Bounded Queries in Recursion Theory

ISBN-13: 9780817639662 / Angielski / Twarda / 1998 / 353 str.

William Levine;Georgia Martin

Opis

One of the major concerns of theoretical computer science is the classifi cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function."

Recenzje

"Ideal for an advanced undergraduate or beginning graduate student who has some exposure to basic computability theory and wants to see what one can do with it. The questions asked are interesting and can be easily understood and the proofs can be followed without a large amount of training in computability theory."

--Sigact News

Spis treści

A: Getting Your Feet Wet.- 1 Basic Concepts.- 1.1 Notation, Conventions, and Definitions.- 1.2 Basic Recursion Theory.- 1.2.1 Recursive and Recursively Enumerable Sets.- 1.2.2 Reductions.- 1.2.3 Jump and the Arithmetic Hierarchy.- 1.2.4 Simulation and Dovetailing.- 1.3 Useful Concepts from Recursion Theory.- 1.3.1 The Recursion Theorem.- 1.3.2 Initial Segment Arguments.- 1.4 Advanced Concepts We Will Need.- 1.5 Exercises.- 1.6 Bibliographic Notes.- 2 Bounded Queries and the Halting Set.- 2.1 Basic Results About K.- 2.2 Exercises.- 2.3 Bibliographic Notes.- 3 Definitions and Questions.- 3.1 Definitions of Classes.- 3.2 Questions We Address.- 3.3 Enumerability and Bounded Queries.- 3.4 Uniformity.- 3.5 Order Notation.- 3.6 Exercises.- 3.7 Bibliographic Notes.- B: The Complexity of Functions.- 4 The Complexity of CnA.- 4.1 A General Lower Bound.- 4.2 Class Separation: FQ(n,A) ? FQ(n+ 1,A).- 4.3 Verbose Sets.- 4.4 Non-superterse Sets Are Semiverbose.- 4.5 Superterse Sets.- 4.6 Semiverbose Sets.- 4.7 Most Sets Are Superterse.- 4.7.1 The Set of Superterse Sets Has Measure 1.- 4.7.2 The Set of Superterse Sets Is Co-meager.- 4.8 Recursively Enumerable Sets.- 4.8.1 Using Queries to Other Sets.- 4.8.2 Using Queries to A.- 4.8.3 An R.E. Terse Set in Every Nonzero R.E. Degree.- 4.9 Exercises.- 4.10 Bibliographic Notes.- 5 #nA and Other Functions.- 5.1 Trees.- 5.2 Ramsey Theory.- 5.3 Lower Bound on the Complexity of #nA.- 5.4 A General Theorem.- 5.5 Strong Class Separation: EN(2n - 1) ? FQ(n, A).- 5.6 An Alternative Proof.- 5.7 Exercises.- 5.8 Bibliographic Notes.- C: The Complexity of Sets.- 6 The Complexity of ODDnA and MODmnA.- 6.1 ODDnA for Semirecursive Sets A.- 6.2 ODDnA for R.E. Sets A.- 6.2.1 A Proof in the Spirit of Theorem 4.1.1.- 6.2.2 A Proof in the Spirit of Theorem 5.3.2.- 6.2.3 A Very Unusual Proof.- 6.3 The Complexity of MODmnA.- 6.4 ODDnA and MODmnA Can Be Easy.- 6.5 Exercises.- 6.6 Bibliographic Notes.- 7 Q Versus QC.- 7.1 Preliminaries.- 7.2 K Is U.C.- 7.3 Nonzero R.E. Degrees R.E.S.N.U.C.- 7.4 An Intermediate R.E. Set That Is U.C.- 7.5 A Completely R.E.S.N.U.C. Degree.- 7.6 Other U.C. Sets.- 7.7 Truth Table Degrees.- 7.8 Exercises.- 7.9 Bibliographic Notes.- 8 Separating and Collapsing Classes.- 8.1 Sets That Are Both Supportive and Parallel Supportive.- 8.2 Sets That Are Supportive but Not Parallel Supportive.- 8.3 Sets That Are Neither Supportive Nor Parallel Supportive.- 8.4 Exercises.- 8.5 Bibliographic Notes.- D: Miscellaneous.- 9 Nondeterministic Complexity.- 9.1 Nondeterministic Computation.- 9.2 Subjective Sets.- 9.3 Locally Subjective Sets.- 9.3.1 K Is Not Locally Subjective.- 9.3.2 There Exist Locally 1-Subjective Sets.- 9.3.3 Characterizing Locally 1-Subjective Sets.- 9.4 Exercises.- 9.5 Bibliographic Notes.- 10 The Literature on Bounded Queries.- References.