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# 15-815 Automated Theorem Proving

# Lecture 1: Introduction

In this first lecture we provide an overview and introduction to the
material and approach taken in this class.

### Summary

Logic is a science studying the principles of reasoning and valid
inference. Automated deduction is concerned with the mechanization of
formal reasoning, following the laws of logic. In this course we study
techniques for automated deduction, both from a high-level and low-level
perspective.

The course begins with a general introduction to logic and means of defining
a variety of logics. This part is based on Gentzen's calculus of natural
deduction. We then show how the problem of proof search for natural
deductions can be rendered in a sequent calculus, also due to Gentzen. Most
forms of automated deduction can be traced to a sequent calculus of one form
or another.

Present theorem proving techniques can be broadly classified into backward
reasoning methods (from the proposed theorem to the axioms) and forward
reasoning methods (from the axiom to the proposed theorem). We will discuss
both, starting from the more intuitive backwards reasoning methods such as
tableaux. We then change our perspective and show how forward reasoning
methods can be made practical using the *inverse method*. Refinements
and efficient implementation techniques for the inverse method will occupy
us most of the semester.

The course is centered on a project, the implementation of a succession
of theorem provers for intuitionistic logic. We chose intuitionistic logic
(rather than classical logic), since it is of central importance in
computer science, and since it exhibits phenomena which are obscured
by special properties which hold only in classical logic.

The class will be organized into several small groups, each of which
implements a complete theorem prover. Starting in week 3, I will set concrete
goals and expect weekly project reports from each member of each group. Up
until the midterm examination on October 8, there will also be weekly
homeworks of a more theoretical nature. Doing the homework and participation
in the projects are essential to achieve the goals of the class.
At the end of the course I would like each student to

- have a thorough understanding of the central techniques in automated
theorem proving,
- know how these techniques can be developed systematically,
- appreciate some low-level issues in the efficient implementation of
theorem provers.

### Reading

- Pages 1-2 of Handout 1 on
*Natural Deduction* (also available in DVI and PDF
formats).

### Links

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Frank Pfenning
fp@cs