Matrix Proof Methods for Modal Logics

We present matrix proof systems for both constantand varying-domain versions of the first-order modal logics K, K4, D, D4, T, 84 and 86 based on modal versions of Herbrand's Theorem specifically formulated to support efficient automated proof search. The systems treat the mil modal language (no normal-forming) and admit straightforward structure sharing implementations. A key fsature of our approach is the use of a specialised unification algorithm to reflect the conditions on the accessibility relation for a given logic. The matrix system for one logic differs from the matrix eystem for another only in the nature of this unification algorithm. In addition, proof search may be interpreted as constructing generalised proof trees in an appropriate tableauor sequent-based proof system. This facilitates the use of the matrix systems within interactive environments. 1 In t roduc t ion . Modal logics are widely used in various branches of artificial intelligence and computer science as logics of knowledge and belief (eg., [Moo80,HM85,Kon84]), logics of programs (eg., [Pne77]), and for specifying distributed and concurrent systems (eg., (HM84,Sti85b). As a consequence, the need arises for proof systems for these logics which facilitate efficient automated proof search. Traditional proof systems for modal logics, such as tableauor sequent-based systems are readily available (eg., [Kan57,Nis83,Fit83]). While these systems are to some extent human-oriented, the proof rules form an inadequate basis for automated proof search since they generate search spaces that contain considerable redundancies. The redundancies arise mainly from the characteristic emphasis on connectives and the proof rules for modal operators and quantifiers. The matrix methods for first-order classical logic, pioneered by Prawits [Pra60], and further developed by Andrews [And81j and Bibel [Bib8l], have been demonstrated to be less redundant than the most efficient of the resolution based methods for that logic [Bib82b|. The methods combine an emphasis on connections (drawn from the resolution methods) with an intensions! notion of a path. In this paper we present matrix proof systems for the modal logics K, K4, D, D4, T, 84 and 85, based on modal versions of Bibel's "computationally improved* Herbrand Theorem for first-order classical logic [Bib82cj. We consider both constantand varying-domain versions of the first-order modal logics. The major features of our approach may be summarised as follows. Validity within a logic is characterised by the existence of a set of connections (pairs of atomic formula occurrences: one positive, one negative) within the formula, with the property that every so-called atomic path through the formula contains (as a subpath) a connection from the set (§ 2.4). Such a set of connections is said to span the formula. For classical propositions! logic this condition suffices [And81,Bib8l|. For first-order logic a substitution (of parameters or terms for variables) must be found under which the (then propositional) connections in the spanning set are simultaneously complementary. Conditions are placed on the substitution that ensure amongst This work was supported in pert by 8ERC grant GR/D/44874 other things that a proof within a particular tableauor sequent-based proof system is constructable from the connections and the substitution [Bib82c, Wal86). This basically amounts to ensuring that the restrictions found on the traditional quantifier rules can be met. For the propositions! modal logics we keep the basic matrix framework but define a notion of complementarity for atomic formulae that ensures the existence of a proof in one of Fitting's prefixed tableau systems [Fit72,Fit83|. This amounts to ensuring that, semantically: the two atomic formulae of a connection can be interpreted as inhabiting the same "possible world," and proof-theoretically: that they can be given the same prefix (| 2.5.1). The key observation is that this can be established by noting the position of the atoms relative to the modal operators in the original formula and utilising a specialised unification algorithm operating over representations of these positions. Clearly, this notion of complementarity is logic-dependent, a dependence which is reflected in the choice of unification algorithm. Lifting these results to first-order constant-domain modal logics is simply a matter of combining this modal notion of complementarity with the first-order notion (§ 2.5.2). For the varying-domain versions we index individual variables with the prefix of their quantifier. The substitution of one variable for another is permitted provided their prefixes can be unified ($ 2.5.2). Checking a formula for validity within a modal logic is therefore reduced to a process of path checking and complementarity tests performed by a specialised unification algorithm (5 3). During this process extra copies may need to be considered of universally quantified formulae and/or formulae dominated by a modal operator of "necessary'' (Q) force. The duplication in both cases is managed by an extension of Bibel's indexing technique or multiplicity [Bib82a] which supports the implementation of the matrix systems using structure-sharing techniques [BM72]. The notions of multiplicity, substitution and spanning sets of connections form the basis of the relationship with Herbrand's Theorem. A number of authors have attempted to adapt computationally oriented proof systems for first-order logic to the modal logics considered here (eg., [Far83,AM66a,Kon86]). We compare our approach favourably to theirs in Section 4. 2 The moda l m a t r i x system*. 2.1 Prel iminar ies.