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Guide to Part C: Recursion Theory
Guide to Part C : Recursion Theory with the cooperation of Y.N. MOSCHOVAKIS C.1. ENDERTON Elements of recursion theory C.2. DAVIS Unsolvable problems C.3. RABIN Decidable theories C.4. SIMPSON Degrees of unsolvability : a survey of results C.5. SHORE a -recursion theory C.6. KECHRISand MOSCHOVAKIS Recursion in higher types C.J. ACZEL A n introduction to inductive definitions C.8. MARTIN Descriptive set theory : projective sets
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Strictly speaking, recursion theory is the study of the class of recursive, or effectively computable, functions and their applications to mathematics. A broader interpretation is when recursion theory is taken to mean the study of the general process of definition by recursion, not just on natural numbers but on all types of mathematical structures. The first four chapters of this Part fit under the narrow definition, the last four under the broader one. The class of recursive functions is defined and studied in Enderton’s introductory chapter. This chapter discusses the arguments for the identification of this class with the “effectively calculable” functions (Church’s Thesis). It also introduces the reader to many of the ways that the basic notions can be applied. 525
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GUIDE TO PART C
The next three chapters take up in depth topics introduced in Enderton’s chapter. Martin Davis’s chapter pursues the uses of the theory of recursive functions for showing that certain classes of problems cannot be effectively decided - t h e word problem for groups being one of the best known. The related problem of decidable versus undecidable theories of first-order logic is discussed in Rabin’s chapter. Among undecidable problems, some are more undecidable than others. The definition of “degree;’ of unsolvability is introduced in Section 8 of Enderton’s paper and a survey of important results on these degrees is given in Simpson’s chapter. Moving to the broader definition of recursion theory we come to Shore’s chapter on the generalization of recursion theory to admissible ordinals. Shore presents a fine introduction to the basic notions and, as a case study, shows what new considerations arise when the Splitting Theorem is generalized to admissible ordinals. The chapter also contains a very useful annotated bibliography to the study of a-recursion theory. The study of Kleene recursion in higher types (recursive functions of functions of functions, say, rather than recursive functions of natural numbers) has always been more or less inaccessible to all but the dedicated specialist - due to the difficulty of the basic papers in the subject. This situation should be remedied in the chapter by Kechris and Moschovakis, where a conceptually simple approach via inductive definability is taken. The study of inductive definitions in general is taken up in Aczel’s chapter. It should interest logicians of all persuasions since it combines the concerns of the recursion-theorist with the vantage points of the modeltheorist and proof-theorist. Martin’s chapter discusses one of the major applications of recursion theory - to descriptive set theory. Here definability considerations over the continuum give rise to a beautiful theory which finds its origins in the French “constructivist” school of Borel, Baire and Lebesgue. We had planned to have a chapter on the more “practical” aspects of recursion theory, those where running times of programs and computational complexity appear, but this chapter did not rnateria1;ze. Among other chapters of the Handbook relevant to recursion theory we mention Statman’s chapter on the equation calculus (in Part D), and Makkai’s chapter on admissible sets (in Part A). The recursion-theorist might also be interested to see some proof-theoretic applications of recursion theory in Feferman’s chapter in Part D.
521
561 595 631 65 3 68 1 739 783
Strictly speaking, recursion theory is the study of the class of recursive, or effectively computable, functions and their applications to mathematics. A broader interpretation is when recursion theory is taken to mean the study of the general process of definition by recursion, not just on natural numbers but on all types of mathematical structures. The first four chapters of this Part fit under the narrow definition, the last four under the broader one. The class of recursive functions is defined and studied in Enderton’s introductory chapter. This chapter discusses the arguments for the identification of this class with the “effectively calculable” functions (Church’s Thesis). It also introduces the reader to many of the ways that the basic notions can be applied. 525
526
GUIDE TO PART C
The next three chapters take up in depth topics introduced in Enderton’s chapter. Martin Davis’s chapter pursues the uses of the theory of recursive functions for showing that certain classes of problems cannot be effectively decided - t h e word problem for groups being one of the best known. The related problem of decidable versus undecidable theories of first-order logic is discussed in Rabin’s chapter. Among undecidable problems, some are more undecidable than others. The definition of “degree;’ of unsolvability is introduced in Section 8 of Enderton’s paper and a survey of important results on these degrees is given in Simpson’s chapter. Moving to the broader definition of recursion theory we come to Shore’s chapter on the generalization of recursion theory to admissible ordinals. Shore presents a fine introduction to the basic notions and, as a case study, shows what new considerations arise when the Splitting Theorem is generalized to admissible ordinals. The chapter also contains a very useful annotated bibliography to the study of a-recursion theory. The study of Kleene recursion in higher types (recursive functions of functions of functions, say, rather than recursive functions of natural numbers) has always been more or less inaccessible to all but the dedicated specialist - due to the difficulty of the basic papers in the subject. This situation should be remedied in the chapter by Kechris and Moschovakis, where a conceptually simple approach via inductive definability is taken. The study of inductive definitions in general is taken up in Aczel’s chapter. It should interest logicians of all persuasions since it combines the concerns of the recursion-theorist with the vantage points of the modeltheorist and proof-theorist. Martin’s chapter discusses one of the major applications of recursion theory - to descriptive set theory. Here definability considerations over the continuum give rise to a beautiful theory which finds its origins in the French “constructivist” school of Borel, Baire and Lebesgue. We had planned to have a chapter on the more “practical” aspects of recursion theory, those where running times of programs and computational complexity appear, but this chapter did not rnateria1;ze. Among other chapters of the Handbook relevant to recursion theory we mention Statman’s chapter on the equation calculus (in Part D), and Makkai’s chapter on admissible sets (in Part A). The recursion-theorist might also be interested to see some proof-theoretic applications of recursion theory in Feferman’s chapter in Part D.