The Functional Approach to Programming

Cover
Description Page
Title Page
Copyright Page
Table of Contents
Preface
Introduction
I Basic Principles
1 Expressions
1.1 Expressions and Definitions
1.1.1 Expressions
1.1.2 Definitions
1.1.3 Local Definitions
1.2 Elementary Types
1.2.1 Integers
1.2.2 Floating-Point Numbers
1.2.3 Floating Mode
1.2.4 Booleans
1.2.5 Character Strings
1.2.6 Characters
1.3 Cartesian Product
1.4 Functions
1.4.1 Defining Simple Functions
1.4.2 Function Expressions
1.4.3 Higher Order Function Expressions
1.4.4 Recursive Functions
1.4.5 Mutually Recursive Functions
1.4.6 Functions Defined by Case
1.4.7 The function Construction
1.4.8 The match with Construction
1.5 Polymorphism
1.5.1 Type Variables
1.5.2 Type Synthesis
1.6 About Syntax
1.6.1 Application Notation
1.6.2 Syntactic Analysis of Caml Expressions
1.6.3 Infix Operators Defined by the User
1.7 Scope of Identifiers
1.8 More about Function Types: Equivalent Function Spaces
1.8.1 The Types of Predefined Operators
1.9 Examples: Expressive Power
1.9.1 Bounded Iterations
1.9.2 Unbounded Iterations
1.9.3 Binary Method
1.9.4 Newtons Method
1.9.5 About Iteration
1.9.6 Summations
1.10 Summary
2 Data Structures
2.1 Record or Named Cartesian Products
2.2 Sums with Constructors
2.2.1 Constant Constructors
2.2.2 Constructors with Arguments
2.2.3 Types with Only One Constructor and Abbreviations ...
2.2.4 Recursive Types
2.2.5 Polymorphic Types
2.2.6 Lists
2.2.7 Recursive Definitions of Values
2.2.8 Abstract Types
2.2.9 Concrete Syntax of Data Structures
2.3 Lists
2.3.1 General Functional for Lists
2.3.2 Partitioning and Sorting
2.3.3 Representing Sets by Lists
2.3.4 Searching in a List
2.4 Summary
3 Semantics
3.1 Evaluation
3.1.1 Evaluation as Rewriting Expressions
3.1.2 Evaluation Strategies
3.1.3 How to Deal with Recursion
3.1.4 Another Way of Handling Recursion
3.1.5 Behavior of Evaluation Processes
3.2 Defining Strategies
3.2.1 The Caml Strategy, or Evaluation by Value
3.2.2 Another Strategy: Delayed Evaluation
3.2.3 Extending Delayed Evaluation to Data Structures
3.3 Program Semantics
3.3.1 Rewrite Semantics
3.3.2 Denotational Semantics
3.4 Proving the Correctness of Programs
3.4.1 Equational Reasoning
3.4.2 Taking Functions as Values into Account
3.4.3 Reasoning by Case
3.4.4 Reasoning by Induction
3.4.5 Proof of Termination
3.4.6 First Example: Mirror Image of Trees
3.4.7 Second Example: Associativity of append
3.4.8 Example: Using Generalization
3.4.9 Example: Using Lemmas
3.4.10 Example: Proof under Hypothesis or Proof by Assumption
3.5 Typing Expressions
3.6 Summary
3.7 To Learn More
4 Imperative Aspects
4.1 Exceptions
4.1.1 Examples of Exceptions
4.1.2 Defining Exceptions
4.1.3 Creating an Exceptional Value
4.1.4 Filtering Exceptional Values
4.1.5 Using Exceptions
4.2 Input and Output
4.2.1 Sequence
4.2.2 The Type unit
4.2.3 I/O Functions
4.2.4 Example: Print Function
4.2.5 Example: Interaction
4.3 Character Streams
4.3.1 Constructing Streams
4.4 Modifiable Data Structures
4.4.1 Vectors
4.4.2 Records with Modifiable Fields
4.4.3 References
4.4.4 Circular Lists
4.4.5 Doubly Linked Lists
4.4.6 Comparing with Pascal, C, and Lisp
4.5 Semantics of Destructive Operations
4.6 Summary
II Applications
5 Formal Terms, Pattern Matching, Unification
5.1 Trees
5.1.1 A Few Useful Functions
5.1.2 Functional for Trees
5.1.3 Occurrences in a Tree
5.1.4 Trees and Abstract Syntax
5.2 Terms with Variables
5.2.1 Introducing Variables
5.2.2 Substitutions
5.2.3 Filtering and Pattern Matching
5.2.4 Unification
5.3 Application: Type Synthesis
5.3.1 Constraints on Types
5.3.2 Generating Constraints
5.3.3 Constraint Resolution
5.4 Summary
5.5 To Learn More 1626 Balanced Trees
6.1 Binary Trees
6.2 Tree Traversals and Morphisms
6.3 Order and Pre-Order Relations
6.4 Binary Search Trees
6.4.1 Searching for an Element
6.4.2 Adding an Element
6.4.3 Adding an Element to the Root
6.4.4 Removing an Element
6.5 Balanced Trees
6.5.1 Rotations
6.5.2 AVL TYees
6.5.3 Adding Elements to an AVL Tree
6.5.4 Removing Elements from an AVL Tree
6.6 Dictionaries
6.7 Ordered Sets
6.8 Functional Queues
6.9 Summary
6.10 To Learn More
7 Graphs and Problem Solving
7.1 Algorithms to Explore Graphs
7.1.1 Naive Breadth-First Search
7.1.2 Naive Depth-First Search
7.1.3 Optimal Breadth-First Search
7.1.4 Variations and Improvements
7.1.5 Optimal Depth-First Search
7.1.6 Exhaustive Search
7.1.7 Computing Connected Components
7.2 The Red Donkey
7.3 The Game of Solitaire
7.4 Summary
7.5 To Learn More
8 Syntactic Analysis
8.1 Regular Expressions
8.2 Context-Free Grammar
8.2.1 Types and Values
8.2.2 Ambiguities
8.3 Recognizers
8.3.1 Recognizer Constructors
8.3.2 Limitations
8.3.3 Recognizers for Regular Expressions
8.3.4 Recognizers for Grammars
8.3.5 Various Derived Forms
8.4 Analysis = Recognition + Values
8.4.1 Constructors for Analyzers
8.4.2 A Lexical Analyzer
8.4.3 Analyzer for Nested Parentheses
8.4.4 Derived Forms and Higher Order Analyzers
8.4.5 An Analyzer for Arithmetic Expressions
8.4.6 Predictive Analysis
8.5 Streams and Pattern Matching
8.5.1 Streams
8.5.2 Stream Pattern Matching
8.5.3 Lexical Analysis by Stream Pattern Matching
8.5.4 Higher Order Analyzers
8.5.5 Syntactic Analysis by Stream Pattern Matching
8.6 Compiling Regular Expressions
8.6.1 Finite Automata
8.6.2 Abstract Syntax Trees for Regular Expressions
8.6.3 Computing Null, First, and Last
8.6.4 Computing follow
8.6.5 Representing Automata
8.6.6 Computing Automata
8.6.7 Interpreting a Deterministic Finite Automaton
8.7 Summary
8.8 To Learn More
8.8.1 Comparing with Other Methods of Syntactic Analysis . . .
8.8.2 Further References
9 Geometry and Drawings
9.1 Meet MLgraph
9.1.1 Geometric Plane
9.1.2 Geometric Elements
9.1.3 Constructing Images
9.2 Drawing Trees
9.2.1 Drawing Principles
9.2.2 Computing the Reduction Coefficients
9.3 Tiling
9.3.1 The Method Used
9.3.2 Generating Transformations
9.3.3 Colored Tilings
9.3.4 Handling Transformations Formally
9.4 Tiling a Hyperbolic Plane
9.4.1 Complex Arithmetic
9.4.2 Hyperbolic Isometries
9.4.3 Eschers Fish
9.4.4 The Isometry Group
9.4.5 Eschers Circle Limit III
9.5 Summary
9.6 To Learn More
9.6.1 About PostScript
9.6.2 About Drawing Trees
9.6.3 About Tilings
10 Exact Arithmetic
10.1 Representing Integers by Lists
10.1.1 Choosing a Base
10.1.2 Operations with the Least Significant Digit at the Head . .
10.1.3 Comparisons as Operations
10.1.4 Division as an Operation
10.1.5 A Type for Natural Numbers (Integers for Counting) ....
10.1.6 Exponentiation
10.1.7 Computing Square Roots
10.1.8 Reading and Printing Natural Numbers
10.2 Other Representations of Natural Numbers
10.2.1 Doubly Linked Circular Lists
10.2.2 Arrays
10.3 Signed Integers, Both Negative and Positive
10.4 Rational Numbers
10.5 An Application: Computing 7r
10.6 Summary
Part III Implementation
11 Evaluation
11.1 Evaluation with the Help of Environments
11.2 Writing an Evaluator in Caml
11.3 Evaluation by Necessity = Lazy Evaluation = Delayed Evaluation
11.4 Summary
11.5 To Learn More
12 Compilation
12.1 A Simplified Model of Computer Memory
12.2 Implementation of Data Structures
12.2.1 Coding Data
12.2.2 CONS as an Operation
12.2.3 Sharing Data
12.2.4 Representing Other Types of Data
12.2.5 Recovering Allocated Memory
12.3 Producing Code
12.3.1 List of Instructions
12.3.2 Principle behind Compilation
12.3.3 About the Compilation Scheme
12.4 Implementation
12.4.1 Code Simulation
12.4.2 Producing Code
12.5 Examples
12.6 Summary
13 Type Synthesis
13.1 Type Rules
13.1.1 Polymorphism by Substitution
13.1.2 Parametric Polymorphism
13.1.3 Generalizing Type Schemes
13.1.4 Instantiation
13.1.5 Type Rules with Parametric Polymorphism
13.2 Writing a Program to Synthesize Types
13.2.1 Generalizing a Type
13.2.2 Instantiation
13.2.3 A Few Utilities
13.2.4 Main Function
13.2.5 Examples
13.3 Synthesizing Types by Destructive Unification
13.3.1 Destructive Unification
13.3.2 Instantiation and Generalization
13.3.3 The Algorithm for Synthesis
13.4 Summary
13.5 To Learn More
Quick Reference to the Syntax of Caml Light
How to Get Caml, MLgraph, and the Examples
References
Index