## Theorem Proving and Equality |
Technische Universität Dresden |

The aim of this lecture is to review the fundamental techniques in the automated theorem proving in the first order logic with equality predicate. We will study completeness of various inference systems starting with Resolution, Ordered Resolution, Paramodulation and Superposition, i.e. saturation-based theorem provers. The main method of proving completeness, which we will use and study, is model generation and reduction of minimal counterexamples.

Prerequisites: an acquaintance with the basics of propositional and first-order logic is expected.We will also refer to the notions from the Term Rewriting Systems.

The lecture takes place in room GRU 359: Friday 11:10-12:40.

**The next lecture will take place in room GRU 359:
Tuesday (10.01.2006) 11:10-12:40.**

There is no exercise group, though some (voluntary) exercises are given in the lecture.

Some slides used in the lecture will be available here.

- slide 1
- example1 (without "usable")
- example2
- example3 (with paramodulation)
- slide 2
- slide 3
- slide 4
- slide 5
- slide 6
- slide 7
- example4
- example5 (with order option cleared)
- slide 8
- slide 9
- slide 10
- slide 11
- slide 12
- slide 13
- slide 14
- slide 15
- slide 16
- example 6
- example 7
- slide 17
- slide 18
- slide 19
- slide 20

Exercises:

- lecture 1 - exercises
- lecture 2 - exercises
- lecture 3 - exercises
- lecture 4 - exercises
- lecture 5 - exercises
- lecture 6 - exercises
- lecture 7 - exercises
- lecture 8 - exercises
- lecture 9 - exercises
- lecture 10 and 11 - exercises
- lecture 12 and 13 - exercises
- lecture 14 - exercises

Computational logic students can earn 3 credits by attending this lecture. The lecture can be used in the module TCSL and IT. In order to get the credits, CL students have to pass a short oral examination at the end of the term.

- L. Bachmair, H. Ganzinger. Resolution Theorem Proving,
Chap. 2 in
*Handbook of Automated Reasoning*, ed. A. Robinson, A. Voronkov, vol.1, North Holland 2001. - R. Nieuwenhuis, A. Rubio.
Paramodulation-Based Theorem Proving.
Chapter of
*Handbook of Automated Reasoning*ed. A. Robinson, A. Voronkov. ISBN 0-444-82949-0. © Elsevier Science and MIT Press, North Holland 2001. - L. Bachmair, H. Ganzinger, Ch. Lynch, W. Snyder.
Basic Paramodulation. in
*Information and Computation*, vol. 121, nr.2, 1995, pp. 172-192. - William McCune. Otter 3.3 Reference Manual, August 2003.
- F. Baader, T. Nipkow.
*Term Rewriting and All That.*ISBN 0-521-77920-0, Cambridge University Press, 1998. - L. Bachmair, N. Dershowitz, D.A.Plaisted. Completion without Failure.
Chap. 1 in
*Resolution of Equations in Algebraic Structures 2: Rewriting Techniques*, H. Ait-Kaci and M. Nivat, eds. pp. 1-30, Academic Press, New York (1989).

Barbara Morawska