Computability and complexity theory:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2011
|
| Edition: | 2. ed. |
| Series: | Texts in computer science
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 298 S. graph. Darst. |
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Record in the Search Index
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| adam_text | Titel: Computability and complexity theory
Autor: Homer, Steven
Jahr: 2011
Contents
1 Preliminaries................................................................ 1
1.1 Words and Languages................................................ 1
1.2 Ä -adic Representation................................................ 2
1.3 Partial Functions...................................................... 3
1.4 Graphs................................................................ 4
1.5 Propositional Logic.................................................. 5
1.5.1 Boolean Functions.......................................... 7
1.6 Cardinality............................................................ 8
1.6.1 OrderedSets................................................ 10
1.7 Elementary Algebra.................................................. 10
1.7.1 Rings and Fields............................................ 10
1.7.2 Groups....................................................... 15
1.7.3 Number Theory............................................. 17
2 Introduction to Computability............................................ 23
2.1 Turing Machines..................................................... 24
2.2 Turing Machine Concepts............................................ 26
2.3 Variations of Turing Machines....................................... 28
2.3.1 Multitape Turing Machines................................ 29
2.3.2 Nondeterministic Turing Machines........................ 32
2.4 Church s Thesis...................................................... 35
2.5 RAMs................................................................. 36
2.5.1 Turing Machines for RAMS............................... 40
3 Undecidability............................................................... 41
3.1 Decision Problems................................................... 41
3.2 Undecidable Problems............................................... 42
3.3 Pairing Functions..................................................... 45
3.4 Computably Enumerable Sets....................................... 47
3.5 Halting Problem, Reductions, and Complete Sets.................. 50
3.5.1 Complete Problems......................................... 52
3.6 S-m-n Theorem....................................................... 53
xiv Contents
3.7 Recursion Theorem.................................................. 56
3.8 Rice s Theorem....................................................... 58
3.9 Turing Reductions and Oracle Turing Machines ................... 60
3.10 Recursion Theorem: Continued..................................... 67
3.11 References............................................................ 71
3.12 Additional Homework Problems.................................... 71
4 Introduction to Complexity Theory...................................... 75
4.1 Complexity Classes and Complexity Measures..................... 76
4.1.1 Computing Functions....................................... 79
4.2 Prerequisites.......................................................... 79
5 Basic Results of Complexity Theory...................................... 81
5.1 Linear Compression and Speedup................................... 82
5.2 Constructible Functions.............................................. 89
5.2.1 Simultaneous Simulation................................... 90
5.3 Tape Reduction....................................................... 93
5.4 Inclusion Relationships.............................................. 99
5.4.1 Relations Between the Standard Classes.................. 107
5.5 Separation Results.................................................... 109
5.6 Translation Techniques and Padding................................ 113
5.6.1 Tally Languages............................................ 116
5.7 Relations Between the Standard Classes: Continued............... 117
5.7.1 Complements of Complexity Classes:
The Immerman-Szelepcsenyi Theorem................... 118
5.8 Additional Homework Problems.................................... 122
6 Nondeterminism and NP-Completeness................................. 123
6.1 Characterizing NP.................................................... 124
6.2 TheClassP........................................................... 125
6.3 Enumerations......................................................... 127
6.4 NP-Completeness.................................................... 129
6.5 The Cook-Levin Theorem........................................... 131
6.6 More NP-Complete Problems....................................... 136
6.6.1 The Diagonal Set Is NP-Complete......................... 137
6.6.2 Some Natural NP-Complete Problems.................... 138
6.7 Additional Homework Problems.................................... 142
7 Relative Computability .................................................... 145
7.1 NP-Hardness.......................................................... 147
7.2 Search Problems...................................................... 150
7.3 The Structure of NP.................................................. 153
7.3.1 Composite Number and Graph Isomorphism............. 157
7.3.2 Reflection................................................... 160
7.4 The Polynomial Hierarchy........................................... 161
7.5 Complete Problems for Other Complexity Classes................. 169
7.5.1 PSPACE..................................................... 169
Contents xv
7.5.2 Exponential Time........................................... 173
7.5.3 Polynomial Time and Logarithmic Space................. 174
7.5.4 A Note on Provably Intractable Problems................. 178
7.6 Additional Homework Problems.................................... 178
8 Nonuniform Complexity................................................... 181
8.1 Polynomial Size Families of Circuits ............................... 184
8.1.1 An Encoding of Circuits.................................... 187
8.1.2 Advice Classes.............................................. 188
8.2 The Low and High Hierarchies...................................... 191
9 Parallelism................................................................... 201
9.1 Alternating Turing Machines........................................ 201
9.2 Uniform Families of Circuits........................................ 209
9.3 Highly Parallelizable Problems...................................... 213
9.4 Uniformity Conditions............................................... 216
9.5 Alternating Turing Machines and Uniform Families of Circuits... 219
10 Probabilistic Complexity Classes......................................... 225
10.1 TheClassPP.......................................................... 225
10.2 TheClassRP......................................................... 229
10.2.1 TheClassZPP.............................................. 230
10.3 TheClassBPP........................................................ 231
10.4 Randomly Chosen Hash Functions.................................. 237
10.4.1 Operators.................................................... 239
10.5 The Graph Isomorphism Problem................................... 242
10.6 Additional Homework Problems.................................... 246
11 Introduction to Counting Classes......................................... 247
11.1 Unique Satisfiability ................................................. 249
11.2 Toda s Theorem...................................................... 253
11.2.1 Results on BPP and © P.................................... 253
11.2.2 The First Part of Toda s Theorem.......................... 257
11.2.3 The Second Part of Toda s Theorem....................... 257
11.3 Additional Homework Problems.................................... 260
12 Interactive Proof Systems.................................................. 261
12.1 The Formal Model ................................................... 261
12.2 The Graph Non-Isomorphism Problem............................. 263
12.3 Arthur-Merlin Games................................................ 265
12.4 IP Is Included in PSPACE ........................................... 267
12.5 PSPACE Is Included in IP........................................... 270
12.5.1 TheLanguageESAT....................................... 270
12.5.2 True Quantified Boolean Formulas........................ 274
12.5.3 The Proof.................................................... 275
12.6 Additional Homework Problems.................................... 282
xvi Contents
References......................................................................... 283
Author Index...................................................................... 289
Subject Index..................................................................... 291
|
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| discipline | Informatik |
| edition | 2. ed. |
| format | Book |
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| physical | XVI, 298 S. graph. Darst. |
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| spelling | Homer, Steven Verfasser aut Computability and complexity theory Steven Homer ; Alan L. Selman 2. ed. New York [u.a.] Springer 2011 XVI, 298 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in computer science Computer science Information theory Computer software Theory of Computation Algorithm Analysis and Problem Complexity Computational complexity Informatik Komplexitätstheorie (DE-588)4120591-1 gnd rswk-swf Berechnungstheorie (DE-588)4005581-4 gnd rswk-swf Berechnungstheorie (DE-588)4005581-4 s DE-604 Komplexitätstheorie (DE-588)4120591-1 s Selman, Alan L. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024768960&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Homer, Steven Selman, Alan L. Computability and complexity theory Computer science Information theory Computer software Theory of Computation Algorithm Analysis and Problem Complexity Computational complexity Informatik Komplexitätstheorie (DE-588)4120591-1 gnd Berechnungstheorie (DE-588)4005581-4 gnd |
| subject_GND | (DE-588)4120591-1 (DE-588)4005581-4 |
| title | Computability and complexity theory |
| title_auth | Computability and complexity theory |
| title_exact_search | Computability and complexity theory |
| title_full | Computability and complexity theory Steven Homer ; Alan L. Selman |
| title_fullStr | Computability and complexity theory Steven Homer ; Alan L. Selman |
| title_full_unstemmed | Computability and complexity theory Steven Homer ; Alan L. Selman |
| title_short | Computability and complexity theory |
| title_sort | computability and complexity theory |
| topic | Computer science Information theory Computer software Theory of Computation Algorithm Analysis and Problem Complexity Computational complexity Informatik Komplexitätstheorie (DE-588)4120591-1 gnd Berechnungstheorie (DE-588)4005581-4 gnd |
| topic_facet | Computer science Information theory Computer software Theory of Computation Algorithm Analysis and Problem Complexity Computational complexity Informatik Komplexitätstheorie Berechnungstheorie |
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