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# 15-815 Automated Theorem Proving

# Lecture 16: Towards Tableaux

The considerations introduced into proof search so far are mostly motivated
by backward theorem proving, where we start with a proposed theorem and work
towards the initial sequents. Nonetheless, with some modifications, the
principal ideas and techniques are shared between different styles of theorem
provers and apply to forward theorem proving as well.

In this lecture we introduce a further improvement which postpones
disjunctive choices (don't know non-determinism) until later in the search, in
the hope they may become irrelevant or otherwise shorten the proof. This is
achieved by postponing the choice of either proving either *A* or
*B* when the succedent is the disjunction *A v B*, leading
to a sequent with multiple conclusions. A similar idea can be applied
to delay the choice for existential instantiation.

Taken a step further, these ideas lead to tableaux provers, such as
those developed Otten and Kreitz, following earlier ideas by Wallen.

- Automated Proof Search in Non-Classical Logics: Efficient Matrix
Proof Methods for Modal and Indutionistic Logics. Lincoln A. Wallen,
MIT Press, 1999.
- A Connection Based Proof Method for Intuitionistic Logic. Jens Otten and
Christoph Kreitz. In P. Baumgartner, R. Hähnle and J. Posegga, editors,
*4th International Workshop on Theorem Proving with Analytic Tableaux and
Related Methods*, LNAI 918, pp. 122-137, Springer Verlag, 1995. (dvi,
89 kbytes)
- T-String-Unification: Unifying Prefixes in Non-Classical Proof Methods.
Jens Otten and Christoph Kreitz. In P. Miglioli, U. Moscato, D. Mundici,
M. Ornaghi, editors,
*5th International*

Workshop on Theorem Proving
with Analytic Tableaux and Related Methods, LNAI 1071, pp. 244-260,
Springer Verlag, 1996. (dvi,
79 kbytes)

### Reading

*Not yet available*
### Key Concepts

- Postponing disjunctive choices
- Multiple succedent sequents
- Tableaux and matrix theorem proving

### Links

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Frank Pfenning
fp@cs