A Computational Introduction to Number Theory and Algebra (V. Shoup)
A Computational Introduction to Number Theory and Algebra (V. Shoup)
A Computational Introduction to Number Theory and Algebra (V. Shoup)
This book introduces the basic concepts from computational number theory
and algebra, including all the necessary mathematical background. Covered topics
are: Basic properties of the integers, Congruences, Computing with large
integers, Euclid’s algorithm, The distribution of primes, Abelian groups, Rings,
Finite and discrete probability distributions, Probabilistic algorithms,
Probabilistic primality testing, Finding generators and discrete logarithms in
Zp, Quadratic reciprocity and computing modular square roots, Modules and vector
spaces, Matrices, Subexponential-time discrete logarithms and factoring,
Polynomial arithmetic and applications.
Frank Stephan's
detailed lecture notes on the theory of computation cover quite a wide spectrum
of issues. The document starts with the basics of sets and regular expressions,
then goes ahead to grammars and the Chomsky hierarchy, helping one in
understanding the structure of languages. Then it discusses finite automaton and
nondeterministic finite automata, giving all details about the processing of
strings by these models. The notes also treat the composition of languages,
normal forms, and algorithms used in computation. Membership testing, whether
deterministic or nondeterministic, is also explained, together with the proof of
how models of computation handle language recognition. Finally, the approach is
important when considering complexity, the problems that turn out undecidable,
showing thus the intrinsic limits of computation. This is an important resource
concerning formal languages, automata theory, and basic bounds of
computability.
Introduction
to the Theory of Computing is a course that undertakes
an intensive study of the underpinnings of the theory of computation. Beginning
with mathematical foundations, the course moves into regular operations and
expressions, and then into proofs on languages being nonregular and other
further treatments on regular languages. Other important topics include parse
trees, ambiguity, Chomsky normal form, pushdown automata, and Turing machines.
Further, the PDF discusses various types of Turing machines, the stack machine
model, and undecidable languages, making it a great starting point in the topic
of computability.
This
book surveys some of the most relevant theoretical concepts with computational
models. The limits of computation, undecidability of the Halting Problem,
several automata models, including both deterministic and nondeterministic
finite-state automata, pushdown automata, and Turing machines, are introduced.
The ending is dedicated to computational complexity, with NP-Completeness,
approximation algorithms, and hardness of approximation.
This book, written for graduates, covers
general subjects on computational models, logic circuits, and memory machines,
with advanced subjects being parallel computation, circuit complexity, and
space-time trade-offs; therefore, it's a very thorough course on computational
models and their complexities.
This lecture
note from S R Engineering College offers a detailed introduction to key concepts
in the Theory of Computation. It begins with Properties of Binary
Operations, exploring fundamental mathematical operations and their
essential properties like associativity and commutativity. The section on
Concatenation Properties covers how strings can be joined and
the characteristics of such operations, including associativity and the identity
element. Finite Automata are thoroughly discussed, explaining
both deterministic and nondeterministic (NFA) models, and their role in
recognizing regular languages. The notes also cover Formal Languages,
categorizing them into regular, context-free, context-sensitive, and recursively
enumerable types based on complexity. Finally, the Pumping Lemma
is introduced as a critical tool for proving the non-regularity and
non-context-freeness of languages by demonstrating how strings in these
languages can be decomposed and manipulated.
This
is an all-inclusive course on computational theory provided in this online
resource by Wikiversity. It begins with Finite State Machines–their
definitions, operations, and minimization techniques. The notes also cover
closure and nondeterminism—how these properties may affect computational
models. Their discussion greatly involves the Pumping Lemma, proving language
property. The book also surveys Context-Free Languages and their connection to
Compilers and introduces Pushdown Machines emphatically and focuses on their
importance in parsing. It contains important material on the CYK algorithm for
parsing and the more basic problems of Undecidability. It also surveys Turing
Machines, the Halting Problem, and more general areas of Complexity Theory,
including Quantified Boolean Formulae, Savitch's Theorem, and Space Hierarchy.
The notes end with the Recursion Theorem, and it can be considered as a
landmark in the theoretical study of the science of computers.