Theory of Computation

Learn the skills and acquire the intuition to assess the theoretical limitations of computer programming Offering an accessible approach to the topic, Theory of Computation focuses on the metatheory of computing and the theoretical boundaries between what various computational models can do and not do from the most general model, the URM (Unbounded Register Machines), to the finite automaton. A wealth of programming-like examples and easy-to-follow explanations build the general theory gradually, which guides readers through the modeling and mathematical analysis of computational phenomena and provides insights on what makes things tick and also what restrains the ability of computational processes. Recognizing the importance of acquired practical experience, the book begins with the metatheory of general purpose computer programs, using URMs as a straightforward, technology-independent model of modern high-level programming languages while also exploring the restrictions of the URM language. Once readers gain an understanding of computability theory including the primitive recursive functions the author presents automata and languages, covering the regular and context-free languages as well as the machines that recognize these languages. Several advanced topics such as reducibilities, the recursion theorem, complexity theory, and Cook's theorem are also discussed. Features of the book include: A review of basic discrete mathematics, covering logic and induction while omitting specialized combinatorial topics A thorough development of the modeling and mathematical analysis of computational phenomena, providing a solid foundation of un-computability The connection between un-computability and un-provability: G del's first incompleteness theorem The book provides numerous examples of specific URMs as well as other programming languages including Loop Programs, FA (Deterministic Finite Automata), NFA (Nondeterministic Finite Automata), and PDA (Pushdown Automata). Exercises at the end of each chapter allow readers to test their comprehension of the presented material, and an extensive bibliography suggests resources for further study. Assuming only a basic understanding of general computer programming and discrete mathematics, Theory of Computation serves as a valuable book for courses on theory of computation at the upper-undergraduate level. The book also serves as an excellent resource for programmers and computing professionals wishing to understand the theoretical limitations of their craft.

Learn the skills and acquire the intuition to assess thetheoretical limitations of computer programmingOffering an accessible approach to the topic, Theory ofComputation focuses on the metatheory of computing and thetheoretical boundaries between what various computational modelscan do and not do--from the most general model, the URM(Unbounded Register Machines), to the finite automaton. A wealth ofprogramming-like examples and easy-to-follow explanations build thegeneral theory gradually, which guides readers through the modelingand mathematical analysis of computational phenomena and providesinsights on what makes things tick and also what restrains theability of computational processes.Recognizing the importance of acquired practical experience, thebook begins with the metatheory of general purpose computerprograms, using URMs as a straightforward, technology-independentmodel of modern high-level programming languages while alsoexploring the restrictions of the URM language. Once readers gainan understanding of computability theory--including theprimitive recursive functions--the author presents automataand languages, covering the regular and context-free languages aswell as the machines that recognize these languages. Severaladvanced topics such as reducibilities, the recursion theoremcomplexity theory, and Cook's theorem are also discussed. Featuresof the book include:* A review of basic discrete mathematics, covering logic andinduction while omitting specialized combinatorial topics* A thorough development of the modeling and mathematical analysisof computational phenomena, providing a solid foundation ofun-computability* The connection between un-computability and un-provability:Gödel's first incompleteness theoremThe book provides numerous examples of specific URMs as well asother programming languages including Loop Programs, FA(Deterministic Finite Automata), NFA (Nondeterministic FiniteAutomata), and PDA (Pushdown Automata). Exercises at the end ofeach chapter allow readers to test their comprehension of thepresented material, and an extensive bibliography suggestsresources for further study.Assuming only a basic understanding of general computerprogramming and discrete mathematics, Theory of Computationserves as a valuable book for courses on theory of computation atthe upper-undergraduate level. The book also serves as an excellentresource for programmers and computing professionals wishing tounderstand the theoretical limitations of their craft.

Preface xi1. Mathematical Foundations 11.1 Sets and Logic; Naïvely 11.2 Relations and Functions 401.3 Big and Small Infinite Sets; Diagonalization 521.4 Induction from a User's Perspective 611.5 Why Induction Ticks 681.6 Inductively Defined Sets1.7 Recursive Definitions of Functions1.8 Additional Exercises 852. Algorithms, Computable Functions and Computations912.1 A Theory of Computability 912.2 A programming Formalism for the Primitive RecursiveFunctions Function Class 1472.3 URM Computations and their Arithmetization 1412.4 A double-recursion that leads outside the PrimitiveRecursive Function Class2.5 Semi-computable Relations: Unsolvability2.6 The Iteration Theorem of Kleene 1722.7 Diagonalization Revisited; Unsolvability via Reductions1752.8 Productive and Creative Sets 2092.9 The Recursion Theorem 2142. 10 Completeness 2172.11 Unprovability from Unsolvability 2212.12 Additional Exercises 2343. A Subset of the URM Language; FA and NFA 2413.1 Deterministic Finite Automata and their Languages 2433.2 Nondeterministic Finite Automata3.3 Regular Expressions 2663.4 Regular Grammars and Languages 2773.5 Additional Exercises 2874. Adding a stack of a NFA: Pushdown Automata4.1 The PDA 2944.2 PDA Computations 2944.3 The PDA-acceptable Languages are the Context Free Languages3054.4 Non-Context Free Languages; Another Pumping Lemma 3124.5 Additional Exercise 3225. Computational Complexity 3255.1 Adding a second stack; Turning Machines 3255.2 Axt, loop program, and Grzegorczyk hierarchies5.3 Additional Exercised