Nullane de tantis gregibus tibi digna videtur? rara avis in terra nigroque simillima cygno. Juvenal Sat. VI 161, 165. 1966-JNC visits AN at CornelI. An idea emerges. 1968-JNC is at V. c. L.A. for the Logic Year. The Los Angeles ma- script appears. 1970-AN visits JNC at Monash. 1971-The Australian manuscript appears. 1972-JNC visits AN at Cornell. Here is the result. We gratefully acknowledge support from Cornell Vniversity, Vni versity of California at Los Angeles, Monash Vniversity and National Science Foundation Grants GP 14363, 22719 and 28169. We are deeply indebted to the many people who have helped uso Amongst the mathe maticians, we are particularly grateful to J.C.E. Dekker, John Myhill, Erik Ellentuck, Peter AczeI, Chris Ash, Charlotte ehell, Ed Eisenberg, Dave Gillam, Bill Gross, Alan Hamilton, Louise Hay, Georg Kreisel, Phil Lavori, Ray Liggett, Al Manaster, Michael D. Morley, Joe Rosen stein, Graham Sainsbury, Bob Soare and Michael Venning. Last, but by no means least, we thank Anne-Marie Vandenberg, Esther Monroe, Arletta Havlik, Dolores Pendell, and Cathy Stevens and the girls of the Mathematics Department of VCLA in 1968 for hours and hours of excellent typing. Thanksgiving November 1972 J.N. Crossley Ithaca, New Y ork Anil Nerode Contents O. Introduction ... 1 Part 1. Categories and Functors 3 1. Categories ... 3 2. Morphism Combinatorial Functors 3 3. Combinatorial Functors ... 18 Part H. Model Theory . . 18 4. Countable Atomic Models 18 5. Copying . 22 6. Dimension ... 26 Part III.

A so-called "effective" algorithm may require arbitrarily large finite amounts of time and space resources, and hence may not be practical in the real world. A "feasible" algorithm is one which only requires a limited amount of space and/or time for execution; the general idea is that a feasible algorithm is one which may be practical on today's or at least tomorrow's computers. There is no definitive analogue of Church's thesis giving a mathematical definition of feasibility; however, the most widely studied mathematical model of feasible computability is polynomial-time computability. Feasible Mathematics includes both the study of feasible computation from a mathematical and logical point of view and the reworking of traditional mathematics from the point of view of feasible computation. The diversity of Feasible Mathematics is illustrated by the. contents of this volume which includes papers on weak fragments of arithmetic, on higher type functionals, on bounded linear logic, on sub recursive definitions of complexity classes, on finite model theory, on models of feasible computation for real numbers, on vector spaces and on recursion theory. The vVorkshop on Feasible Mathematics was sponsored by the Mathematical Sciences Institute and was held at Cornell University, June 26-28, 1989.

The twenty-six papers in this volume reflect the wide and still expanding range of Anil Nerode's work. A conference on Logical Methods was held in honor of Nerode's sixtieth birthday (4 June 1992) at the Mathematical Sciences Institute, Cornell University, 1-3 June 1992. Some of the conference papers are here, but others are from students, co-workers and other colleagues. The intention of the conference was to look forward, and to see the directions currently being pursued, in the development of work by, or with, Nerode. Here is a brief summary of the contents of this book. We give a retrospective view of Nerode's work. A number of specific areas are readily discerned: recursive equivalence types, recursive algebra and model theory, the theory of Turing degrees and r.e. sets, polynomial-time computability and computer science. Nerode began with automata theory and has also taken a keen interest in the history of mathematics. All these areas are represented. The one area missing is Nerode's applied mathematical work relating to the environment. Kozen's paper builds on Nerode's early work on automata. Recursive equivalence types are covered by Dekker and Barback, the latter using directly a fundamental metatheorem of Nerode. Recursive algebra is treated by Ge & Richards (group representations). Recursive model theory is the subject of papers by Hird, Moses, and Khoussainov & Dadajanov, while a combinatorial problem in recursive model theory is discussed in Cherlin & Martin's paper. Cenzer presents a paper on recursive dynamics.

This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.

Recursive Model Theory