Chapter 17 Computation models and function algebras

Studies in Logic and the Foundations of Mathematics

Volumn 140, Issue C, 1999, Pages 589-681

  Clote, P ^a  

^a UNIVERSITY OF MUNICH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33846259502     PISSN: 0049237X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S0049-237X(99)80033-0     Document Type: Chapter

References (147)

1. Ackermann W. Zum Hilbertschen Aufbau der reelen Zahlen. Math. Ann. 99 (1928) 118-133
2. 1 -formulae on finite structures. Ann. Pure Appl. Logic 24 (1983) 1-48
3. Allen B. Arithmetizing uniform NC. Ann. Pure Appl. Logic 53 1 (1991) 1-50
4. Asser G. Primitive recursive word-functions of one variable. In: Börger E. (Ed). Computation Theory and Logic. Lecture Notes in Comput. Sci. Vol. 270 (1987), Springer, Berlin 14-19
5. Babai L., and Fortnow L. Arithmetization: a new method in structural complexity theory. Computational Complexity 1 (1991) 41-66
6. 1 . J. Computer System Sci. 41 3 (1990) 274-306
7. 1 . J. Computer System Sci. 38 (1989) 150-164
8. Beame P.W., Cook S.A., and Hoover H.J. Log depth circuits for division and related problems. SIAM J. Computing 15 (1986) 994-1003
9. 164 pages. Bellantoni S. Predicative recursion and computational complexity. Technical Report 264/92 (1992), University of Toronto, Computer Science Department
10. Bellantoni S. Predicative recursion and the polytime hierarchy. In: Clote P., and Remmel J. (Eds). Feasible Mathematics II (1995), Birkhäuser 15-29
11. Bellantoni S., and Cook S. A new recursion-theoretic characterization of the polytime functions. Computational Complexity 2 (1992) 97-110
12. Translation from Zap. Nauchn. Sem. Lening. Otd. Mat. Inst. V.A. Steklova AN SSSR, Vol. 88, 30-46. Bel'tyukov A. A computer description and a hierarchy of initial Grzegorczyk classes. J. Soviet Math. 20 (1982) 2280-2289 1979
13. Bennett J.H. On Spectra. PhD thesis (1962), Princeton University, Department of Mathematics
14. Bloch S. Function-algebraic characterizations of log and polylog parallel time. Computational Complexity 4 2 (1994) 175-205
15. to appear. Bloch S., Buss J., and Goldsmith J. Sharply bounded alternation within P. submitted for journal publication. Proceedings, DMTCS'96 (1995)
16. MIT Press, Cambridge. Boppana R., and Sipser M. The complexity of finite functions. In: van Leeuwen J. (Ed). Handbook of Theoretical Computer Science Vol. A (1990), Elsevier, Amsterdam 759-804
17. Borodin A. On relating time and space to size and depth. SIAM J. Computing 6 (1973) 733-744
18. 221 pages. Buss S. Bounded Arithmetic, Vol. 3 of Studies in Proof Theory (1986), Bibliopolis
19. Buss S. The polynomial hierarchy and intuitionistic bounded arithmetic. In: Selman A.L. (Ed). Structure in Complexity Theory. Lecture Notes in Comput. Sci. Vol. 223 (1986), Springer, Berlin 77-103
20. Buss S. The boolean formula value problem is in ALOGTIME. Proceedings of the 19th Annual ACM Symposium on Theory of Computing (1987) 123-131
21. Buss S.R. The graph of multiplication is equivalent to counting. Inform. Process. Lett. 41 (1992) 199-201
22. Buss S.R. Algorithms for boolean formula evaluation and for tree contraction. In: Clote P., and Krajíček J. (Eds). Arithmetic, Proof Theory and Computational Complexity (1993), Oxford Univ. Press 96-115
23. Cai J.-Y., and Furst M.L. PSPACE survives three-bit bottlenecks. Proceedings of 3th Annual IEEE Conference on Structure in Complexity Theory (1988) 94-102
24. Chandra A., Kozen D., and Stockmeyer L.J. Alternation. J. Assoc. Comput. Machin. 28 (1981) 114-133
25. Chandra A., Stockmeyer L.J., and Vishkin U. Constant depth reducibility. SIAM J. Computing 13 (1984) 423-439
26. Church A. An unsolvable problem in elementary number theory. Amer. J. Math. 58 (1936) 345-363
27. Clote P. A first order theory for the parallel complexity class NC. Technical Report BCCS-89-01 (1989), Boston College Computer Science Department
28. Clote P. ALOGTIME and a conjecture of S.A. Cook. Ann. Math. Artif. Intell. 6 (1992) 57-106
29. Clote P. A time-space hierarchy between P and PSPACE. Math. Systems Theory 25 (1992) 77-92
30. Clote P. Polynomial size frege proofs of certain combinatorial principles. In: Clote P., and Krajíček J. (Eds). Arithmetic, Proof Theory and Computational Complexity (1993), Oxford Univ. Press 162-184
31. Clote P. A note on the relation between polynomial time functionals and Constable's class K. In: Kleine-Büning (Ed). Computer Science Logic. Lecture Notes in Comput. Sci. (1996), Springer, Berlin
32. Clote P. Nondeterministic stack register machines. Theoret. Computer Sci. A 178 (1997) 37-76
33. Clote P. A safe recursion scheme for exponential time. In: Beklemishev L. (Ed). Logical Foundations of Computer Science, LFCS'97. July 6-12, 1997 in Yaroslavl, Russia (1997), Springer
34. Clote P., Kapron B., and Ignjatovic A. Parallel computable higher type functionals. Proceedings of IEEE 34th Annual Symposium on Foundations of Computer Science. Nov. 3-5, 1993. Palo Alto, CA (1993) 72-83
35. Clote P., and Takeuti G. Exponential time and bounded arithmetic. In: Selman A.L. (Ed). Structure in Complexity Theory. Lecture Notes in Comput. Sci. Vol. 223 (1986), Springer, Berlin 125-143
36. Clote P., and Takeuti G. Bounded arithmetics for NC, ALOGTIME, L and NL. Ann. Pure Appl. Logic 56 (1992) 73-117
37. Clote P., and Takeuti G. First order bounded arithmetic and small boolean circuit complexity classes. In: Clote P., and Remmel J. (Eds). Feasible Mathematics II (1995), Birkhäuser, Boston 154-218
38. Clote P.G. A sequential characterization of the parallel complexity class NC. Technical Report BCCS-88-07 (1988), Department of Computer Science, Boston College
39. k and NC. In: Scott P.J., and Buss S.R. (Eds). Feasible Mathematics (1990), Birkhäuser 49-70
40. Cobham A. The intrinsic computational difficulty of functions. In: Bar-Hillel Y. (Ed). Logic, Methodology and Philosophy of Science II (1965), North-Holland, Amsterdam 24-30
41. Constable R. Type 2 computational complexity. 5th Annual ACM Symposium on Theory of Computing (1973) 108-121
42. Cook S. Computability and complexity of higher type functions. In: Moschovakis Y.N. (Ed). Logic from Computer Science (1992), Springer, Berlin 51-72
43. Cook S.A. The complexity of theorem proving procedures. 3rd Annual ACM Symposium on Theory of Computing (1971) 151-158
44. Cook S.A., and Kapron B.M. Characterizations of the feasible functionals of finite type. In: Scott P.J., and Buss S.R. (Eds). Feasible Mathematics (1990), Birkhäuser 71-98
45. Cook S.A., and Urquhart A. Functional interpretations of feasibly constructive arithmetic. Ann. Pure Appl. Logic 63 2 (1993) 103-200
46. Csillag P. Eine Bemerkung zur Auflösung der eingeschachtelten Rekursion. Acta Sci. Math. Szeged. 11 (1947) 169-173
47. Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Royal Soc. London A 400 (1985) 73-90
48. Fagin R. Generalized first-order spectra and polynomial-time recognizable sets. In: Karp R.M. (Ed). Complexity of Computation. SIAM-AMS Proceedings Vol. 7 (1974) 43-73
49. Journal version to appear in Theoretical Computer Science. Fagin R. Finite-model theory-a personal perspective. In: Abiteboul S., and Kanellakis P. (Eds). Proc. 1990 International Conference on Database Theory. Lecture Notes in Comput. Sci. 470 (1990), Springer, Berlin 3-24
50. Fortune S., and Wyllie J. Parallelism in random access machines. 10th Annual ACM Symposium on Theory of Computing (1978) 114-118
51. Preliminary version. Furst M., Saxe J.B., and Sipser M. Parity circuits and the polynomial time hierarchy. Mathematical Systems Theory 17 (1984) 13-27
52. Furst M., Saxe J.B., and Sipser M. Parity circuits and the polynomial time hierarchy. Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science (1984) 1981
53. Girard J.-Y., Scedrov A., and Scott P. Bounded linear logic. In: Scott P.J., and Buss S.R. (Eds). Feasible Mathematics (1990), Birkhäuser 195-210
54. Gödel K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. J. Monat. Math. Phys. 38 (1931) 173-198
55. Gödel K. Conversation with G. E. Sacks (1975), Institute for Advanced Study
56. Goerdt A. Characterizing complexity classes by general recursive definitions in higher types. Information and Computation 101 2 (1992) 202-218
57. 131 pages. Goldschlager L. Synchronous parallel computation. Technical Report 114 (1977), University of Toronto
58. Goldschlager L. A unified approach to models of synchronous parallel machines. J. Assoc. Comput. Machin. 29 4 (1982) 1073-1086
59. Grzegorczyk A. Some clases of recursive functions. Rozprawy Matematyczne (1953) 4
60. Gurevich Y. Algebras of feasible functions. Proceedings of 24th IEEE Symposium on Foundations of Computer Science (1983) 210-214
61. Gurevich Y., and Shelah S. Fixed-point extensions of first-order logic. Ann. Pure Appl. Logic 32 (1986) 265-280
62. Gurevich Y., and Shelah S. Nearly linear time. Symposium on Logical Foundations of Computer Science. Pereslavl-Zalessky, USSR. Lecture Notes in Comput. Sci. Vol. 363 (1989), Springer, Berlin 108-118
63. Hájek P., and Pudlák P. Metamathematics of First Order Arithmetic (1992), Springer, Berlin
64. Handley W., Paris J.B., and Wilkie A.J. Characterizing some low arithmetic classes. Theory of Algorithms. Colloquia Societatis Janos Bolyai (1984), Akademie Kyado, Budapest 353-364
65. 3 yields ALINTIME, Submitted (1994)
66. n , the semi-group of binary relations on {0, 1, ..., n-1}, Submitted (1994)
67. Hardy G.H. A theorem concerning the infinite cardinal numbers. Quart. J. Math. (1904) 87-94
68. Harnik V. Provably total functions of intuitionistic bounded arithmetic. J. Symbolic Logic 57 2 (1992) 466-477
69. Harrow K. Small Grzegorczyk classes and limited minimum. Z. Math. Logik 21 (1975) 417-426
70. Harrow K. Equivalence of some hierarchies of primitive recursive functions. Z. Math. Logik 25 (1979) 411-418
71. Hilbert D. Über das Unendliche. Math. Ann. 95 (1925) 161-190
72. Hillis W.D., and Steele Jr. G.L. Data parallel algorithms. Commun. ACM 29 12 (1986) 1170-1183
73. Hoover H.J. Computational models for feasible real analysis. In: Buss S.R., and Scott P.J. (Eds). Feasible Mathematics (1990), Birkhäuser 221-238
74. Immerman N. Languages that capture complexity classes. SIAM J. Comput. 16 (1987) 760-778
75. Immerman N. Expressibility and parallel complexity. SIAM J. Comput. 18 3 (1989) 625-638
76. Johannsen J. Schwache Fragmente der Arithmetik und Schwellwertschaltkreise beschränkter Tiefe. PhD thesis (1996), Universität Erlangen-Nürnberg
77. Jones J.P., and Matijasevič Y. A new representation for the symmetric binomial coefficient and its applications. Ann. Sci. Math. Quebec 6 1 (1982) 81-97
78. Jones N.D., and Selman A.L. Turing machines and the spectra of first-order formulas. J. Symbolic Logic 39 (1974) 139-150
79. (In Hungarian with German abstract.). Kálmar L. Egyszerü példa eldönthetetlen arimetikai problémára. Mate és Fizikai Lapok 50 (1943) 1-23
80. Kannan R. Towards separating nondeterministic time from deterministic time. Proceedings of 22nd IEEE Symposium on Foundations of Computer Science (1981) 235-243
81. Preliminary version appeared IEEE Symposium on Foundations of Computer Science, 342-347, 1991. Kapron B., and Cook S. A new characterization of type-2 feasibility. SIAM J. Comput. 25 1 (1996) 117-132
82. Vol. Karp R.M., and Ramachandran V. Parallel algorithms for shared-memory machines. MIT Press, Cambridge. In: van Leeuwen A.J. (Ed). Handbook of Theoretical Computer Science (1990), Elsevier, Amsterdam 871-942
83. Kleene S.C. General recursive functions of natural numbers. Math. Ann. 112 (1936) 727-742
84. Kleene S.C. Lambda-definability and recursiveness. Duke Math. J. 2 (1936) 340-353
85. Ko K.-I. Applying techniques of discrete complexity theory to numerical computation. In: Book R.V. (Ed). Studies in Complexity Theory (1986), Wiley 1-62
86. Krajíček J. Bounded Arithmetic, Propositional Logic, and Complexity Theory (1995), Cambridge Univ. Press
87. Kreisel G., Lacombe D., and Shoenfield J.R. Partial recursive functionals and effective operations. In: Heyting A. (Ed). Constructivity in Mathematics: Proceedings of a colloquium. Amsterdam (1957), North-Holland, Amsterdam 195-207
88. Kutyl{middle dot}owski M. Finite automata, real time processes and counting problems in bounded arithmetics. J. Symbolic Logic 53 1 (1988) 243-258
89. Leivant D. Stratified polymorphism. Proceedings of IEEE 4th Annual Symposium on Logic in Computer Science (1989) 39-47
90. Leivant D. Journal version: Finitely stratified polymorphism. Information and Computation 93 (1991) 93-113
91. Leivant D. A foundational delineation of computational feasibility. Proceedings of IEEE 6th Annual Symposium on Logic in Computer Science (1991)
92. Leivant D. Stratified functional programs and computational complexity. Conference Record of the Twentieth Annual ACM Symposium on Principles of Programming Languages (1993)
93. Leivant D. Ramified recurrence and computational complexity I: word recurrence and poly-time. In: Clote P., and Remmel J. (Eds). Feasible Mathematics II (1994), Birkhäuser 320-343
94. Leivant D. Logic and Computational Complexity. Lecture Notes in Comput. Sci. Vol. 960 (1995), Springer, Berlin
95. Leivant D., and Marion J.-Y. Lambda-calculus characterizations of poly-time. Fundamenta Informaticae 19 (1993) 167-184
96. Lind J.C. Computing in logarithmic space. Technical Report Project MAC Technical Memorandum 52 (1974), Massachusetts Institute of Technology
97. Translation in Mathematical Notes of the Academy of Sciences of the USSR, Plenum Publishing Company. Marčenkov S.S. A superposition basis in the class of Kalmar elementary functions. Mat. Zametki 27 3 (1980) 321-332
98. Mehlhorn K. Polynomial and abstract subrecursive classes. J. Computer System Sci. 12 (1976) 147-178
99. Meyer A.R., and Ritchie D. The complexity of loop programs. Proc. ACM Nat. Conf. (1967) 465-469
100. Monien B. A recursive and grammatical characterization of exponential time languages. Theoret. Computer Sci. 3 (1977) 61-74
101. Muchnick S.S. The vectorized Grzegorczyk hierarchy. Z. Math. Logik. 22 (1976) 441-480
102. Müller H. Klassifizierungen der primitiv rekursiven Funktionen. PhD thesis (1974), Universität Münster
103. Translated in Soviet Math. Dokl., 11, pp. 1462-1465. Nepomnjascii V.A. Rudimentary predicates and turing calculations. Dokl. Acad. Nauk SSSR 195 (1970) 29-35
104. Nerode A., Remmel J., and Scedrov A. Polynomially graded logic I - a graded version of system T. Proceedings of IEEE 4th Annual Symposium on Logic in Computer Science (1989)
105. Nguyen A.P. A formal system for linear space reasoning. Technical Report 300/96 (1996), University of Toronto
106. Niggl K.-H. Subrecursive hierarchies on Scott domains. Arch. Math. Logic 32 (1993) 239-257
107. O'Donnell M.J. Equational Logic as a Programming Language (1985), MIT Press
108. 0 . Talk at meeting LCC, Indianapolis, organizer D. Leivant (1994) 13-16
109. Otto J. Complexity Doctrines. PhD thesis (1995), Department of Mathematics and Statistics, McGill University
110. Otto J. Half tiers and linear space (and time). Talk at DIMACS Workshop on Computational Complexity and Programming Languages, organized by B. M. Kapron and J. Royer. July 25-26, 1996 (1996)
111. Pakhomov S.V. Machine independent description of some machine complexity classes (in Russian). Issledovanija po Konstrukt. Matemat. i Mat. Logike VIII (1979), LOMI 176-185
112. Paris J.B., and Wilkie A.J. Counting problems in bounded arithmetic. In: di Prisco C.A. (Ed). Methods in Mathematical Logic. Lecture Notes in Mathematics, Proceedings of Logic Conference held in Caracas (1983), Springer, Berlin 317-340
113. Péter R. Über die mehrfache Rekursion. Math. Ann. 113 (1936) 489-526
114. 0 and ALOGTIME, Typeset manuscript (1995)
115. Pudlák P. Complexity theory and genetics. Proceedings of 9th Annual IEEE Conference on Structure in Complexity Theory (1994)
116. Ritchie R.W. Classes of predictably computable functions. Trans. Amer. Math. Soc. 106 (1963) 139-173
117. Robinson J. Primitive recursive functions. Bull. Amer. Math. Soc. 53 (1947) 923-943
118. Robinson J. Definability and decision problems in arithmetic. J. Symbolic Logic 14 (1949) 98-114
119. Rose H.E. Subrecursion: Function and Hierarchies. Oxford Logic Guides Vol. 9 (1984), Clarendon Press, Oxford
120. Royer J.S. Semantics vs. Syntax vs. Computation, Typescript, November 29, 1994 (1994)
121. Ruzzo W.L. On uniform circuit complexity. J. Comput. System Sci. 22 (1981) 365-383
122. Savitch W.J. Relationship between nondeterministic and deterministic tape complexities. J. Comput. System Sci. 4 (1970) 177-192
123. Schnorr C.P. Satisfiability is quasilinear complete in NQL. J. Assoc. Comput. Machin. 25 1 (1978) 136-145
124. Scholz H. Ein ungelöstes Problem in der symbolischen Logik. J. Symbolic Logic 17 (1952) 160
125. Schwichtenberg H. Rekursionszahlen und die Grzegorczyk-Hierarchie. Arch. Math. Logik. 12 (1969) 85-97
126. Schwichtenberg H. Primitive recursion on the partial continuous functionals. In: Broy M. (Ed). Informatik und Mathematik (1991), Springer, Berlin 251-259
127. Seth A. There is no recursive axiomatization for feasible functionals of type 2. Proceedings of IEEE 7th Annual Symposium on Logic in Computer Science (1992) 286-295
128. Seth A. Some desirable conditions for feasible functionals of type 2. Proceedings of IEEE 8th Annual Symposium on Logic in Computer Science (1993)
129. Seth A. Turing machine characterizations of feasible functionals of all finite types. In: Clote P., and Remmel J. (Eds). Feasible Mathematics II (1994), Birkhäuser 407-428
130. Shiloach Y., and Vishkin U. Finding the maximum, merging and sorting in a parallel computation model. J. Algorithms 3 (1982) 57-67
131. Shor P. Algorithms for quantum computation: discrete log and factoring. Proceedings of IEEE 35th Annual Symposium on Foundations of Computer Scienc (1994)
132. Simmons H. The realm of primitive recursion. Arch. Math. Logic 27 (1988) 177-188
133. Skolem Th. Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. Skrifter utgit av Videnskapsselskapet, I. Mate, Klasse, 6, Oslo (1923)
134. Smullyan R. Theory of Formal Systems. Annals of Mathematical Studies Vol. 47 (1961), Princeton Univ. Press
135. Stockmeyer L., and Vishkin U. Simulation of parallel random access machines by circuits. SIAM J. Computing 13 (1984) 409-422
136. 0 . In: Leivant D. (Ed). Logic and Computational Complexity. Lecture Notes in Comput. Sci. Vol. 960 (1995), Springer, Berlin 221-252
137. Thompson D.B. Subrecursiveness: machine independent notions of computability in restricted time and storage. Math. Systems Theory 6 (1972) 3-15
138. Townsend M. Complexity for type-2 relations. Notre Dame J. Formal Logic 31 (1990) 241-262
139. Turing A.M. On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc., Ser. 2 42 (1936-1937) 230-265
140. Valiant L. The complexity of computing the permanent. Theoret. Computer Sci. 8 (1979) 189-201
141. MIT Press, Cambridge. van Emde Boas P. Machine models and simulations. In: van Leeuwen J. (Ed). Handbook of Theoretical Computer Science Vol. A (1990), Elsevier, Amsterdam 1-66
142. Vollmer H., and Wagner K. Recursion theoretic characterizations of complexity classes of counting functions. Theoret. Computer Sci. 163 (1996) 245-258
143. Wagner K. Bounded recursion and complexity classes. Lecture Notes in Comput. Sci. Vol. 74 (1979), Springer, Berlin 492-498
144. Wagner K., and Wechsung G. Computational Complexity (1986), Reidel Publishing Co
145. Wilkie A.J. Modèles non standard de l'arithmétique, et complexité algorithmique. In: Wilkie A.J., and Ressayre J.-P. (Eds). Modèles Non Standard en Arithmétique et en Théorie des Ensembles (1983), Publications Mathématiques de l'Université Paris VII 5-45
146. Woods A. Bounded arithmetic formulas and Turing machines of constant alternation. In: Paris J.B., Wilkie A.J., and Wilmers G.M. (Eds). Logic Coloquium 1984 (1986), North-Holland, Amsterdam
147. Wrathall C. Complete sets and the polynomial time hierarchy. Theoret. Computer Sci. 3 (1976) 23-33