Foundations of Computing
Foundations of Computing
An Accessible Introduction to Formal Languages
About the Book
A textbook for upper-division Computer Science majors covering formal languages and automata with an introduction to computability. Intended to give CS majors a solid foundation in the Theory of Computation without being overly formal mathematically, while retaining the rigor of the material. It has been classroom tested since 2016 with good success.
About the Author
Table of Contents
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- Foreword
- Preface
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1 Introduction
- 1.1 Formal Languages
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1.2 Finite State Machines
- Exercises
- Chapter Summary
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I Regular Languages
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2 Finite Automata
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- Where Are We?
- Chapter Objectives
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2.1 Deterministic Finite Automata
- Exercises
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2.2 Non-Deterministic Finite Automata
- Equivalence of NFAs and DFAs
- NFAs and Complements
- Exercises
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2.3 Minimal Automata
- Exercises
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2.4 Machines with Output
- Computer Arithmetic
- Lexical Analysis
- Minimal Mealy Machines
- Exercises
- Chapter Summary
-
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3 Regular Expressions and Grammars
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- Where Are We?
- Chapter Objectives
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3.1 Regular Expressions
- Exercises
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3.2 Equivalence of Regular Expressions and Regular Languages
- From Regular Expression to NFA
- From NFA to Regular Expression
- Exercises
-
3.3 Regular Grammars
- Left-Linear Grammars
- Exercises
- Chapter Summary
-
-
4 Properties of Regular Languages
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- Where Are We?
- Chapter Objectives
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4.1 Closure Properties
- Computing Set Operations
- Exercises
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4.2 Decision Algorithms
- Exercises
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4.3 Infinite Regular Languages and a “Pumping Theorem”
- Exercises
- Chapter Summary
-
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2 Finite Automata
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II Context-Free Languages
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5 Pushdown Automata
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- Where Are We?
- Chapter Objectives
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5.1 Adding a Stack to Finite Automata
- Exercises
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5.2 Pushdown Automata and Determinism
- Exercise
- Chapter Summary
-
-
6 Context-Free Grammars
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- Where Are We?
- Chapter Objectives
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6.1 Context-Free Grammars and Derivations
- Simplifying Grammars
- Exercises
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6.2 Derivation Trees and Ambiguous Grammars
- Operator Precedence
- Operator Associativity
- Expression Trees
- Exercises
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6.3 Equivalence of PDAs and CFGs
- From CFG to PDA
- From PDA to CFG (Special Case)
- From PDA to CFG (General Case)
- Exercises
- Chapter Summary
-
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7 Properties of Context-Free Languages
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- Where Are We?
- Chapter Objectives
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7.1 Chomsky Normal Form
- Removing Lambda
- Removing Unit Productions
- Chomsky Normal Form Rules
- Exercises
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7.2 Closure Properties
- Closure Properties of DCFLs
- Exercises
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7.3 Decision Algorithms
- Is a CFL Empty or Infinite?
- Exercises
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7.4 Infinite CFLs and Another Pumping Theorem
- Exercises
- Chapter Summary
-
-
5 Pushdown Automata
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III Recursively Enumerable Languages
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8 Turing Machines
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- Where Are We?
- Chapter Objectives
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8.1 Prelude
- Post Machines
- Exercise
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8.2 The Standard Turing Machine
- Subroutines
- Halting
- Exercises
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8.3 Variations on Turing Machines
- The Universal Turing Machine
- Non-Deterministic TM = Deterministic TM
- Programming Exercise
- Chapter Summary
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-
9 The Landscape of Formal Languages
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- Where Are We?
- Chapter Objectives
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9.1 Recursively Enumerable Languages
- A Non-Recursive, RE Language
- Context-Sensitive Languages
- Properties of Recursively Enumerable Languages
- Exercises
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9.2 Unrestricted Grammars
- Context-Sensitive Grammars
- Equivalence of Unrestricted Grammars and Turing Machines
- Exercises
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9.3 The Chomsky Hierarchy
- Countable Sets
- Uncountable Sets
- Chapter Summary
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10 Computability
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- Chapter Objectives
- 10.1 The Halting Problem
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10.2 Reductions and Undecidability
- Exercises
- Chapter Summary
-
- Glossary
- Bibliography
- Index
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8 Turing Machines
- Notes
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