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Uniform and non uniform strategies for tableaux calculi for modal logics. (English) Zbl 0826.03006

Logical calculi are formula manipulation methods for proving logical theorems. A very prominent class of logical calculi are the so-called tableaux systems. They can be used to refute formulae (proof by contradiction) or to show consistency by finding a model. The tableaux rules specify how a formula to be refuted must be decomposed into subformulae to be investigated. Tableaux systems usually build trees of formulae. A formula is refuted if each branch in the tree can be ‘closed’ by finding a contradiction.
Tableaux systems for standard propositional logic are confluent. This means the tableaux rules can be applied in an arbitrary order an no backtracking is necessary. For other logics this need not always be the case. A quite pathological case is the modal logic S4. Modal logic is propositional logic extended with the two modal operators box (‘necessarily’) and diamond (‘possibly’). S4 has the ‘transitivity axiom’ box \(p\) implies box box \(p\), which causes a lot of problems for automated reasoning systems. In particular the tableaux systems known so far are no longer confluent because wrong choices may cause the system to loop.
The problem of controlling the search in a tableaux system for propositional modal logics is investigated in the paper. Different backtracking and ordering strategies are proposed. Loop checking is combined with backtracking such that unnecessary choice points are avoided. The ordering strategies control the application of the tableaux rules for cases where different rules can be applied to the same node in the tree. So-called uniform strategies fix the ordering of the rule applications once and for all whereas non-uniform strategies choose the ordering according to the actual situation on the branch. Completeness is proved for the various strategies. They are tested with many examples from the literature and much empirical data is collected.

MSC:

03B35 Mechanization of proofs and logical operations
03B45 Modal logic (including the logic of norms)

Software:

Minlog; TPTP
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