The Work of J. Richard Büchi

Abstract

J. Richard Buchi has done influential work in mathematics, logic, and computer sciences. He is probably best known for using finite automata as combinatorial devices to obtain strong results on decidability and definability in monadic second order theories, and extending the method to infinite combinatorial tools. Many consider his way of describing computations in logical theories as seminal in the area of reduction types. With Jesse Wright, identifying automata with algebras he opened them to algebraic treatment. In a book which I edited after his death he deals with the subject, and with its generalization to tree automata and context-free languages, in a uniform way through semi-Thue systems, aiming for a mathematical theory of terms. Less recognized is his concept of “abstraction” for characterizing structures by their automorphism groups, which he considered basic for a theory of definability. An axiomatic theory of convexity which originated therefrom is partly published jointly with W. Fenton. Also partly published is joint work on formalizing computing and complexity on abstract data types with the present author. Unpublished is, and likely will be, his continued work on an algorithmic version of Gauss’ theory of quadratic forms, which stemmed from his interest in Hilbert’s 10th problem. Results in the existential theory of concatenation, which link the two areas, are published jointly with S. Senger. Saunders MacLane and I edited a volume of Collected Works of Richard Buchi.