From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

An Introduction to Kolmogorov Complexity and Its Applications

A new approach to the problem of evaluating the complexity ("randomness") of finite sequences is presented. The proposed complexity measure is related to the number of steps in a self-delimiting production process by which a given sequence is presumed to be generated. It is further related to the number of distinct substrings and the rate of their occurrence along the sequence. The derived properties of the proposed measure are discussed and motivated in conjunction with other well-established complexity criteria.

On the Complexity of Finite Sequences

A type system is a syntactic method for automatically checking the absence of certain erroneous behaviors by classifying program phrases according to the kinds of values they compute. The study of type systems -- and of programming languages from a type-theoretic perspective -- has important applications in software engineering, language design, high-performance compilers, and security.This text provides a comprehensive introduction both to type systems in computer science and to the basic theory of programming languages. The approach is pragmatic and operational; each new concept is motivated by programming examples and the more theoretical sections are driven by the needs of implementations. Each chapter is accompanied by numerous exercises and solutions, as well as a running implementation, available via the Web. Dependencies between chapters are explicitly identified, allowing readers to choose a variety of paths through the material.The core topics include the untyped lambda-calculus, simple type systems, type reconstruction, universal and existential polymorphism, subtyping, bounded quantification, recursive types, kinds, and type operators. Extended case studies develop a variety of approaches to modeling the features of object-oriented languages.

https://www.seas.upenn.edu/~bcpierce/tapl/contents.pdf

Types and Programming Languages

A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to polynomial-time computable functions ƒr: {1, … , 2k} → {1, … , 2k}. These ƒr's cannot be distinguished from random functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory.

How to construct random functions

Preface These lectures were given in Padova at the Laboratorio per Ricerche di Di-namica dei Sistemi e di Elettronica Biomedica of the Consiglio Nazionale delle Ricerche during the month of June 1980. I am indebted to Dr. Enrico Pagello of that laboratory for the opportunity of so doing. The audience was made up by philosophers, mathematicians and computer scientists. Accordingly, I tried to say something which might be of interest to each of these three categories. Essentially the same lectures, albeit in a somewhat improved and more advanced form, were given later in the same year as part of the meeting on Konstruktive Mengenlehre und Typentheorie which was organized in Munich by Prof. Dr. Helmut Schwichtenberg, to whom I am indebted for the invitation, during the week 29 September – 3 October 1980. The main improvement of the Munich lectures, as compared with those given in Padova, was the adoption of a systematic higher level (Ger. Stufe) notation which allows me to write simply respectively. Moreover, the use of higher level variables and constants makes it possible to formulate the elimination and equality rules for the cartesian product in such a way that they follow the same pattern as the elimination and equality rules for all the other type forming operations. In their new formulation, these rules read Π-elimination c ∈ Π(A, B) (y(x) ∈ B(x) (x ∈ A)) d(y) ∈ C(λ(y)) F(c, d) ∈ C(c) and Π-equality (x ∈ A) b(x) ∈ B(x) (y(x) ∈ B(x) (x ∈ A)) d(y) ∈ C(λ(y)) F(λ(b), d) = d(b) ∈ C(λ(b)) respectively. Here y is a bound function variable, F is a new non-canonical (eliminatory) operator by means of which the binary application operation can be defined, putting Ap(c, a) ≡ F(c, (y) y(a)), and y(x) ∈ B(x) (x ∈ A) is an assumption, itself hypothetical, which has been put within parentheses to indicate that it is being discharged. A program of the new form F(c, d) has value e provided c has value λ(b) and d(b) has value e. This rule for evaluating F(c, d) reduces to the lazy evaluation rule for Ap(c, a) when the above definition is being made. Choosing C(z) to be B(a), thus independent of z, and d(y) to be y(a), the new elimination rule reduces to the old one and the new equality rule to the first of the two old equality rules. Moreover, the second …

Intuitionistic type theory

Kolmogorov has defined the conditional complexity of an object y when the object x is already given to us as the minimal length of a binary program which by means of x computes y on a certain asymptotically optimal machine. On the basis of this definition he has proposed to consider those elements of a given large finite population to be random whose complexity is maximal. Almost all elements of the population have a complexity which is close to the maximal value. In this paper it is shown that the random elements as defined by Kolmogorov possess all conceivable statistical properties of randomness. They can equivalently be considered as the elements which withstand a certain universal stochasticity test. The definition is extended to infinite binary sequences and it is shown that the non random sequences form a maximal constructive null set. Finally, the Kollektivs introduced by von Alises obtain a definition which seems to satisfy all intuitive requirements.

The definition of random sequences

Publisher Summary The theory of types is intended to be a full-scale system for formalizing intuitionistic mathematics as developed. The language of the theory is richer than the languages of traditional intuitionistic systems in permitting proofs to appear as parts of propositions so that the propositions of the theory can express properties of proofs. There are axioms for universes that link the generation of objects and types and play somewhat the same role for the present theory as does the replacement axiom for Zermelo–Fraenkel set theory. The present theory is based on a strongly impredicative axiom that there is a type of all types in symbols. This axiom has to be abandoned, however, after it has been shown to lead to a contraction. This chapter discusses Normalization theorem, which can be strengthened in two ways: it can be made to cover open terms and it can be proved that every reduction sequence starting from an arbitrary term leads to a unique normal term after a finite number of steps. The definition of the notion of convertibility and the proof that an arbitrary term is convertible can no longer be separated because the type symbols and the terms are generated simultaneously.

An Intuitionistic Theory of Types: Predicative Part

Publisher Summary This chapter discusses that relating constructive mathematics to computer programming seems to be beneficial. Among the benefits to be derived by constructive mathematics from its association with computer programming, one is that you see immediately why you cannot rely upon the law of excluded middle: its uninhibited use would lead to programs that one did not know how to execute. By choosing to program in a formal language for constructive mathematics, like the theory of types, one gets access to the conceptual apparatus of pure mathematics, neglecting those parts that depend critically on the law of excluded middle, whereas even the best high level programming languages so far designed are wholly inadequate as mathematical languages. The virtue of a machine code is that a program written in it can be directly read and executed by the machine. The distinction between low and high level programming languages is of course relative to the available hardware. It may well be possible to turn what is now regarded as a high level programming language into machine code by the invention of new hardware.

Constructive mathematics and computer programming

If programming is understood not as the writing of instructions for this or that computing machine but as the design of methods of computation that it is the computer’s duty to execute (a difference that Dijkstra has referred to as the difference between computer science and computing science), then it no longer seems possible to distinguish the discipline of programming from constructive mathematics. This explains why the intuitionistic theory of types (Martin-Lof 1975 In Logic Colloquium 1973 (ed. H. E. Rose & J. C. Shepherdson), pp. 73- 118. Amsterdam: North-Holland), which was originally developed as a symbolism for the precise codification of constructive mathematics, may equally well be viewed as a programming language. As such it provides a precise notation not only, like other programming languages, for the programs themselves but also for the tasks that the programs are supposed to perform. Moreover, the inference rules of the theory of types, which are again completely formal, appear as rules of correct program synthesis. Thus the correctness of a program written in the theory of types is proved formally at the same time as it is being synthesized.

Per Martin-Löf

Papers

Intuitionistic type theory

The definition of random sequences

An Intuitionistic Theory of Types: Predicative Part

Constructive mathematics and computer programming

Constructive mathematics and computer programming