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Computer Science - Theoretical Computer Science | Descriptive Complexity

Descriptive Complexity

Immerman, Neil

1999, XVI, 268 p.

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  • About this textbook

A basic issue in computer science is the complexity of problems. Computational complexity measures how much time or memory is needed as a function of the input problem size. Descriptive complexity is concerned with problems which may be described in first-order logic. By virtue of the close relationship between logic and relational databses, it turns out that this subject has important applications to databases such as analysing the queries computable in polynomial time, analysing the parallel time needed to compute a query, and the analysis of nondeterministic classes. This book is written as a graduate text and so aims to provide a reasonably self-contained introduction to this subject. The author has provided numerous examples and exercises to further illustrate the ideas presented.

Content Level » Research

Keywords » Arithmetic - Counting - Lemma - Switch - Variable - complexity - databases - logic

Related subjects » Information Systems and Applications - Mathematics - Theoretical Computer Science

Table of contents 

1 Background in Logic.- 1.1 Introduction and Preliminary Definitions.- 1.2 Ordering and Arithmetic.- 1.2.1 FO(BIT) = FO(PLUS, TIMES).- 1.3 Isomorphism.- 1.4 First-Order Queries.- 2 Background in Complexity.- 2.1 Introduction.- 2.2 Preliminary Definitions.- 2.3 Reductions and Complete Problems.- 2.4 Alternation.- 2.5 Simultaneous Resource Classes.- 2.6 Summary.- 3 First-Order Reductions.- 3.1 FO ? L.- 3.2 Dual of a First-Order Query.- 3.3 Complete problems for L and NL.- 3.4 Complete Problems for P.- 4 Inductive Definitions.- 4.1 Least Fixed Point.- 4.2 The Depth of Inductive Definitions.- 4.3 Iterating First-Order Formulas.- 5 Parallelism.- 5.1 Concurrent Random Access Machines.- 5.2 Inductive Depth Equals Parallel Time.- 5.3 Number of Variables Versus Number of Processors.- 5.4 Circuit Complexity.- 5.5 Alternating Complexity.- 5.5.1 Alternation as Parallelism.- 6 Ehrenfeucht-Fraïssé Games.- 6.1 Definition of the Games.- 6.2 Methodology for First-Order Expressibility.- 6.3 First-Order Properties Are Local.- 6.4 Bounded Variable Languages.- 6.5 Zero-One Laws.- 6.6 Ehrenfeucht-Fraïssé Games with Ordering.- 7 Second-Order Logic and Fagin’s Theorem.- 7.1 Second-Order Logic.- 7.2 Proof of Fagin’s Theorem.- 7.3 NP-Complete Problems.- 7.4 The Polynomial-Time Hierarchy.- 8 Second-Order Lower Bounds.- 8.1 Second-Order Games.- 8.2 SO?(monadic) Lower Bound on Reachability.- 8.3 Lower Bounds Including Ordering.- 9 Complementation and Transitive Closure.- 9.1 Normal Form Theorem for FO(LFP).- 9.2 Transitive Closure Operators.- 9.3 Normal Form for FO(TC).- 9.4 Logspace is Primitive Recursive.- 9.5 NSPACE[s(n)] = co-NSPACE[s(n)].- 9.6 Restrictions of SO.- 10 Polynomial Space.- 10.1 Complete Problems for PSPACE.- 10.2 Partial Fixed Points.- 10.3 DSPACE[nk] = VAR[k + 1].- 10.4 Using Second-Order Logic to Capture PSPACE.- 11 Uniformity and Precompulation.- 11.1 An Unbounded Number of Variables.- 11.1.1 Tradeoffs Between Variables and Quantifier Depth.- 11.2 First-Order Projections.- 11.3 Help Bits.- 11.4 Generalized Quantifiers.- 12 The Role of Ordering.- 12.1 Using Logic to Characterize Graphs.- 12.2 Characterizing Graphs Using Lk.- 12.3 Adding Counting to First-Order Logic.- 12.4 Pebble Games for Ck.- 12.5 Vertex Refinement Corresponds to C2.- 12.6 Abiteboul-Vianu and Otto Theorems.- 12.7 Toward a Language for Order-Independent P.- 13 Lower Bounds.- 13.1 Håstad’s Switching Lemma.- 13.2 A Lower Bound for REACHa.- 13.3 Lower Bound for Fixed Point and Counting.- 14 Applications.- 14.1 Databases.- 14.1.1 SQL.- 14.1.2 Catalog.- 14.2 Dynamic Complexity.- 14.2.1 Dynamic Complexity Classes.- 14.3 Model Checking.- 14.3.1 Temporal Logic.- 14.4 Summary.- 15 Conclusions and Future Directions.- 15.1 Languages That Capture Complexity Classes.- 15.1.1 Complexity on the Face of a Query.- 15.1.2 Stepwise Refinement.- 15.2 Why Is Finite Model Theory Appropriate?.- 15.3 Deep Mathematical Problems: P versus NP.- 15.4 Toward Proving Lower Bounds.- 15.4.1 Role of Ordering.- 15.4.2 Approximation and Approximability.- 15.5 Applications of Descriptive Complexity.- 15.5.1 Dynamic Complexity.- 15.5.2 Model Checking.- 15.5.3 Abstract State Machines.- 15.6 Software Crisis and Opportunity.- 15.6.1 How can Finite Model Theory Help?.- References.

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