Modelling computing systems mathematics for computer science
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| Sprache: | English |
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Springer
[2013]
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| Schriftenreihe: | Undergraduate topics in computer science
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| adam_text | Undergraduate Topics in Computer Science
Faron
Moiler
-
Georg Strutti
Modelling Computing Systems
Mathematics for Computer Science
UTiCS
We have all experienced delays and frustrations as a result of the notorious computer
glitch. However, the more dependent we become on computational systems in our
daily lives, the more we must ensure that they are safe, reliable and user-friendly.
Ihis engaging textbook presents the fundamental mathematics and modelling
techniques for computing systems in a novel and light-hearted way, which can be
easily followed by students at the very beginning of their university education.
Кел
concepts
aie
taught through a large collection of challenging yet fun mathematical
games and logical puzzles that require no prior knowledge about computers,
lhe
text begins with intuition and examples as a basis from which precise concepts are
then developed; demonstrating how, by working within the confines of a precise
structured method, the occurrence of errors in the system can be drastically reduced.
Topics and features:
•
Introduces important concepts from discrete mathematics as the basis ot
computational thinking, presented in a stimulating and motivating style
•
Demonstrates how game theory provides a paradigm for an intuitive
understanding of the nature of computation
•
Contains more than
400
exercises throughout the text, with detailed
solutions to half of these presented at the end of the book, together with
numerous theorems, definitions and examples
•
Describes
ЛП
approach to the modelling of computing systems based on state
transition systems, exploring the languages and techniques for expressing
and reasoning about systems specifications and concurrent implementations
This clearly written and classroom-tested textbook/reference is essential reading for
first-year undergraduate modules on discrete mathematics and systems modelling.
Prof.
Faron
Moller
is a Professor of Computer Science at Swansea University, L K.
Dr.
Georg
Struth is a Reader in Computer Science at the University of Sheffield, I K.
Computer Science
1-1-84800-:
►
springer.com
CONTENTS v Contents Starred sections are optional and often represent advanced material. xiii Preface 0 Introduction Examples of System Failures ................................................... 0.1.1 Clayton Tunnel Accident ............................................. 0.1.2 USS Scorpion................................................................. 0.1.3 Therac 25 Radiotherapy Machine ................................ 0.1.4 London Ambulance Service . . .’.................................. 0.1.5 Intel Pentium........................................................ 0.1.6 Ariane 5........................................................................... 0.1.7 Needham-Schroeder Protocol....................................... 0.2 System, Model, Abstraction andNotation................................ 0.3 Specification, Implementation andVerification ...................... 0.1 * 1 2 2 4 4 5 6 7 7 9 13 Part I: Mathematics for Computer Science 15 1 Propositional Logic 17 1.1 Propositions and Deductions..................................................... 1.2 The Language of Propositional Logic....................................... 1.2.1 Propositional Variables ................................................. 1.2.2 Negation ......................................................................... 1.2.3 Disjunction...................................................................... 1.2.4 Conjunction...................................................................... 1.2.5 Implication................................................... . 1.2.6
Equivalence..................................................................... 1.2.7 The Syntax of Propositional Logic.............................. 1.2.8 Parentheses and Precedences....................................... 1.2.9 Syntax Trees.................................................................. 1.3 Modelling with Propositional Logic.......................................... 1.4 Ambiguities of Natural Languages ................................ 1.5 Truth Tables............................................................................. 1.6 Equivalences and Valid Arguments................................ 1.7 Algebraic Laws for Logical Equivalences................................. 1.8 Additional Exercises.................................................................. 2 Sets 2.1 2.2 2.3 18 21 22 22 23 25 25 27 27 28 30 32 35 40 45 47 50 57 Set Notation .............................................................................. Membership, Equality and Inclusion....................................... Sets and Properties..................................................................... 2.3.1 Russell’s Paradox............................................................ 57 59 63 64
vi CONTENTS 2.4 * 2.5 2.6 2-7 2.8 2.9 ★ Ά՜ Operations on Sets . . . .......................................................... 2.4.1 Union............................................................................... 2.4.2 Intersection...................................................................... 2.4.3 Difference......................................................................... 2.4.4 Complement.................................................................... 2.4.5 Powerset.................................................... 2.4.6 Generalised Union and Intersection.............................. Ordered Pairs and Cartesian Products.................................... Modelling with Sets................................................................... Algebraic Laws for Set Identities.............................................. Logical Equivalences versus Set Identities.............................. Additional Exercises................................................................... 4r 3 Boolean Algebras and Circuits 65 65 66 67 68 69 72 73 76 79 81 83 87 1 Boolean Algebras.................................................................... 87 Deriving Identities in Boolean Algebras................................. 90 The Duality Principle................................................................ 93 Logic Gates and Digital Circuits.............................................. 95 Making Computers Add............................................................... 100 3.5.1 Binary
Numbers................................................................ 100 3.5.2 Adding Binary Numbers.................................................... 102 3.5.3 Building Half Adders.......................................................... 103 3.5.4 Building Full Adders.......................................................... 104 3.5.5 Putting It All Together.................................................... 105 3.6 Additional Exercises...................................................................... 106 3. 3.2 3.3 3.4 3.5 4 Predicate Logic 109 4.1 Predicates and Free Variables....................................................... 109 4.2 Quantifiers and Bound Variables.................................. Ill 4.2.1 Universal Quantification................ ................................... 113 4.2.2 Existential Quantification.......................... 115 4.2.3 Bounded Quantifications.................................................... 118 4.3 Rules for Quantification................................................................ 120 4.4 Modelling in Predicate Logic ....................................................... 124 4.5 Additional Exercises...................................................................... 127 Ά- 5 Proof Strategies 5.1 5.2 5.3 5.4 5.5 5.6 131 A First Example............................................................................ 132 Proof Strategies for Implication.................................................... 134 Proof Strategies for Negation....................................................... 138 Proof Strategies
for Conjunction and Equivalence......................142 Proof Strategies for Disjunction.................................................... 144 Proof Strategies for Quantifiers.................................................... 147 5.6.1 Universal Quantification................................................... 147
CONTENTS vii 5.7 5.6.2 Existential Quantification.................. 149 5.6.3 Uniqueness........................................................................152 Additional Exercises.................................................................... 153 6 Functions 155 6.1 Basic Definitions.............................................. 155 6.2 One-To-One and Onto Functions............................................... 160 6.3 Composing Functions ............................... 163 6.4 Comparing the Sizes of Sets.......................................................... 166 6.5 The Knaster-Tarski Theorem...................................................... 173 6.6 Additional Exercises..................................................................... 176 •A· Ar 7 Relations 179 7.1 Basic Definitions........................................................................... 179 7.2 Binary Relations ........................ 181 7.2.1 Functions as Binary Relations........................................... 185 7.3 Operations on Binary Relations..................................................... 186 7.3.1 Boolean Operations.......................................................... 186 7.3.2 Inverting Relations ........................................................... 187 7.3.3 Composing Relations.......................................................... 188 7.3.4 The Domain and Range of a Relation............................ 189 7.4 Properties of BinaryRelations ..................................................... 190 7.4.1 Reflexive and Irreflexive
Relations.................................. 190 7.4.2 Symmetric and Antisymmetric Relations ...................... 191 7.4.3 Transitive Relations .......................................................... 191 7.4.4 Orderings Relations.......................................................... 192 7.4.5 Equivalence Relations ....................................................... 193 7.4.6 Equivalence Classes and Partitions............................... 195 7.5 Additional Exercises..................................................................... 197 8 Inductive and Recursive Definitions 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 9 201 Inductively-Defined Sets............................................................... 201 Inductively-Defined Syntactic Sets..............................................205 Backus-Naur Form.........................................................................207 Inductively-Defined Data Types....................................................210 Inductively-Defined Functions.......................................................212 Recursive Functions......................................................................216 Recursive Procedures..................................................................... 218 Additional Exercises..................................................................... 220 Proofs by Induction 223 9.1 Convincing but Inconclusive Evidence........................................ 223 9.2 A Primary School Induction Argument........................................ 227 9.3 The Induction Argument
.............................................................228
viii CONTENTS ~k ★ 9.4 Strong Induction............ .............................................................. 234 9.5 Induction Proofs from Inductive Definitions.............................. 235 9.6 Fun with Fibonacci Numbers...................................................... 237 9.6.1 A Fibonacci Number Test ................................................ 237 9.6.2 A Carrollean Paradox...................................................... 239 9.6.3 Fibonacci Decompositions............ ;................................ 240 9.7 When Inductions Go Wrong......................................................... 241 9.8 Examples of Induction in Computer Science.............................. 244 9.9 Additional Exercises..................................................................... 246 10 Games and Strategies 251 10.1 Strategies for Games-of-No-Chance............................................. 252 10.2 Nim....................... 260 10.3 Fibonacci Nim.................... 262 10.4 Chomp............................................................................................. 264 10.5 Hex................................................................................................ 266 10.6 Bridg-It.......................................................................................... 269 10.7 Additional Exercises................................................ 271 P art II: Modelling Computing Systems 277 11 Modelling Processes 279 11.1 Labelled Transition Systems......................................................... 281 11.2 Computations and Processes
...................................................... 287 11.3 A Language for Describing Processes.......................................... 292 11.3.1 The Nil Process 0............................................................ 292 11.3.2 Action Prefix..................................................................... 293 11.3.3 Process Definitions.............................. 294 11.3.4 Choice................................................................................. 295 11.4 Distinguishing Between Behaviours............................................. 299 11.5 Equality Between Processes......................................................... 302 11.6 Additional Exercises..................................................................... 303 ֊k 12 Distinguishing Between Processes 309 12.1 The Bisimulation Game.............................................................. 309 12.2 Properties of Game Equivalence.................................................. 313 12.3 Bisimulation Relations................................................................. 315 12.4 Bisimulation Colourings.............................................................. 319 12.5 The Bisimulation Game Revisited: To Infinity and Beyond1. . 322 12.5.1 Ordinal Numbers............................................................... 323 12.5.2 Ordinal BisimulationGames ........................................... 324 12.6 Additional Exercises.................................................................... 328
CONTENTS 13 Logical Properties of Processes * 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 ¡X 333 The Mays and Musts of Processes................................................334 A Modal Logic for Properties..................... 336 Negation Is Definable.....................................................................341 The Vending Machines Revisited................................................344 Modal Properties and Bisimulation............................................. 346 Characteristic Formulæ ...............................................................350 Global Semantics.......................................................................... 352 Additional Exercises.....................................................................353 14 Concurrent Processes 357 Synchronisation Merge.............................. 1................................ 357 Counters.................................................................................... . 360 Railway Level Crossing................................................................. 362 Mutual Exclusion...........................................................................365 14.4.1 Dining Philosophers......................................................... 365 14.4.2 Peterson’s Algorithm......................................................... 368 14.5 A Message Delivery System................................. 371 14.6 Alternating Bit Protocol............................................................... 373 14.7 Additional
Exercises..................................................................... 377 14.1 14.2 14.3 14.4 A՜ 15 Temporal Properties 381 15.1 Three Standard Temporal Operators.......................................... 382 15.1.1 Always: DP.................................................................... 382 15.1.2 Possibly: OP.................................................................... 383 15.1.3 Until: PU Q.................................................................... 384 15.2 Recursive Properties..................................................................... 385 15.2.1 Solving Recursive Equations .......................................... 387 15.2.2 Fixed Point Solutions...................................................... 388 15.3 The Modal Mu-Calculus............................................................... 390 15.4 Least versus Greatest Fixed Points............................................. 392 15.4.1 Approximating Fixed Points .......................................... 393 15.5 Expressing Standard Temporal Operators................................. 397 15.5.1 Always: DP..................................................................... 398 15.5.2 Possibly: OP............................................................ · · 398 15.5.3 Until: PU Q..................................................................... 398 15.6 Further Fixed Point Properties................................................... 399 15.7 Additional Exercises..................................................................... 401 Solutions to Exercises 405 Index
493
|
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| discipline | Informatik Mathematik |
| format | Book |
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| genre_facet | Lehrbuch |
| id | DE-604.BV035279238 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T21:24:53Z |
| institution | BVB |
| isbn | 9781848003217 1848003218 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017084475 |
| oclc_num | 255002850 |
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| owner_facet | DE-703 DE-824 DE-91G DE-BY-TUM DE-384 DE-83 DE-11 DE-355 DE-BY-UBR |
| physical | xvi, 500 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate topics in computer science |
| spellingShingle | Moller, Faron 1962- Struth, Georg ca. 20. / 21. Jh Modelling computing systems mathematics for computer science Theoretische Informatik (DE-588)4196735-5 gnd Modellgetriebene Entwicklung (DE-588)4832365-2 gnd Mathematik (DE-588)4037944-9 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| subject_GND | (DE-588)4196735-5 (DE-588)4832365-2 (DE-588)4037944-9 (DE-588)4120591-1 (DE-588)4123623-3 |
| title | Modelling computing systems mathematics for computer science |
| title_auth | Modelling computing systems mathematics for computer science |
| title_exact_search | Modelling computing systems mathematics for computer science |
| title_full | Modelling computing systems mathematics for computer science Faron Moller, Georg Struth |
| title_fullStr | Modelling computing systems mathematics for computer science Faron Moller, Georg Struth |
| title_full_unstemmed | Modelling computing systems mathematics for computer science Faron Moller, Georg Struth |
| title_short | Modelling computing systems |
| title_sort | modelling computing systems mathematics for computer science |
| title_sub | mathematics for computer science |
| topic | Theoretische Informatik (DE-588)4196735-5 gnd Modellgetriebene Entwicklung (DE-588)4832365-2 gnd Mathematik (DE-588)4037944-9 gnd Komplexitätstheorie (DE-588)4120591-1 gnd |
| topic_facet | Theoretische Informatik Modellgetriebene Entwicklung Mathematik Komplexitätstheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017084475&sequence=000001&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=017084475&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mollerfaron modellingcomputingsystemsmathematicsforcomputerscience AT struthgeorg modellingcomputingsystemsmathematicsforcomputerscience |