Chapter 12 Computable rings and fields

Studies in Logic and the Foundations of Mathematics

Volumn 140, Issue C, 1999, Pages 363-447

  Stoltenberg Hansen, V ^a   Tucker, J V ^b  

^a UPPSALA UNIVERSITY   (Sweden)

^b SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956697920     PISSN: 0049237X     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S0049-237X(99)80028-7     Document Type: Chapter

References (235)

1. Abel N.H. Mémoire sur les Équationes Résolubles Algébriques, ou l'on Démontre l'Impossibilité de la Résolution de l'Équation Générale du Cinquième Degré, Christiania (1824)
2. Abel N.H. Mémoire sur les équationes résolubles algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré. In: Sylow L., and Lie S. (Eds). Ouevres Complètes (1881), Cristiania 28-33
3. Aberth O. Computable Analysis (1980), McGraw-Hill, New York
4. Adams W.W., and Loustaunau P. An Introduction to Gröbner Bases. Graduate Studies in Mathematics Vol. 3 (1994), Amer. Math. Soc., Providence, RI
5. In: Adian S.I., Boone W.W., and Higman G. (Eds). Word Problems II (1980), North-Holland, Amsterdam
6. Alton D., and Madison E.W. Computability of Boolean algebras and their extensions. Ann. Math. Logic 6 (1973) 95-128
7. Artin E., and Schreier O. Algebraischen Konstruktion reeler Körper. Abh. Math. Sem. Univ. Hamburg 5 (1927) 85-99
8. Ash C.J., and Downey R.G. Decidable subspaces and recursively enumerable subspaces. J. Symbolic Logic 49 (1984) 1137-1145
9. Asveld P.R.J., and Tucker J.V. Complexity theory and the operational structure of algebraic programming systems. Acta Inform. 17 (1982) 451-476
10. Atiyah M., and MacDonald I. Introduction to Commutative Algebra (1969), Addison-Wesley, New York
11. Avenhaus J., and Madlener K. Subrekursive Komplexität bei Gruppen, I. Gruppen mit vorgeschriebener Komplexität. Acta Inform. 9 (1977) 87-104
12. Avenhaus J., and Madlener K. Subrekursive Komplexität bei Gruppen, II. Der Einbettungssatz von Higman für entscheidbare Gruppen. Acta Inform. 9 (1978) 183-193
13. Avenhaus J., and Madlener K. The Nielsen reduction and P-complete problems in free groups. Theoret. Comput. Sci. 32 (1984) 61-76
14. Avenhaus J., and Madlener K. On the complexity of intersection and conjugacy problems in free groups. Theoret. Comput. Sci. 32 (1984) 279-295
15. Ayoub C.W. On constructing bases for ideals in polynomial rings over the integers. J. Number Theory 17 (1983) 204-225
16. Babbage C. Collected Works. Volume III: Analytical Engine (1989), W. Pickering, London
17. Barwise K.J. Admissible Sets and Structures. Perspectives in Mathematical Logic (1975), Springer, Berlin
18. Barwise K.J. In: Barwise K.J. (Ed). Handbook of Mathematical Logic. Studies in Logic Vol. 90 (1977), North-Holland, Amsterdam
19. Baumslag G., Cannonito F.B., and Miller III C.F. Computable algebra and group embeddings. J. Algebra 69 (1981) 186-212
20. In: Baumslag G., and Miller III C.F. (Eds). Algorithms and Classification in Combinatorial Group Theory (1992), Springer, Berlin
21. Baur W. Rekursive Algebren mit Kettenbedingungen. Zeits. Math. Logik Grundl. Math. 20 (1974) 37-46
22. Becker T., and Weispfenning V. Gröbner Bases (1993), Springer, Berlin
23. Beeson M.J. Foundations of Constructive Mathematics (1985), Springer, Berlin
24. Bergstra J.A., and Tucker J.V. A characterisation of computable data types by means of a finite equational specification method. In: de Bakker J.W., and van Leeuwen J. (Eds). Automata, Languages and Programming (ICALP), 7th Colloquium, Noordwijkerhout. Lecture Notes in Comput. Sci. Vol. 81 (1980), Springer, Berlin 76-90
25. Bergstra J.A., and Tucker J.V. A natural data type with a finite equational final semantics specification, but no effective equational initial specification. Bulletin of the EATCS 11 (1980) 23-33
26. Bergstra J.A., and Tucker J.V. The completeness of the algebraic specification methods for data types. Inform. and Control 54 (1982) 186-200
27. Bergstra J.A., and Tucker J.V. Initial and final algebra semantics for data type specifications: two characterisation theorems. SIAM J. Comput. 12 (1983) 366-387
28. Bergstra J.A., and Tucker J.V. Algebraic specifications of computable and semicomputable data types. Theoret. Comput. Sci. 50 (1987) 137-181
29. Bergstra J.A., and Tucker J.V. Equational specifications for computable data types: 6 hidden functions suffice and other sufficiency bounds. In: Meinke K., and Tucker J.V. (Eds). Many Sorted Logic and its Applications (1993), Wiley 89-102
30. Bergstra J.A., and Tucker J.V. The data type variety of stack algebras. Ann. Pure Appl. Logic 73 (1995) 11-36
31. Bergstra J.A., and Tucker J.V. Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. J. ACM 42 (1995) 1194-1230
32. Blanck J. Domain representability of metric spaces. Ann. Pure Appl. Logic 83 (1996) 225-247
33. Blum L., Cucker F., Shub M., and Smale S. Complexity and Real Computation (1998), Springer, Berlin
34. Blum L., Shub M., and Smale S. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions, and universal machines. Bull. Amer. Math. Soc. 21 (1989) 1-46
35. Boone W.W. The word problem. Ann. of Math. 70 (1959) 207-265
36. Boone W.W. Word problems and recursively enumerable degrees of unsolvability, A sequel on finitely presented groups. Ann. of Math. 84 (1966) 49-84
37. Boone W.W. Word problems and recursively enumerable degrees of unsolvability. An emendation. Ann. of Math. 94 (1971) 389-391
38. In: Boone W.W., Cannonito F.B., and Lyndon R.C. (Eds). Word Problems (1973), North-Holland, Amsterdam
39. Boone W.W., and Higman G. Algebraic characterization of the solvability of the word problem. J. Austral. Math. Soc. 18 (1974) 41-53
40. Bourbaki N. Algèbre Commutative: I. Modules Plats (1961), Hermann, Paris
41. Bridges D., and Richman F. Varieties of Constructive Mathematics. London Mathematical Society Lecture Notes Series Vol. 97 (1987), Cambridge Univ. Press, Cambridge
42. Burris S., and Sankappanavar H.P. A Course in Universal Algebra (1981), Springer, Berlin
43. Cannonito F.B. Hierarchies of computable groups and the word problem. J. Symbolic Logic 31 (1966) 376-392
44. Cannonito F.B. The algebraic invariance of the word problem in groups. In: Boone W.W., Cannonito F.B., and Lyndon R.C. (Eds). Word Problems. Studies in Logic Vol. 71 (1973), North-Holland, Amsterdam 349-364
45. Cannonito F.B., and Gatterdam R.W. The computability of group constructions, part I. In: Boone W.W., Cannonito F.B., and Lyndon R.C. (Eds). Word Problems. Studies in Logic Vol. 71 (1973), North-Holland, Amsterdam 365-400
46. Cenzer D., and Remmel J.B. Polynomial time versus recursive models. Ann. Pure Appl. Logic 54 (1991) 17-58
47. Cenzer D., and Remmel J.B. Polynomial time abelian groups. Ann. Pure Appl. Logic 56 (1992) 313-363
48. Chandler B., and Magnus W. The History of Combinatorial Group Theory: A Case Study in the History of Ideas (1982), Springer, New York
49. Chevalley C. Theory of Lie Groups (1946), Princeton Univ. Press, Princeton, N.J.
50. Clapham C.R.J. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proc. London Math. Soc. 14 3 (1964) 633-676
51. Cohn P.M. Algebra Vol. 2 (1977), Wiley, Chichester
52. Cohn P.M. Universal Algebra (1981), D. Reidel, Dordrecht
53. Collins D.J. Truth table degrees and the Boone groups. Ann. of Math. 94 (1971) 392-396
54. Cutland N.J. Computability: An Introduction to Recursive Function Theory (1980), Cambridge Univ. Press, Cambridge
55. Davis M. Unsolvable problems. In: Barwise K.J. (Ed). Handbook of Mathematical Logic. Studies in Logic Vol. 90 (1977), North-Holland, Amsterdam 567-594
56. Reprinted. Dedekind R. Was sind und was sollen die Zahlen? (1888), Vieweg, Braunschweig
57. Dedekind R. Essays on the Theory of Numbers (1963), Dover, New York
58. Dehn M. Über die Topologie des dreidimensionalen Raumes. Math. Ann. 69 (1910) 137-168
59. Dehn M. Über unendliche diskontinuerliche Gruppen. Math. Ann. 71 (1911) 116-144
60. Dehn M. Papers on Group Theory and Topology (1987), Springer, New York
61. Dobrica V.P. Constructivizable abelian groups. Sibirsk. Mat. Zh. 22 (1981) 208-213
62. Dobrica V.P., Nurtazin A.T., and Khisamiev N.G. Constructive periodic abelian groups. Sibirsk. Mat. Zh. 19 (1978) 1260-1265
63. Domanski B., and Anshel M. The complexity of Dehn's algorithm for word problems in groups. J. Algorithms 6 (1985) 543-549
64. Dyck W. Gruppentheoretische Studien. Math. Ann. 20 (1882) 1-44
65. Edalat A. Domain theory and integration. Theoret. Comput. Sci. 151 (1995) 163-193
66. Edalat A. Dynamical systems, measures and fractals via domain theory. Inform. and Comput. 120 (1995) 32-48
67. Edalat A. Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbolic Logic 3 (1997) 401-452
68. Edwards H.M. Galois Theory. Graduate Texts in Mathematics Vol. 101 (1984), Springer, Berlin
69. Enderton H.B. Elements of recursion theory. In: Barwise K.J. (Ed). Handbook of Mathematical Logic (1977), North-Holland, Amsterdam 527-566
70. Ershov Yu.L. Theorie der Numerierungen I. Zeits. Math. Logik Grundl. Math. 19 (1973) 289-388
71. Ershov Yu.L. Theorie der Numerierungen II. Zeits. Math. Logik Grundl. Math. 21 (1975) 473-584
72. Ershov Yu.L. Theorie der Numerierung III. Zeits Math. Logik Grundl. Math. 23 (1977) 289-371
73. (in Russian). Ershov Yu.L. Theory of Numberings (1977), Nauka, Moscow
74. Fedoryaev S.T. Countability of widths of algebraic reducibility structures for models in some classes. Siberian Adv. Math. 3 (1993) 81-102
75. Feferman S. Impredicativity of the existence of the largest divisible sub-group of an abelian p-group. In: Saracino D.H., and Weispfenning V.B. (Eds). Model Theory and Algebra. Lecture Notes in Mathematics Vol. 498 (1975), Springer, Berlin 117-130
76. Fenstad J.E. General Recursion Theory: An Axiomatic Approach (1980), Springer, Berlin
77. Fridman A.A. Degrees of unsolvability of identity in finitely presented groups. Soviet Math. 3 (1962) 1733-1737
78. Fröhlich A., and Shepherdson J.C. On the factorization of polynomials in a finite number of steps. Math. Z. 62 (1955) 331-334
79. Fröhlich A., and Shepherdson J.C. Effective procedures in field theory. Philos. Trans. Royal Soc. London Ser. A 248 (1956) 407-432
80. Galois E. In: Liouville J. (Ed). Mémoire sur les conditions de résolubilité des équations par radicaux. dated 6 January 1831. Journal de Mathématiques 11 (1846) 381-444
81. Gandy R. The confluence of ideas in 1936. In: Herkin R. (Ed). The Universal Turing Machine (1988), Oxford Univ. Press, Oxford 55-111
82. Gatterdam R.W. The Higman theorem for primitive recursive groups - a preliminary report. In: Boone W.W., Cannonito F.B., and Lyndon R.C. (Eds). Word Problems. Studies in Logic Vol. 71 (1973), North-Holland, Amsterdam 421-425
83. Goncharov S.S. Computable univalent numerations. Algebra i Logika 19 (1980) 507-551
84. Goncharov S.S. The problem of the number of nonautoequivalent constructivisations. Algebra i Logika 19 (1980) 621-639
85. Goncharov S.S. Groups with a finite number of constructivisations. Soviet Math. Doklady 25 (1981) 58-61
86. Goodstein R.L. Recursive Analysis (1961), North-Holland, Amsterdam
87. Grätzer G. Universal Algebra (1979), Springer, Berlin
88. Guichard D. Automorphisms of substructure lattices in recursive algebra. Ann. Pure Appl. Logic 25 (1983) 47-58
89. In: Harrington L.A., Morley M.D., Scedrov A., and Simpson S.G. (Eds). Harvey Friedman's Research on the Foundations of Mathematics (1985), North-Holland, Amsterdam
90. Leipzig. Hensel K. Theorie der Algebraischen Zahlen (1908)
91. Hentzel I.R., and Jacobs D.P. Complexity and unsolvability properties of nilpotency. SIAM J. Comput. 19 (1990) 32-43
92. Henzelt K. Zur Theorie den Polynomideale und Resultanten (Bearbeitet von E. Noether). Math. Ann. 88 (1922) 53-79
93. Hermann G. Die Frage der endlichen vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95 (1926) 736-788
94. Higman G. Subgroups of finitely presented groups. Proc. Royal Soc. London Ser. A 262 (1961) 455-474
95. Hingston P. Effective decomposition in Noetherian rings. In: Crossley J.N. (Ed). Aspects of Effective Algebra (Proceedings of a Conference at Monash University. Australia, 1-4 August, 1979 (1981), Upside Down A Book Company, Steel's Creek, Australia 122-127
96. Jacobsson C., and Stoltenberg-Hansen V. Poincaré-Betti series are primitive recursive. J. London Math. Soc. 31 (1985) 1-9
97. Kalantari I., and Retzlaff A. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. J. Symbolic Logic 42 (1977) 481-491
98. Kaye R. Models of Peano Arithmetic. Oxford Logic Guide Vol. 15 (1991), Oxford Univ. Press, Oxford
99. Khisamiev N.G. Nonconstructivizibility of some ordered fields of real numbers. Sibirsk. Mat. Zh. 28 (1987) 193-195
100. Khisamiev N.G. Constructive abelian p-groups. Siberian Adv. Math. 2 (1992) 68-113
101. Kleene S.C. Origins of recursive function theory. Ann. Hist. Comput. 3 (1981) 52-67
102. Kreitz C., and Weihrauch K. A unified approach to constructive and recursive analysis. In: Börger E., Oberschelp W., Richter M.M., Schinzel B., and Thomas W. (Eds). Computation and Proof Theory. Lecture Notes in Mathematics Vol. 1104 (1984), Springer, Berlin 259-278
103. Kronecker L. Grundzüge einer arithmetischen Theorie der algebraischen Grössen. J. Reine Angew. Math. 92 (1882) 1-122
104. Krull W. Galoische Theorie der unendlichen algebraischen Erweiterungen. Math. Ann. 100 (1928) 687-698
105. Krull W. Über Polynomzerlegung mit endlich vielen Schritten I. Math. Z. 59 (1953) 57-60
106. Krull W. Über Polynomzerlegung mit endlich vielen Schritten II. Math. Z. 59 (1953) 296-298
107. Krull W. Über Polynomzerlegung mit endlich vielen Schritten III. Math. Z. 60 (1954) 109-111
108. Kudinov O.V. Algebraic dependences and reducibilities of constructivizations in universal domains. Siberian Adv. Math. 3 (1993) 121-128
109. Manuscript. Kudinov O.V. An integral domain with finite algorithmic dimension (1996)
110. Kuznetsov A.V. Algorithms as operations in algebraic systems. Uspekhi Mat. Nauk 13 (1958) 240-241
111. König G. Einleitung in die Allgemeine Theorie der Algebraischen Größen, Leipzig (1903)
112. Lachlan A.H., and Madison E.W. Computable fields and arithmetically definable ordered fields. Proc. Amer. Math. Soc. 24 (1970) 803-807
113. Lagrange J.L. Réflexions sur la résolution algébrique des équations. Nouv. Mém. Acad. Berlin, pour les annés 1770/71, Berlin (1772) 203-421
114. Lallement G. Semigroups and Combinatorial Applications (1979), Wiley, Chichester
115. Lang S. Algebra (1965), Addison-Wesley, Reading, MA
116. Lazard D. Algorithmes fondamentaux en algèbre commutative. Asterisque 38-39 (1976) 131-138
117. Lin C. The effective content of Ulm's theorem. In: Crossley J.N. (Ed). Aspects of Effective Algebra (Proceedings of a Conference at Monash University. Australia, 1-4 August, 1979 (1981), Upside Down A Book Company, Steel's Creek, Australia 147-160
118. Lin C. Recursively presented Abelian groups: Effective p-group theory I. J. Symbolic Logic 46 (1981) 617-624
119. Lipton R.J., and Zalcstein Y. Word problems solvable in logspace. J. Assoc. Comput. Mach. 24 (1977) 522-526
120. Lyndon R.C., and Schupp P.E. Combinatorial Group Theory (1977), Springer, Berlin
121. Machtey M., and Young P.R. An Introduction to the General Theory of Algorithms (1978), Elsevier, Amsterdam
122. MacIntyre A. On algebraically closed groups. Ann. of Math. 96 (1972) 53-97
123. Madison E.W. A note on computable real fields. J. Symbolic Logic 35 (1970) 239-241
124. Madison E.W. Some remarks on computable (non-archimedian) ordered fields. J. London Math. Soc. 4 (1971) 304-308
125. Magnus W., Karass A., and Solitar D. Combinatorial Group Theory (1976), Dover
126. Mal'cev A.I. Constructive algebra I. Russian Math. Surv. 16 (1961) 77-129
127. Mal'cev A.I. Algorithms and Recursive Functions (1970), Wolters-Noordhoff, Groningen
128. Mal'cev A.I. The Metamathematics of Algebraic Systems. Collected Papers: 1936-1967 (1971), North-Holland, Amsterdam
129. Mal'cev A.I. Constructive algebras I. In: Wells III B.F. (Ed). The Metamathematics of Algebraic Systems. Collected Papers: 1936-1967 (1971), North-Holland, Amsterdam 148-212
130. Mal'cev A.I. Recursive abelian groups. In: Wells III B.F. (Ed). The Metamathematics of Algebraic Systems. Collected Papers: 1936-1967 (1971), North-Holland, Amsterdam 282-286
131. Markov A.A. On the impossibility of certain algorithms in the theory of associative systems. Dokl. Akad. Sci. USSR 55 (1947) 583-586
132. McAloon K. On the complexity of models of arithmetic. J. Symbolic Logic 47 (1982) 403-415
133. McKenzie R.N., McNulty G.F., and Taylor W.E. Algebras, Lattices, Varieties Vol. 1 (1987), Wadsworth and Brooke/Cole, Monterey
134. Meinke K., and Tucker J.V. Universal algebra. In: Abramsky S., Gabbay D., and Maibaum T.S.E. (Eds). Handbook of Logic in Computer Science, Vol. 1: Mathematical Structures (1992), Oxford Univ. Press, Oxford 189-411
135. Meseguer J., and Goguen J.A. Initiality induction and computability. In: Nivat M., and Reynolds J. (Eds). Algebraic Methods in Semantics (1985), Cambridge Univ. Press, Cambridge 459-541
136. Metakides G., and Nerode A. Recursively enumerable vector spaces. Ann. Math. Logic 11 (1977) 147-171
137. Metakides G., and Nerode A. Effective content of field theory. Ann. Math. Logic 17 (1979) 289-320
138. Metakides G., and Nerode A. Recursion theory on fields and abstract dependence. J. Algebra 65 (1980) 36-59
139. Metakides G., and Nerode A. The introduction of non-recursive methods into mathematics. In: Troelstra A.S., and van Dalen D. (Eds). The L. E. J. Brouwer Centenary Symposium (1982), North-Holland, Amsterdam 319-335
140. Miller III C.F. On Group-Theoretic Decision Problems and their Classification (1971), Princeton Univ. Press
141. Miller III C.F. Decision problems for groups. Surveys and reflections. In: Baumslag G., and Miller III C.F. (Eds). Algorithms and Classification in Combinatorial Group Theory. MSRI Publications Vol. 23 (1992), Springer, Berlin
142. Mines R., Richman F., and Ruitenburg W. A Course in Constructive Algebra (1988), Springer, New York
143. Moldestad J., Stoltenberg-Hansen V., and Tucker J.V. Finite algorithmic procedures and inductive definability. Math. Scand. 46 (1980) 62-76
144. Moldestad J., Stoltenberg-Hansen V., and Tucker J.V. Finte algorithmic procedures and computation theories. Math. Scand. 46 (1980) 77-94
145. Molk J. Sur une notion qui comprend celle de la divisibilité et sur la théorie de l'elimination. Acta Math. 6 (1885) 1-166
146. Moschovakis Y.N. Recursive metric spaces. Fund. Math. 55 (1964) 215-238
147. Moss L., Meseguer J., and Goguen J.A. Final algebras, cosemicomputable algebras, and degrees of unsolvability. Theoret. Comput. Sci. 100 (1992) 267-302
148. Nerode A., and Remmel J.B. Recursion theory on matroids II. In: Chong C.T., and Wicks M.S. (Eds). Proceedings of the Singapore Logic Symposium. Southeast Asian Conference on Logic (1983), North-Holland, Amsterdam 133-184
149. Nerode A., and Remmel J.B. A survey of r.e. substructures. In: Nerode A., and Shore R.A. (Eds). Recursion Theory. Proceedings of Symposia in Pure Mathematics 42 (1985), Amer. Math. Soc., Providence, RI 323-375
150. Nerode A., and Remmel J.B. Complexity theoretic algebra I: Vector spaces over finite fields. Proceedings of the 2nd Annual Conference on Structures in Complexity Theory. IEEE, Silver Spring (1987) 218-239
151. Nerode A., and Remmel J.B. Complexity theoretic algebra II: Boolean algebras. Ann. Pure Appl. Logic 44 (1989) 71-99
152. Nerode A., and Remmel J.B. Complexity-theoretic algebra: Vector space bases. In: Buss S., and Scott P. (Eds). Feasible Mathematics (1990), Springer, Berlin 293-319
153. Nerode A., and Smith R. The undecidability of the lattice of r.e. subspaces. In: Arruda A.I., DiCosta N.C.A., and Sette A.M. (Eds). Proceedings of the Third Brazilian Conference on Mathematical Logic (1982) 245-252
154. Neumann B.H. The isomorphims problem for algebraically closed groups. In: Boone W.W., Cannonito F.B., and Lyndon R.C. (Eds). Word Problems (1973), North-Holland, Amsterdam 553-562
155. (in Russian). Novikov P.S. On the algorithmic unsolvability of the word problem for group theory. Trudy Mat. Inst. Steklov Vol. 44 (1955) 143
156. Novy L. Origins of Modern Algebra (1973), Noordhoff, Leiden
157. Odifreddi P. Classical Recursion Theory. Studies in Logic Vol. 125 (1989), North-Holland, Amsterdam
158. Peacock G. A Treatise on Algebra (1830), J. & J. J. Deighton, Cambridge
159. Phillips I.C.C. Recursion theory. In: Abramsky S., Gabbay D., and Maibaum T.S.E. (Eds). Handbook of Logic in Computer Science, Vol. 1: Mathematical Structures (1992), Oxford Univ. Press, Oxford 79-187
160. Post E.L. Recursive unsolvability of a problem of Thue. J. Symbolic Logic 12 (1947) 1-11
161. Rabin M.O. Computable algebra, general theory and theory of computable fields. Trans. Amer. Math. Soc. 95 (1960) 341-360
162. Rabin M.O. Computable algebraic systems. Summer Institute for Symbolic Logic, Cornell University, 1957. 2nd ed. (1960), Institute for Defence Analyses 134-138
163. Reid C. Hilbert (1970), Springer, Berlin
164. Remmel J.B. Recursive Boolean algebras. In: Monk J.D. (Ed). Handbook of Boolean Algebra Vol. III (1989), North-Holland, Amsterdam 1099-1165
165. Retzlaff A. Direct summands of r.e. vector spaces. Zeits. Math. Logik Grundl. Math. 25 (1979) 363-372
166. Richman F. Constructive aspects of Noetherian rings. Proc. Amer. Math. Soc. 44 (1974) 436-441
167. Rogers H. Theory of Recursive Functions and Effective Computability (1967), McGraw-Hill, New York
168. Rotman J. An Introduction to the Theory of Groups (1995), Allyn and Bacon, Boston
169. Reprinted. Ruffini P. Teoria generale delle equazioni, in cui si dimostra impossibile la soluzion algebraica delle equazioni generali di grado superiore alo quarto 1779. In: Bortolotti E. (Ed). Opere matermatiche (1779), Tomo primo, Palermo 1-324
170. Sacks G.E. Higher Recursion Theory. Perspectives in Mathematical Logic (1990), Springer, Berlin
171. Seidenberg A. Constructions in algebra. Trans. Amer. Math. Soc. 197 (1974) 273-313
172. Seidenberg A. What is Noetherian?. Rendiconti del Seminario Matematico e Fisico di Milano 44 (1974) 55-61
173. Seidenberg A. Survey of constructions in Noetherian rings. In: Nerode A., and Shore R.A. (Eds). Recursion Theory. Proceedings of Symposia in Pure Mathematics Vol. 42 (1985), Amer. Math. Soc., Providence, RI 377-386
174. Shepherdsonn J.C. A non-standard model for a free variable fragment of number theory. Bull Acad. Polon. Sci., Sér. Sci., Math., Astron. Phys. 12 (1964) 79-86
175. Shepherdson J.C. Non-standard models for fragments of number theory. In: Addison J.W. (Ed). Theory of Models (1965), North-Holland, Amsterdam 342-358
176. Shepherdson J.C. Algorithmic procedures, generalised Turing algorithms, and elementary recursion theory. In: Harrington L.A., Morley M.D., Scedrov A., and Simpson S.G. (Eds). Harvey Friedman's Research on the Foundations of Mathematics (1985), North-Holland, Amsterdam 285-308
177. Shore R. Controlling the dependence degree of a recursively enumerable vector space. J. Symbolic Logic 43 (1978) 13-22
178. Simon H.U. Word problems for groups and contexfree recognition. Proceedings of FCT'79 (1979), Akademie-Verlag, Berlin 417-422
179. Smith R.L. Effective valuation theory. In: Crossley J.N. (Ed). Aspects of Effective Algebra (Proceedings of a Conference at Monash University. Australia, 1-4 August, 1979 (1981), Upside Down A Book Company, Steel's Creek, Australia 232-245
180. Smith R.L. Two theorems on autostability in p-groups. In: Lerman M., Schmerl J.H., and Soare R.I. (Eds). Logic Year 1979-80. Lecture Notes in Mathematics Vol. 859 (1981), Springer, Berlin 302-311
181. Soare R. Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic (1987), Springer, Berlin
182. Sperschneider V., and Antonio G. Logic. A Foundation for Computer Science (1991), Addison-Wesley
183. Steinitz E. Algebraische Theorie der Körper. Journal für Mathematik 137 (1910) 167-309
184. Stewart I. Galois Theory (1973), Chapman and Hall
185. Stewart I.A. Complete problems for symmetric logspace involving free groups. Inform. Process. Lett. 40 (1991) 263-267
186. Stewart I.A. Refining known results on the generalized word problem for free groups. Internat. J. Algebra Comput. 2 (1992) 221-236
187. Stillwell J. The word problem and the isomorphism problem for groups. Bull. Amer. Math. Soc. 6 (1982) 33-56
188. Stoltenberg-Hansen V., Lindström I., and Griffor E.R. Mathematical Theory of Domains. Cambridge Tracts in Theoretical Computer Science Vol. 22 (1994), Cambridge Univ. Press
189. Stoltenberg-Hansen V., and Tucker J.V. Computing roots of unity in fields. Bull. London Math. Soc. 12 (1980) 463-471
190. Report 1.85, Leeds. Stoltenberg-Hansen V., and Tucker J.V. Complete Local Rings as, Domains. Centre for Theoretical Computer Science Reports (1985), University of Leeds
191. Stoltenberg-Hansen V., and Tucker J.V. Complete local rings as domains. J. Symbolic Logic 53 (1988) 603-624
192. Stoltenberg-Hansen V., and Tucker J.V. Algebraic equations and fixed-point equations in inverse limits. Theoret. Comput. Sci. 87 (1991) 1-24
193. Stoltenberg-Hansen V., and Tucker J.V. Infinite systems of equations over inverse limits and infinite synchronous concurrent algorithms. In: de Bakker J.W., Rozenberg G., and de Roever W.P. (Eds). Semantics-Foundations and Applications. Lecture Notes in Comput. Sci. Vol. 666 (1993), Springer 531-562
194. Stoltenberg-Hansen V., and Tucker J.V. Effective algebras. In: Abramsky S., Gabbay D.M., and Maibaum T.S.E. (Eds). Handbook of Logic in Computer Scieece, Vol. IV: Semantic Modelling (1995), Oxford Univ. Press, Oxford 357-526
195. to appear. Stoltenberg-Hansen V., and Tucker J.V. Concrete models of computation for topological algebras. Theoret. Comp. Sci. (1999)
196. Sturm C.-F. Mémoire sur la résolution des équations numériques. Annales de Mathématiques Pures et Appliquées 6 (1835) 271-318
197. Soublin J.-P. Anneaux et modules cohérents. J. Algebra 15 (1970) 455-472
198. Suter G.H. Recursive elements and constructive extensions of computable local integral domains. J. Symbolic Logic 38 (1973) 272-290
199. Tennenbaum S. Non-archimedian models for arithmetic. Notices Amer. Math. Soc. 6 (1959) 270
200. Thue A. Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. Skr. Vid. Kristiania, I Mat. Naturv. Klasse 10 (1914) 34
201. Thue A. Axel Thue's Selected Works (1977), University of Oslo Press, Oslo
202. Tretkoff C. Complexity, combinatorial group theory and the language of palutators. Theoret. Comput. Sci. 56 (1988) 253-275
203. ITLI preprint ML-88-04. Troelstra A.S. On the early history of intuitionistic logic (1988), University of Amsterdam
204. ITLI Prerint ML-91-05. Troelstra A.S. history of constructivism in the twentieth century (1991), University of Amsterdam
205. Troelstra A.S., and Van Dalen D. Constructivism in Mathematics. An Introduction, Vol. I, Studies in Logic Vol. 121 (1988), North-Holland, Amsterdam
206. Troelstra A.S., and Van Dalen D. Constructivism in, Mathematics. An Introduction, Vol. II, Studies in Logic Vol. 123 (1988), North-Holland, Amsterdam
207. Tucker J.V. Computabilty as an Algebraic Property. Ph.D. Thesis (1976), School of Mathematics, University of Bristol
208. Tucker J.V. Computability and the algebra of fields: Some affine constructions. J. Symbolic Logic 45 (1980) 103-120
209. Tucker J.V. Computing in algebraic systems. In: Drake F.R., and Wainer S.S. (Eds). Recursion Theory, its Generalisations and Applications. London Mathematical Society Lecture Note Series Vol. 45 (1980), Cambridge Univ. Press, Cambridge 215-235
210. Tucker J.V. Theory of computation and specification over abstract data types and its application. In: Bauer F.L. (Ed). Logic, Algebra and Computation (1991), Springer, Berlin 215-235
211. Tucker J.V., and Zucker J.I. Program Correctness over Abstract Data Types with Error-State Semantics (1988), North-Holland, Amsterdam
212. Tucker J.V., and Zucker J.I. Examples of semicomputable sets of real and complex numbers. In: Myers Jr. J.P., and O'Donnell M.J. (Eds). Constructivity in Computer Science. Lecture Notes in Comp. Sci. Vol. 613 (1991), Springer, Berlin 179-198
213. Tucker J.V., and Zucker J.I. Provable computable selection functions on abstract structures. In: Aczel P., Simmons H., and Wainer S.S. (Eds). Proof Theory (1993), Cambridge Univ. Press, Cambridge 277-306
214. Tucker J.V., and Zucker J.I. Computable functions on stream algebras. In: Schwichtenberg H. (Ed). Proof and Computation (1994), Springer, Berlin 341-383
215. in preparation. Tucker J.V., and Zucker J.I. Computable functions and semicomputable sets on many sorted algebras. In: Abramsky S., Gabbay D., and Maibaum T.S.E. (Eds). Handbook of Logic in Computer Science Vol. V (1999), Oxford Univ. Press
216. Turing A.M. The word problem in semigroups with cancellation. Ann. of Math. 52 (1950) 491-505
217. Valiev M.K. On a theorem of G. Higman. Algebra and Logic 8 (1969) 93-128
218. Valiev M.K. On polynomial reducibility of the word problem under embedding of recursively presented groups in finitely generated groups. In: Becvar J. (Ed). Mathematical Foundations of Computer Science 1975. Lecture Notes in Comp. Sci. Vol. 32 (1975), Springer, Berlin 432-438
219. Van der Waerden B.L. Moderne Algebra. 1st ed. (1930), Springer, Berlin
220. Van der Waerden B.L. Eine Bemerkung über die Unzerlegbarkeit von Polynonen. Math. Ann. 102 (1930) 738-739
221. Vandiver H.S. On the ordering of real algebraic numbers by constructive methods. Ann. of Math. 37 (1936) 7-16
222. Ventsov Yu.G. Computable classes of constructivizations of models of infinite algorithmic dimension. Algebra and Logic 33 (1994) 22-45
223. Waack S. Tape complexity of word problems. In: Gécseg F. (Ed). Fundamentals of Computation Theory, Proceedings FCT'81. Lecture Notes in Comp. Sci. Vol. 117 (1981), Springer, Berlin 467-471
224. Waack S. Raumkomplexität von Wortproblemen endlicher Gruppenpräsentationen. Dissertation A, Berlin (1983)
225. Waack S. On the parallel complexity of linear groups. RARIO Inform. Théro. Appl. 25 (1991) 323-354
226. Weber H. Untersuchungen über die allgemeinen Grundlagen der Galoi'schen Gleichungstheorie. Math. Ann. 43 (1893) 521-549
227. Wechler W. Universal Algebra for Computer Scientists. EATCS Monographs on Theoretical Computer Science Vol. 25 (1992), Springer, Berlin
228. Weihrauch K. Type 2 recursion theory. Theoret. Comput. Sci. 38 (1985) 17-33
229. Weihrauch K. Computability. EATCS Monographs on Theoretical Computer Science Vol. 9 (1987), Springer, Berlin
230. Weihrauch K. Computability on computable metric spaces. Theoret. Comput. Sci. 113 (1993) 191-210
231. manuscript. Weihrauch K. A foundation of computable analysis (1996), Fern Universität Hagen
232. Weihrauch K., and Schreiber U. Embedding metric spaces into cpo's. Theoret. Comput. Sci. 16 (1981) 5-24
233. Wirsing M. Algebraic specifications. In: van Leeuwen J. (Ed). Handbook of Theoretical Computer Science. Vol. B: Formal Models and Semantics (1990), North-Holland 675-788
234. Wüssing H. The Genesis of the Abstract Group Concept (1984), MIT Press, Cambridge
235. Ziegler M. Ein rekursiv aufzahlbaurer btt-Grad, der nicht zum Wort problem einer Gruppe gehört. Zeits. Math. Logik Grundl. Math. 22 (1976) 165-168