Handbook of Logic
in Computer Science
Volume 1
Background: Mathematical Structures
Edited by
S. ABRAMSKY
Professor of Computing Science
DOV M. GABBAY
Professor of Computing Science
and
T. S. E. MAIBAUM
Professor of Foundations of
Software Engineering
Imperial College of Science, Technology and Medicine
London
Volume Coordinator
DOV M. GABBAY
Technische Hochschule Darmstadt
FACHBEREICH INFORMATIK
B I B L I O T H E K
Inventar-Nr.:
Sachgebieta:
Standort:
CLARENDON PRESS
1992
OXFORD
Contents
Valuation systems and consequence
relations
1
2
3
4
5
Introduction
1.1 Logics and computer science
1.2 Summary
,
Valuation systems
2.1 Satisfaction
2.2 Valuation systems
2.3 Modal logic and possible worlds
2.4 Predicate languages
2.5 Summary
Consequence relations and entailment relations
3.1 Consequence relations
3.2 Entailment relations
3.3 The systems C and 54
3.4 Levels of implication
3.5 Consequence operator
3.6 Summary
Proof theory and presentations
4.1 Hilbert presentations
4.2 Natural deduction presentations
4.3 Natural deduction in sequent style
4.4 Intuitionistic logic
4.5 Gentzen sequent calculus for /
4.6 Gentzen sequent calculus for C and 54
4.7 Properties of presentations
Some further topics
5.1 Valuation systems for I
5.2 Maps between logics
5.3 Correspondence theory
5.4 Consistency
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Recursion theory
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Introduction
0.1 Opening remarks
0.2 A taster
0.3 Contents of the chapter
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Contents
Languages and notions of computability
1.1 Data types and coding
1.2 The imperative paradigm
1.3 The functional paradigm
1.4 Recursive functions
1.5 Universality
Computability and non-computability
2.1 Non-computability
2.2 Computability
2.3 Recursive and recursively enumerable sets
2.4 The S-m-n theorem and partial evaluation
2.5 More undecidable problems
2.6 Problem reduction and r.e. completeness
Inductive definitions
3.1 Operators and fixed points
3.2 The denotational semantics of the functional
language FL
3.3 Ordinals
3.4 The general case
Recursion theory
4.1 Fixed point theorems
4.2 Acceptable programming systems
4.3 Recursive operators
4.4 Inductive definitions and logics
Universal algebra
1
2
3
Introduction
1.1 What is universal algebra?
1.2 Universal algebra in mathematics and computer science
1.3 Overview of the chapter
1.4 Historical notes
1.5 Acknowledgements
1.6 Prerequisites
Examples of algebras
2.1 Some basic algebras
2.2 Some simple constructions
2.3 Syntax and semantics of programs
2.4 Synchronous concurrent algorithms
2.5 Algebras and the modularisation of software
Algebras and morphisms
3.1 Signatures and algebras
3.2 Subalgebras
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3.3 Congruences and quotient algebras
3.4 Homomorphisms and isomorphisms
3.5 Direct products
3.6 Abstract data types
Constructions
4.1 Subdirect products, residual and local properties
4.2 Direct and inverse limits
4.3 Reduced products and ultraproducts
4.4 Local and residual properties and approximation
4.5 Remarks on references
Classes of algebras
'
5.1 Free, initial and final algebras
5.2 Equational logic
5.3 Equational Horn logic
5.4 Specification of abstract data types
5.5 Remarks on references
Further reading
6.1 Universal algebra
6.2 Model theory
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Basic category t h e o r y
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Categories, functors and natural transformations
1.1 Types, composition and identities
1.2 Categories
'
1.3 Relating functional calculus and category theory
1.4 Compositionality is functorial
1.5 Natural transformations
On universal definitions: products, disjoint sums and
higher types
2.1 Product types
2.2 Coproducts
2.3 Higher types
2.4 Reasoning by universal arguments
2.5 Another 'universal definition': primitive recursion
2.6 The categorical abstract machine
Universal problems and universal solutions
3.1 On observation and abstraction
3.2 A more categorical point of view
3.3 Universal morphisms
3.4 Adjunction
3.5 On generation
3.6 More examples for separation and generation
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x
Contents
4
Elements and beyond
4.1 Variable elements, variable subsets and representable
functors
4.2 Yoneda's heritage
4.3 Towards an enriched category theory
5 Data structures
5.1 Subtypes
5.2 Limits
5.3 Colimits
6 Universal constructions
6.1 The adjoint functor theorem
6.2 Generation as partial evaluation
6.3 Left Kan extensions, tensors and coends
6.4 Separation by testing
6.5 On bimodules and density
7 Axiomatizing programming languages
7.1 Relating theories of A-calculus
7.2 Type equations and recursion
7.3 Solving recursive equations
8 Algebra categorically
8.1 Functorial semantics
8.2 Enriched functorial semantics
8.3 Monads
9 On the categorical interpretation of calculi
9.1 Category theory as proof theory
9.2 Substitution as predicate transformation
9.3 Theories of equality
9.4 Type theories
10 A sort of conclusion
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11 Literature
11.1 Textbooks
11.2 References
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Topology
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Observable properties
Examples of topological spaces
2.1 Sierpinski space
2.2 Scott Topology
2.3 Spaces of maximal elements. Cantor space
2.4 Alexandroff topology
2.5 Stone spaces
2.6 Spectral spaces
2.7 The reals
Contents
3
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Alternative formulations of topology
3.1 Closed sets
3.2 Neighbourhoods
3.3 Examples
3.4 Closure operators
3.5 Convergence
Separation, continuity and sobriety
4.1 Separation conditions
4.2 Continuous functions
4.3 Predicate transformers and sobriety
4.4 Many-valued functions
Constructions: new spaces from old
5.1 Postscript: effectiveness and representation
Metric Spaces
6.1 Basic definitions
6.2 Examples
6.3 Completeness
6.4 Topology and metric
6.5 Constructions
6.6 A note on uniformities
Compactness
7.1 Compactness and finiteness
7.2 Spectral spaces
7.3 Positive and negative information: patch topology
7.4 Hyperspaces
7.5 Tychonoff's theorem
7.6 Locally compact spaces
7.7 Function spaces
Appendix
-
Model theory and computer science:
An appetizer
1
2
3
Introduction
The set theoretic modelling of syntax and semantics
2.1 First order structures
2.2 The choice of the vocabulary
2.3 Logics
Model theory and computer science
3.1 Computer science
3.2 The birth of model theory
3.3 Definability questions
3.4 Preservation theorems
3.5 Disappointing ultraproducts
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Contents
3.6 Complete theories and elimination of quantifiers
3.7 Spectrum problems
3.8 Beyond first order logic
3.9 The hidden method
3.10 0-1 Laws
Preservation theorems
4.1 Horn formulas
Fast growing functions
5.1 Non-provability results in second order arithmetic
5.2 Non-provability in complexity theory
5.3 Model theory of fast growing functions
Elimination of quantifiers
6.1 Computer aided theorem proving in classical
mathematics
6.2 Tarski's theorem
6.3 Elementary geometry
6.4 Other theories with elimination of quantifiers
Computable logics over finite structures
7.1 Computable logics
7.2 Computable quantifiers
7.3 Computable predicate transformers
7.4 L-Reducibility
7.5 Logics capturing complexity classes
Ehrenfeucht-Fraisse games ,
8.1 The games
8.2 Completeness of the game
8.3 Second order logic and its sublogics
8.4 More non-definability results
8.5 The games and pumping lemmas
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Conclusions
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Author index
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Index
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