Logic and Complexity

By: Richard Lassaigne, Michel de Rougemont

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Hardcover | 20 January 2004

At a Glance

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376 Pages

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23.5 x 15.88 x 2.54

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Logic and Complexity looks at basic logic as it is used in Computer Science, and provides students with a logical approach to Complexity theory. With plenty of exercises, this book presents classical notions of mathematical logic, such as decidability, completeness and incompleteness, as well as new ideas brought by complexity theory such as NP-completeness, randomness and approximations, providing a better understanding for efficient algorithmic solutions to problems.

Divided into three parts, it covers:

- Model Theory and Recursive Functions - introducing the basic model theory of propositional, 1st order, inductive definitions and 2nd order logic. Recursive functions, Turing computability and decidability are also examined.

- Descriptive Complexity - looking at the relationship between definitions of problems, queries, properties of programs and their computational complexity.

- Approximation - explaining how some optimization problems and counting problems can be approximated according to their logical form.

Logic is important in Computer Science, particularly for verification problems and database query languages such as SQL. Students and researchers in this field will find this book of great interest.

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Non-Fiction Mathematics Mathematical Foundation Mathematical Logic Computing & I.T.Computer Science Mathematical Theory of Computation Computer Architecture & Logic Design Probability & Statistics Discrete Mathematics Optimisation Linear Programming

Calculus & Mathematical Analysis Databases Data Capture & Analysis Computer Programming & Software Development

Introduction	p. 1
Basic model theory and computability	p. 3
Propositional logic	p. 5
Propositional language	p. 5
Construction of formulas	p. 5
Proof by induction	p. 7
Decomposition of a formula	p. 7
Semantics	p. 9
Tautologies. Equivalent formulas	p. 10
Logical consequence	p. 11
Value of a formula and substitution	p. 11
Complete systems of connectives	p. 15
Normal forms	p. 15
Disjunctive and conjunctive normal forms	p. 15
Functions associated to formulas	p. 16
Transformation methods	p. 17
Clausal form	p. 19
OBDD: Ordered Binary Decision Diagrams	p. 20
Exercises	p. 23
Deduction systems	p. 25
Examples of tableaux	p. 25
Tableaux method	p. 27
Trees	p. 28
Construction of tableaux	p. 29
Development and closure	p. 30
Completeness theorem	p. 31
Provable formulas	p. 31
Soundness	p. 31
Completeness	p. 32
Natural deduction	p. 33
Compactness theorem	p. 36
Exercices	p. 38
First-order logic	p. 41
First-order languages	p. 41
Construction of terms	p. 42
Construction of formulas	p. 43
Free and bound variables	p. 44
Semantics	p. 45
Structures and languages	p. 45
Structures and satisfaction of formulas	p. 46
Valid and equivalent formulas	p. 48
Substitution	p. 50
Prenex formulas and Skolem forms	p. 51
Prenex formulas and normal forms	p. 51
Skolem forms	p. 53
Exercises	p. 55
Completeness of first order logic	p. 57
Natural deduction sytem	p. 57
Soundness	p. 59
Completeness	p. 59
Examples of theories	p. 61
Successor function	p. 62
Discrete successor function	p. 62
Graphs	p. 63
Robinson's arithmetic	p. 64
Exercises	p. 65
Models of computation	p. 67
General model of computation	p. 67
Turing machines	p. 68
Deterministic Turing machines	p. 68
Non-deterministic Turing machines	p. 71
Machines with multiple tapes	p. 74
Turing machine with oracles	p. 76
Alternating machines	p. 76
Other sequential models of computation	p. 78
Finite automata	p. 78
RAM: Random Access Machines	p. 80
Exercises	p. 82
Recursion and decidability	p. 83
Recursive functions	p. 83
Primitive recursive functions	p. 83
Construction of primitive recursive functions	p. 84
Coding sequences	p. 86
Non-primitive recursion	p. 87
Recursive functions	p. 88
Church's thesis	p. 89
Recursion and computation	p. 90
Recursive functions are Turing computable	p. 90
Turing computable functions are recursive	p. 91
Universal Turing machine	p. 93
Recursive systems	p. 95
Equivalence with the recursive functions	p. 98
Implementation of recursive functions	p. 98
Recursion and decidability	p. 101
Recursive and recursively enumerable sets	p. 101
The halting problem	p. 103
Reductions	p. 104
The s-m-n and Rice's theorems	p. 105
Exercises	p. 107
Incompleteness of Peano arithmetic	p. 109
Arithmetic theory and representable functions	p. 109
Peano arithmetic	p. 109
Arithmetical proofs	p. 110
Non-standard models of arithmetic	p. 111
Representable functions	p. 111
Coding arithmetical proofs	p. 114
Coding terms and formulas	p. 115
Coding sequences of formulas	p. 116
Arithmetical theorems are recursively enumerable	p. 116
Undecidability and incompleteness	p. 117
The validity problem over finite structures	p. 118
The arithmetical hierarchy	p. 120
Exercises	p. 127
Descriptive Complexity	p. 129
Complexity: time and space	p. 131
Complexity classes	p. 132
Complexity of decision problems	p. 132
Examples of decision problems	p. 135
Complexity of search problems	p. 136
Hierarchies for time and space	p. 139
Simulation of several tapes	p. 139
The role of constants	p. 140
Relations between classes	p. 140
Hierarchies	p. 141
Non-deterministic space classes	p. 143
Other models and measures	p. 147
Random access machines	p. 147
Complexity on general structures	p. 148
Average and smoothed complexities	p. 148
Other measures	p. 148
Exercises	p. 150
First-order definability	p. 151
Explicit definitions	p. 151
Definability on a class of structures	p. 152
Partial elementary equivalence	p. 153
Ehrenfeucht-Fraïssé games	p. 154
Logical characterization of r-equivalence	p. 156
Applications of Ehrenfeucht-Fraïssé games	p. 158
Relational structures	p. 160
0-1 Law	p. 162
Loś-Vaught's test	p. 163
The theory of almost all relational structures	p. 163
Convergence and 0-1 law	p. 164
Implicit definitions	p. 166
Craig's interpolation theorem	p. 166
Beth's definability theorem	p. 168
Finite structures	p. 169
Exercises	p. 171
Inductive definitions and second-order logic	p. 175
Inductive definitions	p. 175
Inductive systems	p. 176
Fixed point logic	p. 178
Substitution	p. 179
Stage comparison	p. 180
Fixed point hierarchy	p. 182
Datalog	p. 184
Second-order logic	p. 186
Second-order formulas and interpretation	p. 186
Inductive definitions and second-order logic	p. 187
Exercises	p. 189
Time complexity : the classes P and NP	p. 191
The class NP and NP-complete problems	p. 191
Polynomial time reductions	p. 192
NP-complete problems	p. 192
Logical characterization of NP	p. 199
Examples of definable NP problems	p. 202
The logical characterizations of P and FP classes	p. 204
P: the class of global inductive relations	p. 204
FP: the class of global recursive functions	p. 208
Exercises	p. 211
Models of parallel computations	p. 213
The boolean circuits	p. 213
Complexity in parallel models	p. 217
Other circuits	p. 219
Examples of NC problems	p. 219
Matrix multiplication	p. 219
Csanky's matrix inversion algorithm	p. 220
The Parallel RAM	p. 223
Exercises	p. 226
Space complexity: the classes L, FL, NL and PSPACE	p. 229
The classes L, NL and FL	p. 230
NL-complete problems	p. 230
Alternation with space constraints	p. 231
Logical characterization of NL	p. 232
Logical characterization of FL	p. 236
Sequential space and parallel time	p. 241
The parallel time thesis	p. 243
P-complete problems	p. 243
The class PSPACE	p. 244
The quantified boolean formulas and the class AP	p. 244
Stochastic satisfiability	p. 246
Problems with uncertainty	p. 247
Exercises	p. 249
Definability of optimization and counting problems	p. 251
Optimization problems	p. 251
Counting problems	p. 257
The counting classes	p. 257
Definability of counting functions	p. 259
Exercises	p. 262
Approximation and classes beyond NP	p. 263
Probabilistic Classes	p. 265
Probabilistic decision classes	p. 265
Error amplification	p. 268
Comparing PP and #P	p. 269
Other probabilistic classes	p. 269
Examples of probabilistic algorithms	p. 270
Random walk in an undirected graph	p. 270
Perfect matching in a bipartite graph	p. 272
Solovay-Strassen primality test	p. 275
Exercises	p. 277
Probabilistic verification	p. 279
Interactive proofs	p. 280
Examples of interactive protocols	p. 281
The problem of non-isomorphism of graphs	p. 281
A protocol for the permanent	p. 283
A protocol for QBF	p. 285
Multiprover protocols	p. 289
Probabilistic checkable proofs PCP	p. 290
A PCP (n³, 1) for 3SAT	p. 293
Exercises	p. 297
Approximation	p. 299
Optimization problems	p. 299
Examples of approximation algorithms	p. 301
Approximation of optimization problems	p. 304
Reductions and completeness	p. 309
Non-approximability	p. 311
Counting problems	p. 314
Approximation of counting problems	p. 314
Approximation of #DNF	p. 315
Approximate verification	p. 317
Exercises	p. 320
Classes beyond	p. 323
The polynomial hierarchy	p. 323
Second-order definability	p. 323
Relative computability	p. 324
Alternating machines	p. 326
Generalizations of NP	p. 327
The classes <$>oplus<$> P, PP and BPP	p. 328
Universal hashing	p. 331
The Valiant-Vazirani theorem	p. 333
Toda's theorem	p. 334
Exponential time	p. 338
Exercises	p. 342
Bibliography	p. 343
List of Figures	p. 349
Index	p. 353
Table of Contents provided by Publisher. All Rights Reserved.

Logic and Complexity

At a Glance

Hardcover

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