Limits of computation: from a programming perspective
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Switzerland
Springer
[2016]
|
| Series: | Undergraduate topics in computer science
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xviii, 348 Seiten Illustrationen, Diagramme |
| ISBN: | 9783319278872 |
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| 245 | 1 | 0 | |a Limits of computation |b from a programming perspective |c Bernhard Reus |
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Record in the Search Index
| _version_ | 1804176171665981440 |
|---|---|
| adam_text | Contents
1 Limits? What Limits?................................................. 1
1.1 Physical Limits of Computation................................ 2
1.1.1 Fundamental Engineering Constraints to
Semiconductor Manufacturing and Scaling............... 2
1.1.2 Fundamental Limits to Energy Efficiency............... 3
1.1.3 Fundamental Physical Constraints on Computing
in General............................................ 3
1.2 The Limits Addressed.......................................... 4
1.2.1 Computability Overview................................ 4
1.2.2 Complexity Overview................................... 6
References.......................................................... 9
Part I Computability
2 Problems and Effective Procedures................................... 13
2.1 On Computability............................................. 14
2.1.1 Historical Remarks................................. . 14
2.1.2 Effective Procedures................................. 16
2.2 Sets, Relations and Functions................................ 17
2.2.1 Sets................................................. 17
2.2.2 Relations............................................ 21
2.2.3 Functions............................................ 22
2.2.4 Partial Functions.................................... 22
2.2.5 Total Functions...................................... 23
2.3 Problems..................................................... 24
2.3.1 Computing Solutions to Problems...................... 26
References......................................................... 27
xiii
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32
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67
70
The WHILE-Language..........................
3.1 The Data Type of Binary Trees........
3.2 WHILE-Syntax.........................
3.2.1 Expressions...................
3.2.2 Commands......................
3.2.3 Programs......................
3.2.4 A Grammar for WHILE...........
3.2.5 Layout Conventions and Brackets
3.3 Encoding Data Types as Trees.........
3.3.1 Boolean Values................
3.3.2 Lists and Pairs...............
3.3.3 Natural Numbers...............
3.3.4 Finite Words..................
3.4 Sample Programs......................
3.4.1 Addition......................
3.4.2 List Reversal.................
3.4.3 Tail Recursion................
3.4.4 Analysis of Algorithms........
References..................................
Semantics of while..........................
4.1 Stores...............................
4.2 Semantics of Programs................
4.3 Semantics of Commands................
4.4 Semantics of Expressions.............
References..................................
Extensions of while.........................
5.1 Equality.............................
5.2 Literals ............................
5.2.1 Number Literals............... -
5.2.2 Boolean Literals..............
5.3 Adding Atoms.........................
5.4 List Constructor.....................
5.5 Macro Calls..........................
5.6 Switch Statement.....................
References..................................
Programs as Data Objects....................
61 Interpreters Formally....................
6.2 Abstract Syntax Trees................
6.3 Encoding of WHILE-ASTs in D..........
Reference...................................
Contents
xv
7 A Self-interpreter for WHILE........................................ 71
7.1 A Self-interpreter for WHILE -Programs with One Variable. ... 72
7.1.1 General Tree Traversal for ASTs.................... 72
7.1.2 The STEP Macro....................................... 73
7.2 AS elf- in terpreter for WHILE............................... 81
7.2.1 Store Manipulation Macros............................ 83
References.......................................................... 86
8 An Undecidable (Non-computable) Problem............................. 87
8.1 WHILE-Computability and Decidability......................... 87
8.2 Hie Plaiting Problem for WHILE............................... 89
8.3 Diagonalisation and the Barber “Paradox”..................... 90
8.4 Proof of the Undecidability of the Halting Problem........... 92
References.......................................................... 95
9 More Undecidable Problems........................................... 97
9.1 Semi-decidability of the Halting Problem..................... 97
9.2 Rice’s Theorem............................................... 99
9.3 The Tiling Problem.......................................... 101
9.4 Problem Reduction........................................... 103
9.5 Other (Famous) Undecidable Problems......................... 105
9.6 Dealing with Undecidable Problems........................... 106
9.7 A Fast-Growing Non-computable Function...................... 107
References......................................................... Ill
10 Self-referencing Programs.......................................... 113
10.1 The S -m-n Theorem.......................................... 114
10.2 Kleene’s Recursion Theorem.................................. 116
10.3 Recursion Elimination....................................... 118
References......................................................... 121
11 The Church-Turing Thesis........................................... 123
11.1 The Thesis.................................................. 124
11.2 Semantic Framework for Machine-Like Models.................. 125
11.3 Turing Machines TM.......................................... 126
11.4 GOTO-Language............................................... 129
11.5 Register Machines RAW and SRAM.............................. 131
11.6 Counter Machines CM......................................... 134
11.7 Cellular Automata........................................... 135
11.7.1 2D: Game of Life.................................... 138
11.7.2 ID: Rule 110........................................ 140
11.8 Robustness of Computability................................ 141
11.8.1 The Crucial Role of Compilers....................... 141
11.8.2 Equivalence of Models............................... 142
References......................................................... 147
XVI
Contents
Part n Complexity
12 Measuring Time Usage............................................ 151
12.1 Unit-Cost Time Measure.................................... 152
12.2 Time Measure for WHILE.................................... 154
12.3 Comparing Programming Languages Considering Time.......... 157
References...................................................... 160
13 Complexity Classes.............................................. 161
13.1 Runtime Bounds............................................ 162
13.2 Time Complexity Classes................................... 163
13.3 Lifting Simulation Properties to Complexity Classes....... 165
13.4 Big-O and LittLe-o........................................ 166
References...................................................... 171
14 Robustness of P................................................. 173
14.1 Extended Church-Turing Thesis............................. 174
14.2 Invariance or Cook’s Thesis............................... 174
14.2.1 Non-sequential Models............................. 175
14.2.2 Evidence for Cook’s Thesis........................ 176
14.2.3 Linear Time....................................... 178
14.3 Cobham-Edmonds Thesis..................................... 179
References...................................................... 181
15 Hierarchy Theorems.............................................. 183
15.1 Linear Time Hierarchy Theorems............................ 184
15.2 Beyond Linear Time........................................ 189
15.3 Gaps in the Hierarchy..................................... 192
References...................................................... 193
16 Famous Problems in P............................................ 195
16.1 Decision Versus Optimisation Problems..................... 197
16.2 Predecessor Problem....................................... 198
16.3 Membership Test for a Context Free Language............... 201
16.4 Primality Test............................................ 202
16.5 Graph Problems............................................ 203
16.5.1 Reachability in a Graph........................... 203
16.5.2 Shortest Paths in a Graph......................... 204
16.5.3 Maximal Matchings................................. 206
16.5.4 Min-Cut and Max-Flow.............................. 207
16.5.5 The Seven Bridges of Königsberg................... 208
16.6 Linear Programming........................................ 210
References...................................................... 215
17 Common Problems Not Known to Be in P............................ 217
17.1 The Travelling Salesman Problem (TSP)..................... 218
17.2 The Graph Colouring Problem............................... 220
Contents xvii
17.3 Max-Cut Problem............................................. 221
17.4 The 0-1 Knapsack Problem.................................... 222
17.5 Integer Programming Problem................................. 223
17.6 Does Not Being in P Matter?................................. 224
References........................................................ 226
18 The One-Million-Dollar Question................................... 227
18.1 The Complexity Class NP..................................... 228
18.2 Nondeterannistic Programs................................... 229
18.2.1 Time Measure of Nondeterminisäc Programs............ 231
18.2.2 Some Basic Facts About NP........................... 233
18.3 Robustness of NP............................................ 234
18.4 Problems m NP............................................... 235
18.5 The Biggest Open Problem in (Theoretical)
Computer Science............................................ 237
References........................................................ 239
19 How Hard Is a Problem?............................................ 241
19.1 Reminder: Effective Reductions.............................. 242
19.2 Polynomial Time Reduction................................... 242
19.3 Hard Problems............................................... 245
References........................................................ 249
20 Complete Problems................................................. 251
20.1 A First NP-complete Problem................................. 252
20.2 More NP-complete Problems................................... 255
20.3 Puzzles and Games........................................... 256
20.3.1 Chess............................................... 258
20.3.2 Sudoku.............................................. 259
20.3.3 Tile-Matching Games................................. 260
20.4 Database Queries............................................ 261
20.5 Policy Based Routing........................................ 264
20.6 “Limbo** Problems........................................... 266
20.7 Complete Problems in Other Classes.......................... 268
20.7.1 P-complete.......................................... 268
20.7.2 RTEcomplete......................................... 269
References........................................................ 273
21 How to Solve NP-Complete Problems................................. 275
21.1 Exact Algorithms............................................ 276
21.2 Approximation Algorithms.................................... 276
21.3 Parallelism................................................. 281
21.4 Randomization............................................... 282
21.4.1 The Class RP........................................ 282
21.4-2 Probabilistic Algorithms.......................... 284
xviii Contents
21.5 Solving the Travelling Salesman Problem.................... 285
21.5.1 Exact Solutions.................................... 285
21.5.2 Approximative Solutions............................ 286
21.6 When Bad Complexity is Good News........................... 289
References....................................................... 295
22 Molecular Computing............................................. 299
22.1 The Beginnings of DNA Computing............................ 300
22.2 DNA Computing Potential.................................... 301
22.3 DNA Computing Challenges................................... 302
22.4 Abstract Models of Molecular Computation................... 302
22.4.1 Chemical Reaction Networks (CRN).................. 303
22.4.2 CRNs as Effective Procedures....................... 305
22.4.3 Are CRNs Equivalent to Other Notions
of Computation?.................................... 308
22.4.4 Time Complexity for CRNs........................... 309
22.4.5 Implementing CRNs.................................. 309
References....................................................... 315
23 Quantum Computing............................................... 317
23.1 Molecular Electronics...................................... 318
23.2 The Mathematics of Quantum Mechanics....................... 319
23.3 Quantum Computability and Complexity....................... 321
23.4 Quantum Algorithms......................................... 323
23.4.1 Shor’s Algorithm................................... 323
23.4.2 Grover’s Algorithm................................. 324
23.5 Building Quantum Computers................................. 325
23.6 Quantum Computing Challenges............................... 325
23.7 To Boldly Go............................................... 326
References.................... ..............-.................. 328
Further Reading—Computability and Complexity Textbooks.......... 331
Glossary........................................................... 335
Index
341
Undergraduate Topics in Computer Science
Series Editor: Ian Mackie
Bernhard Reus
This textbook discusses the most fundamental and puzzling questions about the
foundations of computing. In 23 lecture-sized chapters it provides an exciting tour
through the most important results in the field of computability and time complexity
u Rices Theorem, Kleenes Recursion Theorem, the
Church-Hiring Thesis, Hierarchy Theorems, and Cook-Levins Theorem, Each chapter
contains classroom-tested material, including examples and exercises. Links between
adjacent chapters provide a coherent narrative.
Fundamental results are explained lucidly by means of programs written in a simple,
high-level imperative programming language, which only requires basic mathematical
knowledge. Throughout the book, the impact of the presented results on the entire
field of computer science is emphasised. Examples range from program analysis to
networking, from database programming to popular games and puzzles. Numerous
biographical footnotes about the famous scientists who developed the subject are also
included.
Limits if Computation offers a thorough, yet accessible, introduction to computability
and complexity for the computer science student of the 21st century.
|
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| id | DE-604.BV043523761 |
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| indexdate | 2024-07-10T07:27:57Z |
| institution | BVB |
| isbn | 9783319278872 |
| language | English |
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| physical | xviii, 348 Seiten Illustrationen, Diagramme |
| publishDate | 2016 |
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| publisher | Springer |
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| series2 | Undergraduate topics in computer science |
| spelling | Reus, Bernhard 1965- Verfasser (DE-588)1111581339 aut Limits of computation from a programming perspective Bernhard Reus Switzerland Springer [2016] © 2016 xviii, 348 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Undergraduate topics in computer science Mathematik (DE-588)4037944-9 gnd rswk-swf Informatik (DE-588)4026894-9 gnd rswk-swf Algorithmische Programmierung (DE-588)4293504-0 gnd rswk-swf Algorithmische Programmiersprache (DE-588)4141876-1 gnd rswk-swf Algorithmische Programmierung (DE-588)4293504-0 s DE-604 Informatik (DE-588)4026894-9 s Mathematik (DE-588)4037944-9 s Algorithmische Programmiersprache (DE-588)4141876-1 s 1\p DE-604 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028939622&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028939622&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Reus, Bernhard 1965- Limits of computation from a programming perspective Mathematik (DE-588)4037944-9 gnd Informatik (DE-588)4026894-9 gnd Algorithmische Programmierung (DE-588)4293504-0 gnd Algorithmische Programmiersprache (DE-588)4141876-1 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4026894-9 (DE-588)4293504-0 (DE-588)4141876-1 |
| title | Limits of computation from a programming perspective |
| title_auth | Limits of computation from a programming perspective |
| title_exact_search | Limits of computation from a programming perspective |
| title_full | Limits of computation from a programming perspective Bernhard Reus |
| title_fullStr | Limits of computation from a programming perspective Bernhard Reus |
| title_full_unstemmed | Limits of computation from a programming perspective Bernhard Reus |
| title_short | Limits of computation |
| title_sort | limits of computation from a programming perspective |
| title_sub | from a programming perspective |
| topic | Mathematik (DE-588)4037944-9 gnd Informatik (DE-588)4026894-9 gnd Algorithmische Programmierung (DE-588)4293504-0 gnd Algorithmische Programmiersprache (DE-588)4141876-1 gnd |
| topic_facet | Mathematik Informatik Algorithmische Programmierung Algorithmische Programmiersprache |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028939622&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028939622&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT reusbernhard limitsofcomputationfromaprogrammingperspective |