Gentzen-type systems and resolution rules part I propositional logic

4 Computational modal logic

“…every valid formula φ has a derivation in which only (negated or unnegated) sub-formulae of φ occur [51]. Calculi for modal logics using the inverse method have been developed in [139,140,190]. The inverse method has been shown to be suitable for efficient modal logic theorem proving and is amenable to optimisations [190].…”

“…In the late 1980s and early 1990s various direct resolution methods for modal logics have been investigated [1,10,33,46,59,61,72,79,126,139,140]. According to [139] a resolution method for a logic L is determined by specifying (i) a class of formulae called clauses, (ii) a reduction method which allows us to transform any formula of L into a finite set of clauses, (iii) a calculus consisting of a set of resolution rules for deriving clauses (and possibly redundancy elimination and simplification rules), and (iv) a derivation process which starts from an initial set of clauses and constructs a sequence of derivable clauses.…”

“…According to [139] a resolution method for a logic L is determined by specifying (i) a class of formulae called clauses, (ii) a reduction method which allows us to transform any formula of L into a finite set of clauses, (iii) a calculus consisting of a set of resolution rules for deriving clauses (and possibly redundancy elimination and simplification rules), and (iv) a derivation process which starts from an initial set of clauses and constructs a sequence of derivable clauses. One can then define a modal resolution method to be a resolution method in which clauses are formulae of the modal logic L under consideration.…”

A resolution theorem prover for intuitionistic logic

“…[6] and [7]) is a forward-chaining proof search method. Search starts with the set of axioms and produces new sequents from the already derived ones by applying the sequent calculus rules in the "downwards" direction, until the formula we want to prove is eventually derived.…”

“…Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [17], [5], [1], [7], [18], [10], [16], [12], [13], [3], [8].…”