Automated reasoning

Reasoning is the ability to make inferences, and automated reasoning is concerned with the building of computing systems that automate this process. Although the overall goal is to mechanize different forms of reasoning, the term has largely been identified with valid deductive reasoning as practiced in mathematics and formal logic. (cited from Encyclopedia of Philosophy)

Bourbaki
Nicolas Bourbaki is the collective allonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. (cited from the wikipedia article Bourbaki)

Bourbaki wanted to create a work that would be an essential tool for all mathematicians. Their aim was to create something logically ordered, starting with a strong foundation and building continuously on it. The foundation that they chose was set theory which would be the first book in a series of 6 that they named “éléments de mathématique”(with the 's' dropped from mathématique to represent their underlying belief in the unity of mathematics). (cited from PlanetMath)

The QED Project
In the mid 90s of the last century an international initiative, called the QED Project (WP), published an emphatic manifest still representing the spirit of many researchers of the automated reasoning community. Although the project does not exist any longer its generall spirit is still alive in many current automated reasoning projects. For that this qed manifest is worth reading.

Summary of the QED Manifest:

''The aim of the QED project is to build a single, distributed, computerized repository that rigorously represents all important, established mathematical knowledge. The construction of this system will be a scientific undertaking of significant proportions, requiring the cooperation of many mathematicians, computer scientists, research groups, research agencies, universities, and corporations. This system will have benefits for mathematics, science, technology, and education.''

History of Formalized Mathematics
The article Formalized Mathematics gives an excellent overview of the formalization efforts in mathematics with all its ups and downs including recent projects such as QED. It also envisages the future of formalized mathematics. The author John Harrison is a researcher at Intel Corporation and a former member of the Automated Reasoning Group at the Computer Laboratory of the University of Cambridge.

Terminological Notes
The notion of "automated reasoning" has various (more or less) synonyms. Instead of "reasoning" one finds quite often "theorem proving" or "deduction". And instead "meachanized" is often used instead of "automated". Googling these word combinations will cover all what is written on this page.

Categories for Deduction Systems
As "automated reasoning" is quite a general term. It might be usefull to classify deduction systems according to these questions:
 * What logical language does it support?
 * Which logical calculus does it implement?
 * What logical problem does it solve?

Examples for these categories:
 * Logical language: Propositional, first order, higher order, sorted, intuitionistic, temporal, modal, ...
 * Logical calculus: Hilbert, Natural Deduction, Sequent, Tableau, Resolution, Matrix, ...
 * Logical problem: Model checking, model generation, theorem proving, proof verification, program verification.

List of Deduction Systems
This list does not claim to be complete. Only prominent examples are listed here. A more comprehensive, however pretty out-dated list of deduction systems can be found here.

There are also some reasoners dedicated to Description logic which are listed there.