iProverModulo 2.5+0.1

G. Burel
Samovar, ENSIIE, University Paris-Saclay, France

Architecture

iProverModulo [Bur11] is an extension of iProver [Kor08] to integrate Polarized resolution modulo [Dow10]. Polarized resolution modulo consists in presenting the theory in which the problem has to be solved by means of polarized rewriting rules instead of axioms. It can also be seen as a combination of the set-of-support strategy and selection of literals.

iProverModulo consists of two tools: First, autotheo is a theory preprocessor that converts the axioms of the input into rewriting rules that can be used by Polarized resolution modulo. Second, these rewriting rules are handled by a patched version of iProver 2.5 which integrates Polarized resolution modulo. The integration of polarized resolution modulo in iProver only affects its ordered resolution calculus, so that the instantiation calculus is untouched.

iProverModulo 2.5+0.1 outputs a proof that is made of two parts: First, autotheo prints a derivation of the transformation of the axioms into rewriting rules. This derivation is in TSTP format and includes the CNF conversions obtained from E. Second, the modified version of iProver outputs a proof in TSTP format from this set of rewriting rules and the other input formulas.

Strategies

Autotheo is first run to transform the formulas of the problem whose role is "axiom" into polarized rewriting rules. Autotheo offers a set of strategies to that purpose. For the competition, the Equiv and the ClausalAll strategies will be used. The former strategy orients formulas intuitively depending of their shape. It may be incomplete, so that the prover may give up in certain cases. However, it shows interesting results on some problems. The second strategy should be complete, at least when equality is not involved. The rewriting system for the first strategy is tried for half the time given for the problem, then the prover is restarted with the second strategy if no proof has been found.

The patched version of iProver is run on the remaining formulas modulo the rewriting rules produced by autotheo. No scheduling is performed. To be compatible with Polarized resolution modulo, literals are selected only when they are maximal w.r.t. a KBO ordering, and orphans are not eliminated. To take advantage of Polarized resolution modulo, the resolution calculus is triggered more often than the instantiation calculus, on the contrary to the original iProver.

Normalization of clauses w.r.t. the term rewriting system produced by autotheo is performed by transforming these rules into an OCaml program, compiling this program, and dynamically linking it with the prover.

Implementation

iProverModulo is available as a patch to iProver. The most important additions are the plugin-based normalization engine and the handling of polarized rewriting rules. iProverModulo is available from 
  http://www.ensiie.fr/~guillaume.burel/blackandwhite_iProverModulo.html.en
Since iProverModulo needs to compile rewriting rules, an OCaml compiler is also provided.

Autotheo is available independently from iProverModulo from 

    http://www.ensiie.fr/~guillaume.burel/blackandwhite_autotheo.html.en
Autotheo uses E to compute clausal normal form of formula. The version of E it uses is very slightly modified to make it print the CNF derivation even if no proof is found.

Both of autotheo and iProver are written in OCaml.

For the SLD division, iProverModulo uses the CNF transformation tool provided with the Logtk library [Cru14].

Expected Competition Performance

Although iProverModulo is now based on version 2.5 of iProver, no great improvement of performance is expected compared to CASC-25, since only the resolution part of iProver, which is relatively stable, is modified.

References

Bur11
Burel G. (2011), Experimenting with Deduction Modulo, Bjorner N., Sofronie-Stokkermans V., Proceedings of the 23rd International Conference on Automated Deduction (Wroclaw, Poland), 162-176, Lecture Notes in Artificial Intelligence 6803, Springer-Verlag.
Cru14
Cruanes S. (2014), Logtk: A Logic ToolKit for Automated Reasoning and its Implementation, Schulz S., de Moura L., Konev B., 4th Workshop on Practical Aspects of Automated Reasoning (Vienna, Austria), 39-49, ePiC Series 31, EasyChair.
Dow10
Dowek G. (2010), Polarized Resolution Modulo, Calude C., Sassone V., Theoretical Computer Science, 182-196, IFIP Advances in Information and Communication Technology 323, Springer-Verlag.
Kor08
Korovin K. (2008), iProver - An Instantiation-Based Theorem Prover for First-order Logic (System Description), Armando A., Baumgartner P., Dowek G., Proceedings of the 4th International Joint Conference on Automated Reasoning (Sydney, Australia), 292-298, Lecture Notes in Artificial Intelligence 5195, Springer-Verlag.