CLASSICAL RECURSION THEORY VOLUME II

P.G. ODIFREDDI University of Turin

Turin, Italy

ELSEVIER AMSTERDAM • LAUSANNE • NEW YORK • OXFORD • SHANNON • SINGAPORE • TOKYO

Contents

Preface vii

Introduction 1 What is in the Book 1 How to Use the Book 3 Notation and Conventions 5

VII THEORIES OF RECURSIVE FUNCTIONS 9 VII.l Measures of Complexity 10

Static complexity measures * 11 Shortening proofs by adding axioms * 17 Definition of a dynamic complexity measure 19 First properties of dynamic complexity measures 23

VII.2 Speed of Computations 26 Upper and lower bounds of the complexities of a function . . . 26 The Compression Theorem * 32 Functions with best complexity 34 The Speed-Up Theorem 38

VII.3 Complexity Classes 47 Hierarchies for the recursive functions: successor levels • . . . . 50 Hierarchies for the recursive functions: limit levels 55 Hierarchies for the recursive functions: exhaustiveness * . . . . 59 Names for complexity classes * 61

VII.4 Time and Space Measures 67 One-tape Turing machines 67 Other Turing machine modeis * 70 Linear Speed-Up 76 Hierarchy theorems 80 Space versus time 84 Nondeterministic Turing machines 86

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Alternating Turing machines * 95 VII.5 Inductive Inference 96

Historical background • 98 Identification by next value 101 Identification by consistent explanation 104 Identification by reliable explanation 111 Identification by explanation 116 Identification by explanation with finite errors 126 Behaviorally correct (consistent) identification 131 Behaviorally correct identification with finite errors 138 The world of inference classes • 141 Degrees of inferability • 141

VIII HIERARCHIES OF RECURSIVE FUNCTIONS 145 VIII. 1 Small Time and Space Bounds * 147

Realtime 147 Constant space 150 Logarithmic space 160

VIII.2 Deterministic Polynomial Time 163 Polynomial time computable functions 163 Closure properties 164 Alternative characterizations 170 Rate of growth 177 Feasibly computable functions * 178 TheclassP 181 Logarithmic space again and beyond * 184 A look inside P * 187 Polynomial time degrees • 188

VIII.3 Nondeterministic Polynomial Time * 197 The class NP 198 Deterministic polynomial time again * 198 NP sets as analogues of r.e. sets: successes 199 NP sets as analogues of r.e. sets: failures * 208 Relativizations 213 Polynomial time degrees again * 222

VIII.4 The Polynomial Time Hierarchy * 223 Truth in Bounded Quantifier Arithmetic 224 Types of bounded quantifiers * 225 The Polynomial Time Hierarchy 226 Second-Order Logic on finite domains • 227 The levels of the Polynomial Time Hierarchy 228 Relativizations 231

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VIII.5 Polynomial Space * 235 Linear space * 235 The class PSPACE 237 Polynomial time again * 240 The Polynomial Time Hierarchy again 242 Relativizations 244 A look inside PSPACE * 248

VIII.6 Exponential Time and Space 249 Exponential time 250 Beyond exponential time * 255 Exponential space 259 Superexponential bounds * 261

VIII.7 Elementary Functions 264 Elementary functions 264 Closure properties 265 Alternative characterizations 269 Computation time and space 272 Rate of growth 275 Hierarchies 278 The diagonal function * 279 Relativizations 284

VIII.8 Primitive Recursive Functions 286 Primitive recursive functions 286 Closure properties 287 Alternative characterizations 295 Computation time and space 297 Rate of growth 298 Hierarchies 301 The diagonal function • 308

VIII.9 eo-Recursive Functions 314 Multiple recursive functions * 315 Ordinal recursion 316 eo-recursive functions 318 Closure properties 319 Alternative characterizations 322 Functions provably total in Peano Arithmetic * 324 Rate of growth 326 Hierarchies 338 The diagonal function * 342 A natural stopping point * 355

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IX RECURSIVELY ENUMERABLE SETS 357 IX. 1 Global Properties of Recursive Sets 358

Characterizations of the lattice of recursive sets 358 The complexity of the theory of recursive sets 361 Homogeneity 361 Automorphisms 362 Absolute definability 363 Ultrafilters and modeis of fragments of Arithmetic * 364

IX.2 Local Properties of R.E. Sets 368 Splitting theorems 368 Hyperhypersimple sets 374 R-maximal sets 376 Maximal sets 379 Sets without maximal supersets 385 The world of simple sets * 393

IX.3 Global Properties of R.E. Sets 401 The complexity of the theory of r.e. sets * 401 Absolute definability 403 Homogeneity 411 Automorphisms 414 Orbits* 422 Definable and invariant classes of r.e. degrees * 423

IX.4 Complexity of R.E. Sets 424 Nonspeedable sets 424 Effectively speedable sets 431 Existence theorems 435 Complexity sequences • 438

IX.5 Inductive Inference of R.E. Sets * 447 Identification by explanation of sets 447 Identification by explanation of partial functions 452

X RECURSIVELY ENUMERABLE DEGREES 455 X I The Finite Injury Priority Method 456

Motivation 456 Embeddability results 459 Sacks agreement method 464 A tactical Variation * 467

X.2 Effective Baire Category * 467 Categorical formulation of finite injury arguments 468 The permitting method 470 Degrees r.e. in smaller degrees * 471

Contents xm

X.3 The Infinite Injury Priority Method 473 Thicksubsets 473 The Injury Lemma 474 The Hat Trick 476 The Window Lemma 478 The Gate Opening Lemma • 480 The Thickness Lemma 482 Formal Systems and r.e. sets * 485

X.4 The Priority Method 487 Historical remarks * 487 The Tree Method 488 Admissibility * 496 Bounded Arithmetic • 501 Avoiding the priority method 505 A review of Solutions to Post's Problem * 509

X.5 Many-One Degrees 511 Incomparable degrees 511 Upward density 513 Greatest lower bounds 514 Least upper bounds 518 Connections between structure and degrees * 519 Minimal degrees 526 Initial segments 530 Global properties 534

X.6 Turing Degrees 538 Incomparable degrees 539 Density 541 Greatest lower bounds 543 Least upper bounds 551 Lattice embeddings • 559 Definability from parameters 565 The complexity of the theory of r.e. degrees 572 Absolute definability * 578 Homogeneity • 579 Automorphisms * 580

X.7 Comparison of Degree Theories * 582 One-one degrees 582 Bounded truth-table degrees 585 Truth-table degrees 593 Elementary inequivalences 601

X.8 Structure Inside Degrees * 603 Inside many-one degrees 603

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Inside truth-table degrees 605 Inside weak truth-table degrees 611 Inside Turing degrees 612

X.9 Index Sets • 618 Complexity of index sets 619 Specific index sets 620 Applications of index sets to degrees 632 Global structure 634

XI LIMIT SETS 641 XI.l Jump Classes 641

Domination properties 641 Jump classes 646 Hops 650 Jump inversion below 0' 656 Generalized jump classes * 660

XI.2 1-Generic Degrees 662 Füll approximation arguments 664 Permitting below r.e. degrees • 667 Permitting below GL2 degrees * 669 Permitting below 1-generic degrees below 0' * 671

XI.3 Structure Theory 676 The Diamond Theorem 676 Incomparable degrees 679 The Capping Theorem 683 The Cupping Theorem 687 The Complementation Theorem 689 Exact pairs and ideals 690

XI.4 Minima] Degrees 695 Methodology * 695 Minimal degrees below 0' 698 Pull approximation arguments • 702 Permitting below r.e. degrees * 705 The initial Segments of the degrees below 0' * 708

XI.5 Global Properties 708 Definability from parameters 708 The complexity of the theory of degrees below 0' 714 Absolute definability * 718 Homogeneity * 719 Automorphisms * 720 Subclasses of degrees below 0' • 722 Definability of 0' in the degrees * 726

Contents xv

XI.6 Many-One Degrees * 729 Partial many-one reducibility 729 The mini-jump Operator 732 A2 many-one degrees 734

XII ARITHMETICAL SETS 737 XII.l Forcing in Arithmetic 737

Definition of forcing 738 Generic sets 741 Genericity without forcing * 744 An alternative approach to forcing * 747 History of the notion of forcing * 747 Product forcing * 748 Local forcing on trees 751

XII.2 Applications of Forcing 753 Turing degrees 753 Arithmetical reducibilities * 755 Arithmetical definability * 758 Implicit arithmetical definability • 760 w-Hops 765

XII.3 Turing Degrees of Arithmetical Sets 769 Local and global properties 769 Cones of minimal covers 773 Definability of the Turing degrees of arithmetical sets 776

XIII ARITHMETICAL DEGREES 781 XIII.l The Theory of Arithmetical Degrees 781

The finite extension method 782 The tree method 784 Arithmetical jump 787 Global properties 787 Arithmetical degrees below 0^ 791

XIII.2 An Analogue of R.E. Sets 794 Basic properties 794 Representation of infinite hops 796 The complexity of w-r.e.a. sets 797

XIII.3 An Analogue of Post's Problem 798 A first approximation to the construction 800 The real construction 804 The basic module of the construction 806 The ingredients of the construction 809 An analogue of the Friedberg-Muchnik Theorem 809

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XIII.4 An Analogue of the Jump Classes 811 The main result 811 Jump classes 815 Jump inversion 816

XIII.5 Comparison with the R.E. Degrees 817 Minimal pairs 817 The Diamond Theorem 824

XIV ENUMERATION DEGREES 827 XIV.l Enumeration Degrees 827

Total degrees 829 XIV.2 The Theory of Enumeration Degrees 833

Minimal pairs 834 Minimal Covers 836 Global properties 842

XIV.3 Enumeration Degrees below 0'e 847 Density 849 Greatest lower bounds 851 Least upper bounds 853 Lattice embeddings 853 Global properties 856

XIV.4 A Model of the Lambda Calculus * 857

Bibliography 863

Notation Index 923

Subject Index 929