The Work of Anil Nerode: A Retrospective

Anil Nerode’s work divides into a number of periods though, inevitably, there are overlaps and cross-currents. In this survey we indicate what we consider the highlights so far in mathematics and computer science. (We have not included his work in environmental science policy for the United States Environmental Protection Agency, except insofar as it is listed in the bibliography at the end (Loehr, Goldstein, Nerode & Risser [1992]).) We distinguish six periods or areas: 1. His thesis and early work in automata theory and recursion theory, 2. Isols, 3. Undecidability, 4. Recursive algebra, 5. Polynomial-time structures and 6. Computer science. From the authors’ personal experience, we know that much of Nerode’s effort has gone into developing foundations for various subjects. This, of course, means that a lot of his work has involved not only proving theorems but searching for the appropriate basic definitions which allow an area to be developed. Even though our space is limited, we hope to give some sense of this foundational work in this survey. Finally we should note that Nerode has made contributions to logic in general by his influence on other researchers in the field. This not only includes the 34 Ph.D. students that he has supervised to date, but also many researchers in logic who have benefited from his suggestions for areas of research to pursue, or approaches to a problem, as well as his enthusiastic encouragement of their efforts.

[1]  John T. Baldwin,et al.  On strongly minimal sets , 1971, Journal of Symbolic Logic.

[2]  Joseph R. Schoenfield.  Recursion theory , 1993 .

[3]  Anil Nerode,et al.  Some Lectures on Modal Logic , 1991 .

[4]  S. S. Goncharov,et al.  Autostability of models , 1980 .

[5]  Adrian Walker,et al.  Towards a Theory of Declarative Knowledge , 1988, Foundations of Deductive Databases and Logic Programming..

[6]  Krzysztof R. Apt,et al.  Logic Programming , 1990, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[7]  David W. Etherington.  Reasoning With Incomplete Information , 1988 .

[8]  A. R. Turquette,et al.  Logic, Semantics, Metamathematics , 1957 .

[9]  Anil Nerode,et al.  Computing Definite Logic Programs by Partial Instantiation , 1994, Ann. Pure Appl. Log..

[10]  J. C. E. Dekker,et al.  Countable Vector Spaces with Recursive Operations. Part I , 1969, J. Symb. Log..

[11]  Julia F. Knight.  Degrees of Types and Independent Sequences , 1983, J. Symb. Log..

[12]  Terrence Millar,et al.  Homogeneous models and decidability , 1980 .

[13]  Jeffrey B. Remmel,et al.  Recursion theory on orderings. II , 1980, Journal of Symbolic Logic.

[14]  The uniqueness of finite division for recursive equivalence types , 1961 .

[15]  Anil Nerode,et al.  Some lectures on intuitionistic logic , 1990 .

[16]  J. B. Remmel,et al.  Complementation in the lattice of subalgebras of a Boolean algebra , 1980 .

[17]  Erik Ellentuck.  A Coding Theorem for Isols , 1970, J. Symb. Log..

[18]  S. S. Goncharov,et al.  Some properties of the constructivization of Boolean algebras , 1975 .

[19]  Robert G. Jeroslow,et al.  Computation-oriented reductions of predicate to propositional logic , 1988, Decis. Support Syst..

[20]  Leo Harrington.  Recursively Presentable Prime Models , 1974, J. Symb. Log..

[21]  Terrence Millar,et al.  Counter-examples via model completions , 1981 .

[22]  Robert I. Soare,et al.  Degrees of members of Π10 classes , 1972 .

[23]  Emil L. Post.  Recursively enumerable sets of positive integers and their decision problems , 1944 .

[24]  Terrence Millar.  Omitting Types, Type Spectrums, and Decidability , 1983, J. Symb. Log..

[25]  Jeffrey B. Remmel.  Recursively rigid Boolean algebras , 1987, Ann. Pure Appl. Log..

[26]  A. Nerode,et al.  Additive relations among recursive equivalence types , 1965 .

[27]  S. S. Goncharov.  Strong constructivizability of homogeneous models , 1978 .

[28]  I. Kalantari.  Major Subsets in Effective Topology , 1982 .

[29]  Christopher J. Ash,et al.  Labelling Systems and r.e. Structures , 1990, Ann. Pure Appl. Log..

[30]  Some aspects of generalized computability , 1973 .

[31]  Jeffrey B. Remmel,et al.  Classification of Degree Classes Associated with r.e. Subspaces , 1989, Ann. Pure Appl. Log..

[32]  Anil Nerode.  EXTENSIONS TO ISOLS , 1961 .

[33]  Jeffrey B. Remmel,et al.  Cancellation Laws for Polynomial-Time p-Isolated Sets , 1992, Ann. Pure Appl. Log..

[34]  Anil Nerode,et al.  Generic objects in recursion theory , 1985 .

[35]  John Newsome Crossley,et al.  A logical calculus for polynomial-time realizability , 1994, Methods Log. Comput. Sci..

[36]  A. S. Denisov.  HomogeneousO′-elements in structural pre-orders , 1989 .

[37]  S. S. Goncharov,et al.  The quantity of nonautoequivalent constructivizations , 1977 .

[38]  Matthew J. Hassett.  Recursive Equivalence Types and Groups , 1969, J. Symb. Log..

[39]  A. Nerode,et al.  Recursion Theory on Matroids II , 1983 .

[40]  J. C. E. Dekker,et al.  Recursive Equivalence Types , 1962 .

[41]  Julia F. Knight,et al.  Expansions of models and turing degrees , 1982, Journal of Symbolic Logic.

[42]  A. Nerode.  A decision method for $p$-adic integral zeros of diophantine equations , 1963 .

[43]  J. B. Remmel.  Combinational functors on co-r.e. structures , 1976 .

[44]  Iraj Kalantari,et al.  Simplicity in Effective Topology , 1982, J. Symb. Log..

[45]  A. Nerode,et al.  Effective content of field theory , 1979 .

[46]  J. Guckenheimer,et al.  Simulation for hybrid systems and nonlinear control , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[47]  Iraj Kalantari,et al.  Maximality in Effective Topology , 1983, J. Symb. Log..

[48]  Anil Nerode.  Diophantine correct non-standard models in the isols , 1966 .

[49]  A. Nerode,et al.  Some Stone spaces and recursion theory , 1959 .

[50]  John Newsome Crossley,et al.  Recursive categoricity and recursive stability , 1986, Ann. Pure Appl. Log..

[51]  Jeffrey B. Remmel.  A r-Maximal Vector Space not contained in any Maximal Vector Space , 1978, J. Symb. Log..

[52]  Anil Nerode,et al.  Second order logic and first order theories of reducibility orderings , 1980 .

[53]  A. Grzegorczyk.  On the definitions of computable real continuous functions , 1957 .

[54]  A. Nerode,et al.  Recursion Theory on Fields and Abstract Dependence , 1980 .

[55]  Jeffrey B. Remmel,et al.  Degrees of Recursively Enumerable Topological Spaces , 1983, J. Symb. Log..

[56]  S. S. Goncharov,et al.  Constructive models of complete solvable theories , 1973 .

[57]  Jon Doyle,et al.  A Truth Maintenance System , 1979, Artif. Intell..

[59]  S. S. Goncharov,et al.  Autostability and computable families of constructivizations , 1975 .

[60]  C. J. Ash,et al.  Persistently finite, persistently arithmetic theories , 1983 .

[61]  Terrence Millar.  Vaught's Theorem Recursively Revisited , 1981, J. Symb. Log..

[62]  Functorial properties of algebraic closure and Skolemization , 1981 .

[63]  S. S. Goncharov,et al.  A totally transcendental decidable theory without constructivizable homogeneous models , 1980 .

[64]  Murray Rosenblatt,et al.  Errett Bishop: Reflections on him and his research , 1985 .

[65]  Extension methods in cardinal arithmetic , 1970 .

[66]  Anil Nerode,et al.  Polynomially isolated sets , 1990 .

[67]  Andre Scedrov,et al.  Bounded Linear Logic: A Modular Approach to Polynomial-Time Computability , 1992, Theor. Comput. Sci..

[68]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[69]  A. Nerode.  Logic and foundations , 1975 .

[70]  Jeffrey B. Remmel.  Recursively categorical linear orderings , 1981 .

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[72]  Terrence Millar.  Foundations of recursive model theory , 1978 .

[73]  Keith L. Clark,et al.  Negation as Failure , 1987, Logic and Data Bases.

[74]  Rodney G. Downey.  Bases of Supermaximal Subspaces and Steinitz Systems II , 1986, Math. Log. Q..

[75]  Rodney G. Downey.  Undecidability of L(F∞) and other lattices of r.e. substructures , 1986, Ann. Pure Appl. Log..

[76]  A. Nerode,et al.  An Algebraic Proof of Kirchhoff's Network Theorem , 1961 .

[77]  Jeffrey B. Remmel,et al.  The undecidability of the lattice of R.E. closed subsets of an effective topological space , 1987, Ann. Pure Appl. Log..

[78]  A. Nerode,et al.  Recursively enumerable vector spaces , 1977 .

[79]  J. Shepherdson,et al.  Effective procedures in field theory , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[80]  Anil Nerode,et al.  Complexity Theoretic Algebra I Vector Spaces over Finite Fields , 2008 .

[81]  McDermottDrew.  Nonmonotonic Logic II , 1982 .

[82]  Yu. L. Ershov,et al.  Skolem functions and constructive models , 1973 .

[83]  Gerald E. Sacks,et al.  Recursion Theory Week , 1985 .

[84]  J. C. E. Dekker.  Two notes on vector spaces with recursive operations , 1971, Notre Dame J. Formal Log..

[85]  R. Soare,et al.  Π⁰₁ classes and degrees of theories , 1972 .

[86]  Michael R. Fellows,et al.  An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations , 1989, 30th Annual Symposium on Foundations of Computer Science.

[87]  Andrzej Grzegorczyk.  On the definition of computable functionals , 1955 .

[88]  Jeffrey B. Remmel,et al.  Recursive isomorphism types of recursive Boolean algebras , 1981, Journal of Symbolic Logic.

[89]  Anil Nerode,et al.  Logical Foundations of Computer Science--Tver '92: Second International Symposium, Tver, Russia, July 20-24, 1992 : Proceedings , 1992 .

[90]  Julia F. Knight.  Constructions by Transfinitely Many Workers , 1990, Ann. Pure Appl. Log..

[91]  John Myhill.  Elementary properties of the group of isolic integers , 1962 .

[92]  Iraj Kalantari,et al.  Recursive Constructions in Topological Spaces , 1979, J. Symb. Log..

[93]  T. S. Millar.  Type structure complexity and decidability , 1982 .

[94]  E. Ellentuck.  The Universal Properties of Dedekind Finite Cardinals , 1965 .

[95]  R A Shore.  The homogeneity conjecture. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[96]  A. Nerode,et al.  Composita, equations, and freely generated algebras , 1959 .

[97]  S. S. Goncharov.  Totally transcendental theory with a nonconstructivizable prime model , 1980 .

[98]  Rodney G. Downey,et al.  Automorphisms of supermaximal subspaces , 1985, Journal of Symbolic Logic.

[99]  R. Downey.  Nowhere simplicity in matroids , 1983, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[100]  H. Keisler,et al.  The Kleene Symposium , 1980 .

[101]  Elementary differences between the isols and the co-simple isols , 1967 .

[102]  Combinatorial series and recursive equivalence types , 1966 .

[103]  Alfred B. Manaster.  Higher-order indecomposable isols , 1966 .

[104]  S. S. Goncharov,et al.  Examples of nonautostable systems , 1984 .

[105]  A. Nerode.  Non-linear combinatorial functions of isols , 1965 .

[106]  Douglas A. Cenzer,et al.  Polynomial-Time versus Recursive Models , 1991, Ann. Pure Appl. Log..

[107]  Victor W. Marek,et al.  Nonmonotonic logic - context-dependent reasoning , 1997, Artificial intelligence.

[108]  The Representation of Cardinals in Models of Set Theory , 1968 .

[109]  E. Barker.  Intrinsically gs;0alpha; relations , 1988, Ann. Pure Appl. Log..

[110]  Thomas G. McLaughlin.  Regressive Sets and the Theory of Isols , 1982 .

[111]  Erik Ellentuck.  A Choice Free Theory of Dedekind Cardinals , 1969, J. Symb. Log..

[112]  Jeffrey B. Remmel,et al.  Maximal and Cohesive vector spaces , 1977, Journal of Symbolic Logic.

[113]  Terrence Millar.  A Complete, Decidable Theory with Two Decidable Models , 1979, J. Symb. Log..

[114]  S. S. Goncharov,et al.  Numerations of saturated and homogeneous models , 1980 .

[115]  Drew McDermott,et al.  Nonmonotonic Logic II: Nonmonotonic Modal Theories , 1982, JACM.

[116]  Louise Hay.  The Co-Simple Isols , 1966 .

[117]  Charles H. Applebaum.  Isomorphisms of ω-groups , 1971, Notre Dame J. Formal Log..

[118]  Julia F. Knight,et al.  Saturation of homogeneous resplendent models , 1986, Journal of Symbolic Logic.

[119]  Anil Nerode,et al.  Computing Circumscriptive Databases: I. Theory and Algorithms , 1995, Inf. Comput..

[120]  J. C. E. Dekker,et al.  Partial Recursive Functions and omega-Functions , 1970, J. Symb. Log..

[122]  J. Remmel.  Recursively enumerable Boolean algebras , 1978 .

[123]  Christopher J. Ash.  Stability of recursive structures in arithmetical degrees , 1986, Ann. Pure Appl. Log..

[124]  Hal A. Kierstead,et al.  Indiscernibles and Decidable Models , 1983, J. Symb. Log..

[125]  Julia F. Knight,et al.  Pairs of Recursive Structures , 1990, Ann. Pure Appl. Log..

[126]  M. Rabin.  Computable algebra, general theory and theory of computable fields. , 1960 .

[127]  Jeffrey B. Remmel,et al.  Polynomial-time combinatorial operators are polynomials , 1990 .

[128]  Christopher J. Ash.  Categoricity in hyperarithmetical degrees , 1987, Ann. Pure Appl. Log..

[129]  J. Remmel,et al.  Recursively presented games and strategies , 1992 .

[130]  A. Nerode,et al.  Modelling hybrid systems as games , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[131]  A. T. Nurtazin,et al.  Strong and weak constructivization and computable families , 1974 .

[132]  Terry Millar.  Bad Models in Nice Neighborhoods , 1986, J. Symb. Log..

[133]  John Newsome Crossley,et al.  Constructive Order Types, II , 1966, J. Symb. Log..

[134]  Jeffrey B. Remmel.  Co-Hypersimple Structures , 1976, J. Symb. Log..

[135]  Erik Ellentuck.  Degrees of isolic theories , 1973, Notre Dame J. Formal Log..

[136]  Johann Eder,et al.  Logic and Databases , 1992, Advanced Topics in Artificial Intelligence.

[137]  Julia A. Knight.  Effective Construction of Models , 1986 .

[138]  Jeffrey B. Remmel,et al.  R-maximal Boolean algebras , 1979, Journal of Symbolic Logic.

[139]  Terrence Millar.  Decidability and the number of countable models , 1984, Ann. Pure Appl. Log..

[140]  Richard Lawrence Smith,et al.  Two theorems on autostability in p-Groups , 1981 .

[141]  Anil Nerode,et al.  The Introduction of Non-Recursive Methods into Mathematics* , 1982 .

[142]  Serge Grigorieff,et al.  Every recursive linear ordering has a copy in DTIME-SPACE(n,log(n)) , 1990, Journal of Symbolic Logic.

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[145]  Drew McDermott,et al.  Non-Monotonic Logic I , 1987, Artif. Intell..

[146]  J. Shepherdson,et al.  On the factorisation of polynomials in a finite number of steps , 1955 .

[147]  Terrence Millar.  Recursive Categoricity and Persistence , 1986, J. Symb. Log..

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[150]  H. Kierstead,et al.  DEGREES OF INDISCERNIBLES IN DECIDABLE MODELS , 1985 .

[151]  Wolfgang Maass.  Recursively Enumerable Generic Sets , 1982, J. Symb. Log..

[152]  Anil Nerode,et al.  Reducibility orderings: Theories, definability and automorphisms☆ , 1980 .

[153]  Christine Froidevaux,et al.  Negation by Default and Unstratifiable Logic Programs , 1991, Theor. Comput. Sci..

[154]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

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[156]  Robert L. Grossman,et al.  The realization of input-output maps using bialgebras , 2020, 2007.09526.

[157]  Julia F. Knight.  A Metatheorem for Constructions by Finitely Many Workers , 1990, J. Symb. Log..

[158]  Anil Nerode,et al.  Implementing Stable Semantics by Linear Programming , 1993, LPNMR.

[159]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[160]  Peter Aczel.  Term declaration logic and generalised composita , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[161]  Victor W. Marek,et al.  Relating Autoepistemic and Default Logics , 1989, KR.

[162]  Julia A. Knight.  Degrees of models with prescribed Scott set , 1987 .

[163]  Jeffrey B. Remmel,et al.  Recursion theory on orderings. I. A model theoretic setting , 1979, Journal of Symbolic Logic.

[164]  Anil Nerode,et al.  Implementing deductive databases by linear programming , 1992, PODS.

[165]  Douglas A. Cenzer,et al.  Polynomial-Time Abelian Groups , 1992, Ann. Pure Appl. Log..

[166]  Сергей Савостьянович Гончаров.  О числе неавтоэквивалентных конструктивизаций , 1977 .

[167]  W. Thomson,et al.  The fair allocation of an indivisible good when monetary compensations are possible , 1993 .

[168]  Andre Scedrov,et al.  Bounded Linear Logic: A Modular Approach to Polynomial Time Computability: Extended Abstract , 1990 .

[169]  Rodney G. Downey,et al.  Decidable subspaces and recursively enumerable subspaces , 1984, Journal of Symbolic Logic.

[170]  J. Baldwin.  Recursion Theory and Abstract Dependence , 1982 .

[171]  Terrence Millar.  Prime Models and Almost Decidability , 1986, J. Symb. Log..

[172]  Victor W. Marek,et al.  Autoepistemic logic , 1991, JACM.

[173]  R. Goodstein.  Boolean algebra , 1963 .

[174]  C. Ash,et al.  Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees , 1986 .

[175]  Jeffrey B. Remmel,et al.  Proofs, programs, and run times , 1994, Methods Log. Comput. Sci..

[176]  Victor W. Marek,et al.  How Complicated is the Set of Stable Models of a Recursive Logic Program? , 1992, Ann. Pure Appl. Log..

[177]  A. Nerode,et al.  Linear automaton transformations , 1958 .

[178]  Geoffrey R. Hird.  Recursive Properties of Relations on Models , 1993, Ann. Pure Appl. Log..

[179]  С.С. Гончаров,et al.  Конструктивные модели полных разрешимых теорий. , 1973 .

[180]  J. Remmel.  Recursion theory on algebraic structures with independent sets , 1980 .

[181]  A decidable countably categorical model without nontrivial recursive automorphisms , 1989 .

[182]  Anil Nerode,et al.  Complexity-Theoretic Algebra II: Boolean Algebras , 1989, Ann. Pure Appl. Log..

[183]  Julia F. Knight,et al.  Degrees coded in jumps of orderings , 1986, Journal of Symbolic Logic.

[184]  C. H. Applebaum.  omega-Homomorphisms and omega-Groups , 1971, J. Symb. Log..

[185]  A. Nerode,et al.  The Work of Harvey Friedman1) , 1985 .

[186]  Rodney G. Downey.  Co-Immune Subspaces and Complementation in V , 1984, J. Symb. Log..