Hintikka Multiplicities in Matrix Decision Methods for Some Propositional Modal Logics

This work is a study of the inter-translatability of two closely related proof methods, i.e. tableau (or sequent) and connection based, in the case of the propositional modal logics K, K4, T, S4, paying particular attention to the relation between matrix multiplicity and multiple use of 0-formulae (contractions) in tableaux/sequent proofs. The motivation of the work is the following. Since the role of a multi-plicity in matrix methods is the encoding of the number of copies of a given formula that are needed in order to prove a valid formula, it is important to nd upper bounds for multiplicities in order to reduce as much as possible the search space for proofs. Moreover, it is obviously a crucial issue if the matrix method is to be used as a decision method. We exploit previous results establishing upper bounds on the number of contractions in tableau/sequent proofs 4], in order to establish upper bounds for multiplicities in matrix systems. We obtain two kinds of upper bounds: in function of the size of the formula to be proved and in function of the number of the atomic paths through the unindexed formula-tree. Such bounds may be non-optimal. However, the method used to establish them may be useful for obtaining ner upper bounds.