
15815 Automated Theorem Proving

Contents  Groups  Methods  Tasks  References  Implementations 

We will form groups to design and implement theorem provers for firstorder intuitionistic logic. In each week (Thursday to Thursday) there should be a welldefined plan for the coming week and a result to be reported at the end of the week. The plan for the next week and the report for the previous week should be emailed to the instructor prior to Thursday's lecture. Each week someone else should write and send the plan and report.
Red  Green  Blue 

Deepak Garg Tom Murphy Greg Price Jason Reed Noam Zeilberger 
Kevin Donnelly Tyler Gibson Neel Krishnaswami Stephen Magill Sungwoo Park 
Gosia Biernacka Darek Biernacki Don Smith 
Week 1 Plans Week 1 Report Week 2 Plans Week 2 Report Week 3 Plans Week 3 Report Week 4 Plans Week 4 Report Week 5 Plans Week 5 Report 
Week 1 Plans Week 1 Report Week 2 Plans Week 2 Report Week 3 Plans Week 3 Report (PS) Week 4 Plans 
Week 1 Plans Week 1 Report Week 2 Plans Week 2 Report Week 3 Plans Week 3 Report Week 4 Plans Week 4 Report Week 5 Plans Week 5 Report Week 6 Report 
Each group should implement either a prover based on backward search or a prover based on forward search. Below are some possible intermediate goals for each type of prover, not necessarily in the right temporal order.
For each prover or group a number of tasks arise. You should try to rotate between tasks so everyone in the group gets some exposure to various aspects of the prover.
Use the firstorder formula (FOF) syntax of the TPTP (Thousands of Problems for Theorem Provers) library. I hope we can start a section on intuitionistic theorems in the library. Note that all Horn problems in the library should be valid problems for your prover (perhaps after syntactic translation).
The search engine includes the basic inference mechanism, termination criteria, and (in the first order case) unification and equational reasoning.
Besides the usual testing, our theorem provers will produce proofs terms much must be independently verifiable by a bidirectional checker. In addition, there should be some regression testing methods.
Various choices, heuristic, or stages in the prover should be evaluated against each other. This is should employ some problems as gleaned from the TPTP as well as some of your own theories. Some sources:
Besides the weekly plan and summary, there should be sufficient documentation of the code to make it readable. Also, there should be a final summary paper describing all aspects of the project. This includes the interface, the inference engine, the validation techniques, the sample suites and the empirical results.
Besides the lecture notes, here are further references that may be useful to each kind of prover.
These are only a few pointers, including one implementation home page. Further references can be found in the papers below and among the handouts.
[T96]  Tanel Tammet. A Resolution Theorem Prover for Intuitionistic Logic Draft manuscript, 1996. Preliminary version appeared at CADE13, pp.216, Springer Verlag LNCS 1104, 1996. Implementation home page  
[DV01a] 
Anatoli Degtyarev and Andrei Voronkov. The inverse method. Handbook of Automated Reasoning, Vol.1, pp.179272, Elsevier Science and MIT Press, 2001. Corrigendum for pp.200201. 
These are only a few pointers, including two currently available implementations. Further references can be found in the papers below and among the handouts.
[O97]  Jens Otten. ileanTAP: An Intuitionistic Theorem Prover. In D.Galmiche, editor, Proceedings of the International Conference on Tableaux and Related Systems (TABLEAUX'97), pp.307312, Springer Verlag LNAI 1227, 1997. Implementation home page 
[SFH92]  Dan Sahlin, Torkel Franzén, Seif Haridi. An Intuitionistic Predicate Logic Theorem Prover Journal of Logic and Computation, Vol.2(5), pp.619656, 1992. Earlier version available as Technical Report SICS8901, Swedish Institute of Computer Science, April 1989. Implementation home page 
[W01] 
Arild Waaler. Connections in nonclassical logics. Handbook of Automated Reasoning, Vol.2, pp.14871578, Elsevier Science and MIT Press, 2001. 
[W90] 
Lincoln Wallen. Automated Proof Search in NonClassical Logics. MIT Press, 1990. 
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