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: Gödel'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.

In the (meta)theory of computing, the fundamental questions of the limitations of computing are addressed.   These limitations, which are intrinsic rather than technology dependent, may immediately rule out the existence of algorithmic solutions for some problems while for others they rule out efficient solutions.  The author's approach is anchored on the concrete (and assumed) practical knowledge about general computer programming, attained readers in a first year programming course, as well as the knowledge of discrete mathematics at the same level.  The book develops the metatheory of general computing and builds on the reader's prior computing experience.  Metatheory via the programming formalism known as Shepherdson-Sturgis Unbounded Register Machines (URM)--a straightforward abstraction of modern high-level programming languages--is developed.  Restrictions of the URM programming language are also discussed.  The author has chosen to focus on the high-level language approach of URMs as opposed to the Turing Machine since URMs relate more directly to programming learned in prior experiences.  The author presents the topics of automata and languages only after readers become familiar, to some extent, with the (general) computability theory including the special computability theory of more "practical" functions, the primitive recursive functions.  Automata are presented as a very restricted programming formalism, and their limitations (in expressivity) and their associated languages are studied.  In addition, this book contains tools that, in principle, can search a set of algorithms to see whether a problem is solvable, or more specifically, if it can be solved by an algorithm whose computations are efficient.  Chapter coverage includes: Mathematical Background; Algorithms, Computable Functions, and Computations; A Subset of the URM Language: FA and NFA; and Adding a Stack to an NFA: Pushdown Automata.

In the (meta)theory of computing, the fundamental questions of the limitations of computing are addressed.   These limitations, which are intrinsic rather than technology dependent, may immediatly rule out the existence of algorithmic solutions for some problems while for others they rule out efficient solutions.  The author's approach is anchored on the concrete (and assumed) practical knowledge about general computer programming, attained readers in a first year programming course, as well as the knowledge of discrete mathematics at the same level.  The book develops the metatheory of general computing and builds on the reader's prior computing experience.  Metatheory via the programming formalism known as Shepherdson-Sturgis Unbounded Register Machines (URM)--a straightforward abstraction of modern highlevel programming languages--is developed.  Restrictions of the URM programming language are also discussed.  The author has chosen to focus on the highlevel language approach of URMs as opposed to the Turing Machine since URMs relate more directly to programming learned in prior experiences.  The author presents the topics of automata and languages only after readers become familiar, to some extent, with the (general) computability theory including the special computability theory of more practical functions, the primitive recursive functions.  Automata are presented as a very restricted programming formalism, and their limitations (in expressivity) and their associated languages are studied.  In addition, this book contains tools that, in principle, can search a set of algorithms to see whether a problem is solvable, or more specifically, if it can be solved by an algorithm whose computations are efficient.  Chapter coverage includes: Mathematical Background; Algorithms, Computable Functions, and Computations; A Subset of the URM Language: FA and NFA; and Adding a Stack to an NFA: Pushdown Automata.