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## Section: Application Domains

### Integrating a model checker and a theorem prover

The goal of combining model checking with inductive and co-inductive theorem is appealing. The strengths of systems in these two different approaches are strikingly different. A model checker is capable of exploring a finite space automatically: such a tool can repeatedly explore all possible cases of a given computational space. On the other hand, a theorem prover might be able to prove abstract properties about a search space. For example, a model checker could attempt to discover whether or not there exists a winning strategy for, say, tic-tac-toe while an inductive theorem prover might be able to prove that if there is a winning strategy for one board then there is a winning strategy for any symmetric version of that board. Of course, the ability to combine proofs from these systems could drastically reduce the amount of state exploration and verification of proof certificates that are needed to prove the existence of winning strategies.

Our first step to providing an integration of model checking and (inductive) theorem proving was the development of a strong logic, that we call $𝒢$, which extends intuitionistic logic with notions of least and greatest fixed points. We had developed the proof theory of this logic in earlier papers [4] [56]. We have now recently converted the Bedwyr system so that it formally accepts almost all definitions and theorem statements that are accepted by the inductive theorem prover Abella. Thus, these two systems are proving theorems in the same logic and their results can now be shared.

Bedwyr's tabling mechanism has been extended so that its it can make use of previously proved lemmas. For instance, when trying to prove that some board position has a winning strategy, an available stored lemma can now be used to obtain the result if some symmetric board position is already in the table.

Heath and Miller have shown how model checking can be seen as constructing proof in (linear) logic [64]. For more about recent progress on providing checkable proof certificates for model checking, see the web site for Bedwyr http://slimmer.gforge.inria.fr/bedwyr/.