An introduction to simple type theory, containing 200 exercises with complete solutions.

From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography

The first contemporary comprehensive treatment of optimization without derivatives. This text explains how sampling and model techniques are used in derivative-free methods and how they are designed to solve optimization problems. It is designed to be readily accessible to both researchers and those with a modest background in computational mathematics.

Programming is the art of designing efficient algorithms that meet their specifications. There are two factors by which algorithms may be judged - their correctness and their performance. This text discusses the calculational style of programming where programs are derived from their specification by means of formula manipulation.

This book provides new presentations of standard computational models that help avoid pitfalls of the conventional description methods. It also includes novel approaches to some of the topics that students normally find the most challenging. The presentations have evolved in response to student feedback over many years of teaching and have been well received by students.The book covers the topics suggested in the ACM curriculum guidelines for the course on ?Theory of Computation?, and in the course on ?Foundations of Computing? in the model liberal arts curriculum. These are standard courses for upper level computer science majors and beginning graduate students.The material in this area of computing is intellectually deep, and students invariably find it challenging to master. This book blends the three key ingredients for successful mastery. The first is its focus on the mingling of intuition and rigor that is required to fully understand the area. This is accomplished not only in the discussion and in examples, but also especially in the proofs. Second, a number of practical applications are presented to illustrate the capacity of the theoretical techniques to contribute insights in a variety of areas; such presentations greatly increase the reader's motivation to grasp the theoretical material. The student's active participation is the third and final major element in the learning process, and to this end an extensive collection of problems of widely differing difficulty is incorporated.

A very active field of research is emerging at the frontier of statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. This book sets up a common language and pool of concepts, accessible to students and researchers from each of these fields.

This volume contains a selection of papers presented at the Seventh Logic Programming Conference that took place in Tokyo, April 11-14, 1988. It is the successor to the previous conference proceedings published as Lecture Notes in Computer Science Volumes 221, 264 and 315. The book covers various aspects of logic programming such as foundations, programming languages/systems, concurrent programming, knowledge bases, applications of computer-aided reasoning and natural language processing. The papers on foundations present theoretical results on "narrowing", a proof strategy for proving properties of Prolog programs based on inductionless induction and several issues in nonmonotonic reasoning. Of special interest to mathematicians is the paper on computer-aided reasoning, which describes a system for assisting human reasoning. Natural language application papers treat the lexical analysis of Japanese sentences, a system that generates a summary of a given sentence and a new knowledge representation formalism suited for representing dynamic behavior by extending the frame system.