메뉴 건너뛰기




Volumn 136, Issue 1-2, 2005, Pages 189-218

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

Author keywords

Analytic combinatorics; Independence results for systems of arithmetic; Logical limit laws; Ordinal analysis; Proof theoretic ordinals; Tauberian theory

Indexed keywords


EID: 23944507284     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apal.2005.05.012     Document Type: Article
Times cited : (24)

References (76)
  • 1
    • 0040628911 scopus 로고
    • Exact asymptotic formulas for the coefficients of nonmodular functions
    • G. Almkvist Exact asymptotic formulas for the coefficients of nonmodular functions J. Number Theory 38 2 1991 145-160
    • (1991) J. Number Theory , vol.38 , Issue.2 , pp. 145-160
    • Almkvist, G.1
  • 2
    • 0038357340 scopus 로고    scopus 로고
    • An introduction to finitary analyses of proof figures
    • London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge
    • T. Arai An introduction to finitary analyses of proof figures in: Sets and Proofs Leeds, 1997 in: London Math. Soc. Lecture Note Ser. vol. 258 1999 Cambridge Univ. Press Cambridge 1-25
    • (1999) Sets and Proofs Leeds 1997 , vol.258 , pp. 1-25
    • Arai, T.1
  • 3
    • 23944514007 scopus 로고    scopus 로고
    • 0
    • Hiroshima (Preprint)
    • 0, Hiroshima, 2000 (Preprint)
    • (2000)
    • Arai, T.1
  • 4
    • 3543056122 scopus 로고    scopus 로고
    • Proof theory for theories of ordinals. I. Recursively Mahlo ordinals
    • T. Arai Proof theory for theories of ordinals. I. Recursively Mahlo ordinals Ann. Pure Appl. Logic 122 1-3 2003 1-85
    • (2003) Ann. Pure Appl. Logic , vol.122 , Issue.1-3 , pp. 1-85
    • Arai, T.1
  • 5
    • 0037526347 scopus 로고    scopus 로고
    • Dynamic ordinal analysis
    • A. Beckmann Dynamic ordinal analysis Arch. Math. Logic 42 4 2003 303-334
    • (2003) Arch. Math. Logic , vol.42 , Issue.4 , pp. 303-334
    • Beckmann, A.1
  • 6
    • 0032487260 scopus 로고    scopus 로고
    • Applications of cut-free infinitary derivations to generalized recursion theory
    • Oberwolfach, 1996 Conference on Computability Theory
    • A. Beckmann W. Pohlers Applications of cut-free infinitary derivations to generalized recursion theory in: Conference on Computability Theory Oberwolfach, 1996 Ann. Pure Appl. Logic 94 1-3 1998 7-19
    • (1998) Ann. Pure Appl. Logic , vol.94 , Issue.1-3 , pp. 7-19
    • Beckmann, A.1    Pohlers, W.2
  • 7
    • 2442575027 scopus 로고    scopus 로고
    • Provability algebras and proof-theoretic ordinals, I
    • L.D. Beklemishev Provability algebras and proof-theoretic ordinals, I Ann. Pure Appl. Logic 128 2004 103-123
    • (2004) Ann. Pure Appl. Logic , vol.128 , pp. 103-123
    • Beklemishev, L.D.1
  • 8
    • 79960825006 scopus 로고    scopus 로고
    • The Worm principle
    • Department of Philosophy, Utrecht University March
    • L. Beklemishev The Worm principle in: Logic Group Preprint Series vol. 219 2003 Department of Philosophy, Utrecht University March
    • (2003) Logic Group Preprint Series , vol.219
    • Beklemishev, L.1
  • 9
    • 0039342466 scopus 로고    scopus 로고
    • How to characterize provably total functions by the Buchholz operator method
    • Gödel'96 Brno, 1996 Springer, Berlin
    • B. Blankertz A. Weiermann How to characterize provably total functions by the Buchholz operator method in: Gödel'96 Brno, 1996 Lecture Notes Logic vol. 6 1996 Springer Berlin 205-213
    • (1996) Lecture Notes Logic , vol.6 , pp. 205-213
    • Blankertz, B.1    Weiermann, A.2
  • 11
    • 84981406734 scopus 로고
    • A uniform approach to fundamental sequences and hierarchies
    • W. Buchholz A. Cichon A. Weiermann A uniform approach to fundamental sequences and hierarchies Math. Logic Quarterly 40 1994 273-286
    • (1994) Math. Logic Quarterly , vol.40 , pp. 273-286
    • Buchholz, W.1    Cichon, A.2    Weiermann, A.3
  • 12
    • 0005146817 scopus 로고    scopus 로고
    • Number Theoretic Density and Logical Limit Laws
    • American Mathematical Society
    • S.N. Burris Number Theoretic Density and Logical Limit Laws in: Mathematical Surveys and Monographs vol. 86 2001 American Mathematical Society
    • (2001) Mathematical Surveys and Monographs , vol.86
    • Burris, S.N.1
  • 14
    • 0002193763 scopus 로고
    • A short proof of two recently discovered independence results using recursion theoretic methods
    • E.A. Cichon A short proof of two recently discovered independence results using recursion theoretic methods Proc. Amer. Math. Soc. 87 1983 704-706
    • (1983) Proc. Amer. Math. Soc. , vol.87 , pp. 704-706
    • Cichon, E.A.1
  • 15
    • 0002554229 scopus 로고
    • Some theorems concerning prime numbers
    • M. Cramer Some theorems concerning prime numbers Ark. Mat. Astr. Fys. 15 1920 1-32
    • (1920) Ark. Mat. Astr. Fys. , vol.15 , pp. 1-32
    • Cramer, M.1
  • 16
    • 0000166697 scopus 로고
    • On Mahler's partition problem
    • Reprinted from: Koninklijke Nederlandsche Akademie van Wetenschappen, Ser. A
    • N.G. de Bruijn On Mahler's partition problem Indagationes Math. 10 1948 210-220. Reprinted from: Koninklijke Nederlandsche Akademie van Wetenschappen, Ser. A.
    • (1948) Indagationes Math. , vol.10 , pp. 210-220
    • de Bruijn, N.G.1
  • 18
    • 84960588672 scopus 로고
    • Combinatorial theorems on classifications of subsets of a given set
    • P. Erdös R. Rado Combinatorial theorems on classifications of subsets of a given set Proc. London Math. Soc. Ser. III 2 1952 417-439
    • (1952) Proc. London Math. Soc. Ser. III , vol.2 , pp. 417-439
    • Erdös, P.1    Rado, R.2
  • 19
    • 1642303516 scopus 로고
    • Infinitary properties, local functors, and systems of ordinal functions
    • Springer, Berlin Conference in Mathematical Logic - London'70 Proc. Conf., Bedford Coll., London, 1970
    • S. Feferman Infinitary properties, local functors, and systems of ordinal functions in: Conference in Mathematical Logic - London'70 Proc. Conf., Bedford Coll., London, 1970 in: Lecture Notes in Math. vol. 255 1972 Springer Berlin 63-97
    • (1972) Lecture Notes in Math. , vol.255 , pp. 63-97
    • Feferman, S.1
  • 20
    • 84963043400 scopus 로고
    • On Gaps between squarefree numbers II
    • (2)
    • M. Filaseta O. Trifonov On Gaps between squarefree numbers II J. London Math. Soc. (2) 45 2 1992 215-221
    • (1992) J. London Math. Soc. , vol.45 , Issue.2 , pp. 215-221
    • Filaseta, M.1    Trifonov, O.2
  • 21
    • 23944503994 scopus 로고    scopus 로고
    • The average case of algorithms
    • Available via WWW at
    • P. Flajolet R. Sedgewick The average case of algorithms. Available via WWW at http://algo.inria.fr/flajolet/Publications/publist.html
    • Flajolet, P.1    Sedgewick, R.2
  • 22
    • 23944462091 scopus 로고    scopus 로고
    • FOM: Urbana thoughts
    • Posting from Saturday 10-th, June 2000
    • H. Friedman FOM: Urbana thoughts, Posting from Saturday 10-th, June 2000 http://www.cs.nyu.edu/pipermail/fom/2000-June/004098.html
    • Friedman, H.1
  • 24
    • 49049145760 scopus 로고
    • 1-logic. I. Dilators
    • 1-logic. I. Dilators Ann. Math. Logic 21 2-3 1981 75-219 1982
    • (1981) Ann. Math. Logic , vol.21 , Issue.2-3 , pp. 75-219
    • Girard, J.Y.1
  • 25
    • 0009177395 scopus 로고    scopus 로고
    • On the contour of random trees
    • B. Gittenberger On the contour of random trees SIAM J. Discrete Math. 12 4 1999 434-458
    • (1999) SIAM J. Discrete Math. , vol.12 , Issue.4 , pp. 434-458
    • Gittenberger, B.1
  • 26
    • 1842779976 scopus 로고    scopus 로고
    • ABC allows us to count squarefrees
    • A. Granville ABC allows us to count squarefrees Int. Math. Res. Not. 19 1998 991-1009
    • (1998) Int. Math. Res. Not. , vol.19 , pp. 991-1009
    • Granville, A.1
  • 28
    • 84960557830 scopus 로고
    • Asymptotic formulae for the distribution of integers of various types
    • G.H. Hardy S. Ramanujan Asymptotic formulae for the distribution of integers of various types Proc. London Math. Soc. 1917
    • (1917) Proc. London Math. Soc.
    • Hardy, G.H.1    Ramanujan, S.2
  • 29
    • 84960612244 scopus 로고
    • Asymptotic formulae in combinatory analysis
    • G.H. Hardy S. Ramanujan Asymptotic formulae in combinatory analysis Proc. London Math. Soc. 17 1918 75-115
    • (1918) Proc. London Math. Soc. , vol.17 , pp. 75-115
    • Hardy, G.H.1    Ramanujan, S.2
  • 30
    • 0037028518 scopus 로고    scopus 로고
    • Two applications of analytic functors. Theories of types and proofs, Tokyo, 1997
    • R. Hasegawa Two applications of analytic functors. Theories of types and proofs, Tokyo, 1997 Theoret. Comput. Sci. 272 1-2 2002 113-175
    • (2002) Theoret. Comput. Sci. , vol.272 , Issue.1-2 , pp. 113-175
    • Hasegawa, R.1
  • 31
    • 0001833630 scopus 로고
    • The number of primes in a short interval
    • D.R. Heath Brown The number of primes in a short interval J. Reine Angew. Math. 389 1988 22-63
    • (1988) J. Reine Angew. Math. , vol.389 , pp. 22-63
    • Heath Brown, D.R.1
  • 33
    • 0002663026 scopus 로고
    • Une thorie combinatoire des series formelles
    • A. Joyal Une thorie combinatoire des series formelles Adv. Math. 42 1 1981 1-82
    • (1981) Adv. Math. , vol.42 , Issue.1 , pp. 1-82
    • Joyal, A.1
  • 34
    • 38249036243 scopus 로고
    • On Gödel incompleteness and finite combinatorics
    • A. Kanamori K. McAloon On Gödel incompleteness and finite combinatorics Ann. Pure Appl. Logic 33 1987 23-41
    • (1987) Ann. Pure Appl. Logic , vol.33 , pp. 23-41
    • Kanamori, A.1    McAloon, K.2
  • 35
    • 23944526551 scopus 로고
    • Extensions of arithmetic for proving termination of computations
    • C.F. Kent B.R. Hodgson Extensions of arithmetic for proving termination of computations J. Symbolic Logic 54 1989 779-794
    • (1989) J. Symbolic Logic , vol.54 , pp. 779-794
    • Kent, C.F.1    Hodgson, B.R.2
  • 36
    • 0003944554 scopus 로고
    • Abstract Analytic Number Theory
    • North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc. New York
    • J. Knopfmacher Abstract Analytic Number Theory in: North-Holland Mathematical Library vol. 12 1975 North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc. New York
    • (1975) North-Holland Mathematical Library , vol.12
    • Knopfmacher, J.1
  • 38
    • 0040597014 scopus 로고
    • Weak asymptotic properties of partitions
    • E.E. Kohlbecker Weak asymptotic properties of partitions Trans. Amer. Math. Soc. 88 1958 346-365
    • (1958) Trans. Amer. Math. Soc. , vol.88 , pp. 346-365
    • Kohlbecker, E.E.1
  • 40
    • 2942595333 scopus 로고    scopus 로고
    • Enumeration des 2-arbres k-gonaux
    • B. Chauvin P. Flajolet (Eds) Birkhäuser Verlag Basel, Switzerland
    • G. Labelle C. Lamathe P. Leroux Enumeration des 2-arbres k-gonaux in: B. Chauvin P. Flajolet (Eds) Trends in Mahematics 2002 Birkhäuser Verlag Basel, Switzerland 95-109
    • (2002) Trends in Mahematics , pp. 95-109
    • Labelle, G.1    Lamathe, C.2    Leroux, P.3
  • 41
    • 23944479075 scopus 로고    scopus 로고
    • Phase transitions in axiomatic thought
    • Ph.D. Thesis, Münster
    • G. Lee, Phase transitions in axiomatic thought. Ph.D. Thesis, Münster, 2005
    • (2005)
    • Lee, G.1
  • 42
    • 0012889843 scopus 로고
    • On undecidability of the weakened Kruskal theorem
    • Arcata CA, 1985 RI Amer. Math. Soc. Providence
    • M. Loebl J. Matoušek On undecidability of the weakened Kruskal theorem in: Logic and Combinatorics Arcata, CA, 1985 1987 Amer. Math. Soc. Providence, RI 275-280
    • (1987) Logic and Combinatorics , pp. 275-280
    • Loebl, M.1    Matoušek, J.2
  • 43
    • 0009827561 scopus 로고
    • A natural rooted tree enumeration by prime factorization
    • D.W. Matula A natural rooted tree enumeration by prime factorization SIAM Rev. 10 1968
    • (1968) SIAM Rev. , vol.10
    • Matula, D.W.1
  • 45
    • 0000902966 scopus 로고
    • The number of trees
    • R. Otter The number of trees Ann. Math. 49 1948 583-599
    • (1948) Ann. Math. , vol.49 , pp. 583-599
    • Otter, R.1
  • 46
    • 0039789249 scopus 로고
    • Partition functions whose logarithms are slowly oscillating
    • S. Parameswaran Partition functions whose logarithms are slowly oscillating Trans. Amer. Math. Soc. 100 1961 217-240
    • (1961) Trans. Amer. Math. Soc. , vol.100 , pp. 217-240
    • Parameswaran, S.1
  • 47
    • 0033453899 scopus 로고
    • Growth of finitely generated polynilpotent lie algebras and groups, generalized partitions, and functions analytic in the unit circle
    • V.M. Petrogradsky Growth of finitely generated polynilpotent lie algebras and groups, generalized partitions, and functions analytic in the unit circle Internat. J. Algebra Comput. 1991 179-212
    • (1991) Internat. J. Algebra Comput. , pp. 179-212
    • Petrogradsky, V.M.1
  • 48
    • 23944499285 scopus 로고
    • Ordinals connected with formal theories for transfinitely iterated inductive definitions
    • W. Pohlers Ordinals connected with formal theories for transfinitely iterated inductive definitions J. Symbolic Logic 43 2 1978 161-182
    • (1978) J. Symbolic Logic , vol.43 , Issue.2 , pp. 161-182
    • Pohlers, W.1
  • 50
    • 23944473350 scopus 로고
    • Cut elimination for impredicative infinitary systems. II. Ordinal analysis for iterated inductive definitions
    • W. Pohlers Cut elimination for impredicative infinitary systems. II. Ordinal analysis for iterated inductive definitions Arch. Math. Logik Grundlag. 22 1-2 1982 69-87
    • (1982) Arch. Math. Logik Grundlag. , vol.22 , Issue.1-2 , pp. 69-87
    • Pohlers, W.1
  • 51
    • 0003362273 scopus 로고
    • Proof theory. An introduction
    • Berlin: Springer-Verlag
    • W. Pohlers Proof theory. An introduction Lecture Notes in Mathematics vol. 1407 1989 Springer-Verlag Berlin
    • (1989) Lecture Notes in Mathematics , vol.1407
    • Pohlers, W.1
  • 52
    • 84963069472 scopus 로고
    • On the Partition Function p(n)
    • H. Rademacher On the Partition Function p(n) Proc. London Math. Soc. 43 1937 241-254
    • (1937) Proc. London Math. Soc. , vol.43 , pp. 241-254
    • Rademacher, H.1
  • 53
    • 84974065271 scopus 로고
    • 1-CA and related systems
    • 1 - CA and related systems Bull. Symbolic Logic 1 4 1995 468-485
    • (1995) Bull. Symbolic Logic , vol.1 , Issue.4 , pp. 468-485
    • Rathjen, M.1
  • 54
    • 2442548033 scopus 로고    scopus 로고
    • The realm of ordinal analysis
    • Sets and Proofs Leeds, 1997 Cambridge Univ. Press Cambridge
    • M. Rathjen The realm of ordinal analysis Sets and Proofs Leeds, 1997 London Math. Soc. Lecture Note Ser. vol. 258 1999 Cambridge Univ. Press Cambridge 219-279
    • (1999) London Math. Soc. Lecture Note Ser. , vol.258 , pp. 219-279
    • Rathjen, M.1
  • 55
    • 17444392810 scopus 로고    scopus 로고
    • 1 -comprehension
    • doi:10.1007/s00153-004-0232-4 Publisher: Springer-Verlag GmbH ISSN: 0933-5846 (Paper) 1432-0665 (Online) Issue: Online First September 1, 2004
    • 1-comprehension Arch. Math. Logic 2004 doi:10.1007/ s00153-004-0232-4 Publisher: Springer-Verlag GmbH ISSN: 0933-5846 (Paper) 1432-0665 (Online) Issue: Online First September 1, 2004
    • (2004) Arch. Math. Logic
    • Rathjen, M.1
  • 56
    • 22044451832 scopus 로고    scopus 로고
    • An ordinal analysis of stability
    • M. Rathjen An ordinal analysis of stability Arch. Math. Logic 44 1 2005 1-62
    • (2005) Arch. Math. Logic , vol.44 , Issue.1 , pp. 1-62
    • Rathjen, M.1
  • 57
    • 38249004336 scopus 로고
    • Proof-theoretic investigations on Kruskal's theorem
    • M. Rathjen A. Weiermann Proof-theoretic investigations on Kruskal's theorem Ann. Pure Appl. Logic 60 1993 49-88
    • (1993) Ann. Pure Appl. Logic , vol.60 , pp. 49-88
    • Rathjen, M.1    Weiermann, A.2
  • 58
    • 0039956445 scopus 로고
    • Asymptotic results for partitions. I. The distribution of certain integers
    • L.B. Richmond Asymptotic results for partitions. I. The distribution of certain integers J. Number Theory 8 4 1976 372-389
    • (1976) J. Number Theory , vol.8 , Issue.4 , pp. 372-389
    • Richmond, L.B.1
  • 59
    • 0002872653 scopus 로고
    • Über die Anzahl der Primzahlen unter einer gegebenen Grösse
    • B. Riemann Über die Anzahl der Primzahlen unter einer gegebenen Grösse Mon. Not. Berlin Akad. 1859 671-680
    • (1859) Mon. Not. Berlin Akad. , pp. 671-680
    • Riemann, B.1
  • 60
    • 23944523541 scopus 로고
    • Bounds for the closure ordinals of replete monotonic increasing functions
    • D. Schmidt Bounds for the closure ordinals of replete monotonic increasing functions J. Symbolic Logic 40 3 1975 305-316
    • (1975) J. Symbolic Logic , vol.40 , Issue.3 , pp. 305-316
    • Schmidt, D.1
  • 63
    • 77956710554 scopus 로고
    • Non-provability of certain combinatorial properties of finite trees
    • North-Holland Amsterdam
    • S.G. Simpson Non-provability of certain combinatorial properties of finite trees in: Harvey Friedman's Research on the Foundations of Mathematics 1985 North-Holland Amsterdam 87-117
    • (1985) Harvey Friedman's Research on the Foundations of Mathematics , pp. 87-117
    • Simpson, S.G.1
  • 64
    • 0006990695 scopus 로고
    • Ordinal numbers and the Hilbert basis theorem
    • S. Simpson Ordinal numbers and the Hilbert basis theorem J. Symbolic Logic 53 3 1988 961-974
    • (1988) J. Symbolic Logic , vol.53 , Issue.3 , pp. 961-974
    • Simpson, S.1
  • 65
    • 85012573117 scopus 로고
    • The consistency strength of some finite forms of the Higman and Kruskal theorems
    • North-Holland Amsterdam
    • R. Smith The consistency strength of some finite forms of the Higman and Kruskal theorems in: Harvey Friedman's Research on the Foundations of Mathematics 1985 North-Holland Amsterdam 119-136
    • (1985) Harvey Friedman's Research on the Foundations of Mathematics , pp. 119-136
    • Smith, R.1
  • 66
    • 0003707214 scopus 로고
    • Weylsche Exponentialsummen in der neueren Zahlentheorie
    • Berlin
    • A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963
    • (1963)
    • Walfisz, A.1
  • 67
    • 23944455053 scopus 로고
    • Bounds for the closure ordinals of essentially monotonic increasing functions
    • A. Weiermann Bounds for the closure ordinals of essentially monotonic increasing functions J. Symbolic Logic 58 2 1993 664-671
    • (1993) J. Symbolic Logic , vol.58 , Issue.2 , pp. 664-671
    • Weiermann, A.1
  • 68
    • 21844525711 scopus 로고
    • Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones
    • A. Weiermann Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones Arch. Math. Logic 34 5 1995 313-330
    • (1995) Arch. Math. Logic , vol.34 , Issue.5 , pp. 313-330
    • Weiermann, A.1
  • 69
    • 0029637528 scopus 로고
    • Termination proofs for term rewriting systems by lexicographic path orderings imply multiply recursive derivation lengths
    • A. Weiermann Termination proofs for term rewriting systems by lexicographic path orderings imply multiply recursive derivation lengths Theoret. Comput. Sci. 139 1-2 1995 355-362
    • (1995) Theoret. Comput. Sci. , vol.139 , Issue.1-2 , pp. 355-362
    • Weiermann, A.1
  • 70
    • 0040350318 scopus 로고    scopus 로고
    • How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: A case study
    • A. Weiermann How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: A case study J. Symbolic Logic 63 4 1998 1348-1370
    • (1998) J. Symbolic Logic , vol.63 , Issue.4 , pp. 1348-1370
    • Weiermann, A.1
  • 71
    • 6344226984 scopus 로고    scopus 로고
    • 0
    • Mathematics and Computer science, II Versailles, 2002 Birkhäuser Basel
    • 0 in: Mathematics and Computer science, II Versailles, 2002 Trends Math. 2002 Birkhäuser Basel 527-539
    • (2002) Trends Math. , pp. 527-539
    • Weiermann, A.1
  • 72
    • 0037356292 scopus 로고    scopus 로고
    • An application of graphical enumeration to PA
    • A. Weiermann An application of graphical enumeration to PA J. Symbolic Logic 68 1 2003 5-16
    • (2003) J. Symbolic Logic , vol.68 , Issue.1 , pp. 5-16
    • Weiermann, A.1
  • 73
    • 23944433603 scopus 로고    scopus 로고
    • An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions
    • A. Weiermann An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions Discrete Math. Comput. Sci. 6 2003 133-142
    • (2003) Discrete Math. Comput. Sci. , vol.6 , pp. 133-142
    • Weiermann, A.1
  • 74
    • 0742271609 scopus 로고    scopus 로고
    • A classification of rapidly growing Ramsey functions
    • A. Weiermann A classification of rapidly growing Ramsey functions Proc. Amer. Math. Soc. 132 2 2004 553-561
    • (2004) Proc. Amer. Math. Soc. , vol.132 , Issue.2 , pp. 553-561
    • Weiermann, A.1
  • 75
    • 0031475943 scopus 로고    scopus 로고
    • Coloring rules for finite trees, and probabilities of monadic second order sentences
    • A.R. Woods Coloring rules for finite trees, and probabilities of monadic second order sentences Random Structures Algorithms 10 1997 453-485
    • (1997) Random Structures Algorithms , vol.10 , pp. 453-485
    • Woods, A.R.1
  • 76
    • 0012790163 scopus 로고
    • Asymptotic estimation of the number of trees
    • (in Japanese)
    • M. Yamashita Asymptotic estimation of the number of trees Trans. IECE Japan 62-A 1979 128-135 (in Japanese).
    • (1979) Trans. IECE Japan , vol.62 A , pp. 128-135
    • Yamashita, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.