A Programming Approach to Computability

Computability theory is at the heart of theoretical computer science. Yet, ironically, many of its basic results were discovered by mathematical logicians prior to the development of the first stored-program computer. As a result, many texts on computability theory strike today's computer science students as far removed from their concerns. To remedy this, we base our approach to computability on the language of while-programs, a lean subset of PASCAL, and postpone consideration of such classic models as Turing machines, string-rewriting systems, and p. -recursive functions till the final chapter. Moreover, we balance the presentation of un solvability results such as the unsolvability of the Halting Problem with a presentation of the positive results of modern programming methodology, including the use of proof rules, and the denotational semantics of programs. Computer science seeks to provide a scientific basis for the study of information processing, the solution of problems by algorithms, and the design and programming of computers. The last 40 years have seen increasing sophistication in the science, in the microelectronics which has made machines of staggering complexity economically feasible, in the advances in programming methodology which allow immense programs to be designed with increasing speed and reduced error, and in the develop ment of mathematical techniques to allow the rigorous specification of program, process, and machine.

Inhalt
1 Introduction.- 1.1 Partial Functions and Algorithms.- 1.2 An Invitation to Computability Theory.- 1.3 Diagonalization and the Halting Problem.- 2 The Syntax and Semantics of while-Programs.- 2.1 The Language of while-Programs.- 2.2 Macro Statements.- 2.3 The Computable Functions.- 3 Enumeration and Universality of the Computable Functions.- 3.1 The Effective Enumeration of while-Programs.- 3.2 Universal Functions and Interpreters.- 3.3 String-Processing Functions.- 3.4 Pairing Functions.- 4 Techniques of Elementary Computability Theory.- 4.1 Algorithmic Specifications.- 4.2 The s-m-n Theorem.- 4.3 Undecidable Problems.- 5 Program Methodology.- 5.1 An Invitation to Denotational Semantics.- 5.2 Recursive Programs 110 5.3* Proof Rules for Program Properties.- 6 The Recursion Theorem and Properties of Enumerations.- 6.1 The Recursion Theorem.- 6.2 Model-Independent Properties of Enumerations.- 7 Computable Properties of Sets (Part 1).- 7.1 Recursive and Recursively Enumerable Sets.- 7.2 Indexing the Recursively Enumerable Sets.- 7.3 Gödel's Incompleteness Theorem.- 8 Computable Properties of Sets (Part 2).- 8.1 Rice's Theorem and Related Results.- 8.2 A Classification of Sets.- 9 Alternative Approaches to Computability.- 9.1 The Turing Characterization.- 9.2 The Kleene Characterization.- 9.3 Symbol-Manipulation Systems and Formal Languages.- References.- Notation Index.- Author Index.