10, 206-210 (1974).
    • (1974) Problems of Information Transmission , vol.10 , pp. 206-210
    • Levin, L.A.1
  • 32
    • 0026851355 scopus 로고
    • Almost everywhere high nonuniform complexity
    • J. H. Lutz, Almost everywhere high nonuniform complexity. J. Computer and System Sciences 44, 220-258 (1992).
    • (1992) J. Computer and System Sciences , vol.44 , pp. 220-258
    • Lutz, J.H.1
  • 33
    • 0000923207 scopus 로고    scopus 로고
    • The quantitative structure of exponential time
    • (L. A. Hemaspaandra and A. L. Selman, eds.), Springer-Verlag, New York et al.
    • J. H. Lutz, The quantitative structure of exponential time. In: Complexity Theory Retrospective II (L. A. Hemaspaandra and A. L. Selman, eds.), pp. 225-254 (Springer-Verlag, New York et al. 1997).
    • (1997) Complexity Theory Retrospective II , pp. 225-254
    • Lutz, J.H.1
  • 36
    • 0345227319 scopus 로고    scopus 로고
    • Dimension in complexity classes
    • J. H. Lutz, Dimension in complexity classes. SIAM J. Computing 32, 1236-1259 (2003). A preliminary version appeared in: Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity, 158-169 (2000).
    • (2003) SIAM J. Computing , vol.32 , pp. 1236-1259
    • Lutz, J.H.1
  • 38
    • 0344118726 scopus 로고    scopus 로고
    • The dimensions of individual strings and sequences
    • J. H. Lutz, The dimensions of individual strings and sequences. Information and Computation 187, 49-79 (2003).
    • (2003) Information and Computation , vol.187 , pp. 49-79
    • Lutz, J.H.1
  • 40
    • 18344371384 scopus 로고
    • The definition of random sequences
    • P. Martin-Löf, The definition of random sequences. Information and Control 9, 602-619 (1966).
    • (1966) Information and Control , vol.9 , pp. 602-619
    • Martin-Löf, P.1
  • 43
    • 0037120704 scopus 로고    scopus 로고
    • A kolmogorov complexity characterization of constructive hausdorff dimension
    • E. Mayordomo, A Kolmogorov complexity characterization of constructive Hausdorff dimension. Information Processing Letters 84, 1-3 (2002).
    • (2002) Information Processing Letters , vol.84 , pp. 1-3
    • Mayordomo, E.1
  • 44
    • 0004122682 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • C. A. Rogers, Hausdorff Measures (Cambridge University Press, Cambridge 1998).
    • (1998) Hausdorff Measures
    • Rogers, C.A.1
  • 45
    • 0004492226 scopus 로고
    • Coding of combinatorial sources and hausdorff dimension
    • B. Ya. Ryabko, Coding of combinatorial sources and Hausdorff dimension. Soviet Mathematics Doklady 30, 219-222 (1984).
    • (1984) Soviet Mathematics Doklady , vol.30 , pp. 219-222
    • Ryabko, B.Ya.1
  • 47
    • 0004018518 scopus 로고
    • Algorithmic approach to the prediction problem
    • B. Ya. Ryabko, Algorithmic approach to the prediction problem. Problems of Information Transmission 29, 186-193 (1993).
    • (1993) Problems of Information Transmission , vol.29 , pp. 186-193
    • Ryabko, B.Ya.1
  • 48
    • 0000797906 scopus 로고
    • The complexity and effectiveness of prediction problems
    • B. Ya. Ryabko, The complexity and effectiveness of prediction problems. Journal of Complexity 10, 281-295 (1994).
    • (1994) Journal of Complexity , vol.10 , pp. 281-295
    • Ryabko, B.Ya.1
  • 49
    • 0000583779 scopus 로고
    • A unified approach to the definition of random sequences
    • C. P. Schnorr, A unified approach to the definition of random sequences. Mathematical Systems Theory 5, 246-258 (1971).
    • (1971) Mathematical Systems Theory , vol.5 , pp. 246-258
    • Schnorr, C.P.1
  • 51
    • 0015902552 scopus 로고
    • Process complexity and effective random tests
    • C. P. Schnorr, Process complexity and effective random tests. J. Computer and System Sciences 7, 376-388 (1973).
    • (1973) J. Computer and System Sciences , vol.7 , pp. 376-388
    • Schnorr, C.P.1
  • 52
    • 0345106453 scopus 로고
    • A survey of the theory of random sequences
    • (R. E. Butts and J. Hintikka, eds.), D. Reidel, Dordrecht
    • C. P. Schnorr, A survey of the theory of random sequences. In: Basic Problems in Methodology and Linguistics (R. E. Butts and J. Hintikka, eds.), pp. 193-210 (D. Reidel, Dordrecht 1977).
    • (1977) Basic Problems in Methodology and Linguistics , pp. 193-210
    • Schnorr, C.P.1
  • 53
    • 84938487169 scopus 로고
    • The synthesis of two-terminal switching circuits
    • C. E. Shannon, The synthesis of two-terminal switching circuits. Bell System Technical J. 28, 59-98 (1949).
    • (1949) Bell System Technical J. , vol.28 , pp. 59-98
    • Shannon, C.E.1
  • 54
    • 0344360940 scopus 로고
    • The frequency approach to the definition of a random sequence
    • In Russian.
    • A. Kh. Shen', The frequency approach to the definition of a random sequence. Semiotika i Informatika 18, 14-42 (1982). (In Russian.)
    • (1982) Semiotika I Informatika , vol.18 , pp. 14-42
    • Shen, A.Kh.1
  • 55
    • 0002394549 scopus 로고
    • On relations between different algorithmic definitions of randomness
    • A. Kh. Shen', On relations between different algorithmic definitions of randomness. Soviet Mathematics Doklady 38, 316-319 (1989).
    • (1989) Soviet Mathematics Doklady , vol.38 , pp. 316-319
    • Shen, A.Kh.1
  • 56
    • 11944250673 scopus 로고    scopus 로고
    • R. M. Solovay 1975. Reported in [9].
    • R. M. Solovay 1975. Reported in [9].
  • 57
    • 38249002230 scopus 로고
    • Kolmogorov complexity and hausdorff dimension
    • L. Staiger, Kolmogorov Complexity and Hausdorff Dimension. Information and Computation 103, 159-194 (1993).
    • (1993) Information and Computation , vol.103 , pp. 159-194
    • Staiger, L.1
  • 58
    • 0038856793 scopus 로고    scopus 로고
    • A tight upper bound on Kolmogorov complexity and uniformly optimal prediction
    • L. Staiger, A tight upper bound on Kolmogorov complexity and uniformly optimal prediction. Theory of Computing Systems 31, 215-229 (1998).
    • (1998) Theory of Computing Systems , vol.31 , pp. 215-229
    • Staiger, L.1
  • 59
    • 0003944635 scopus 로고    scopus 로고
    • How much can you win when your adversary is handicapped
    • Kluwer Academic Press
    • L. Staiger, How much can you win when your adversary is handicapped. In: Numbers, Information and Complexity, pp. 403-412 (Kluwer Academic Press, 2000).
    • (2000) Numbers, Information and Complexity , pp. 403-412
    • Staiger, L.1
  • 60
    • 0842335187 scopus 로고    scopus 로고
    • Constructive dimension equals Kolmogorov complexity
    • University of Auckland, January
    • L. Staiger, Constructive dimension equals Kolmogorov complexity. Technical Report CDMTCS-210, University of Auckland, January 2003.
    • (2003) Technical Report , vol.CDMTCS-210
    • Staiger, L.1
  • 61
    • 0000144097 scopus 로고
    • Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
    • D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica 153, 259-277 (1984).
    • (1984) Acta Mathematica , vol.153 , pp. 259-277
    • Sullivan, D.1
  • 62
    • 10444239935 scopus 로고    scopus 로고
    • A generalization of Chaitin's halting probability Ω and halting self-similar sets
    • K. Tadaki, A generalization of Chaitin's halting probability Ω and halting self-similar sets. Hokkaido Mathematical Journal 31, 219-253 (2002).
    • (2002) Hokkaido Mathematical Journal , vol.31 , pp. 219-253
    • Tadaki, K.1
  • 63
    • 11944262892 scopus 로고    scopus 로고
    • Complexity and randomness
    • University of Auckland, March
    • S. A. Terwijn, Complexity and randomness. Technical Report CDMTCS-212, University of Auckland, March 2003.
    • (2003) Technical Report , vol.CDMTCS-212
    • Terwijn, S.A.1
  • 66
    • 77951203397 scopus 로고
    • The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms
    • A. K. Zvonkin and L. A. Levin, The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25, 83-124 (1970).
    • (1970) Russian Mathematical Surveys , vol.25 , pp. 83-124
    • Zvonkin, A.K.1    Levin, L.A.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.