If S ⊆{0,1}; * and S ′ = {0,1} * \sb S are both recognized within a certain nondeterministic time bound T then, in not much more time, one can write down tautologies A n → A′ n with unique interpolants I n that define S ∩{0,1} n ; hence, if one can rapidly find unique interpolants, then one can recognize S within deterministic time T p for some fixed p \s>0. In general, complexity measures for the problem of finding (...) unique interpolants in sentential logic yield new relations between circuit depth and nondeterministic Turing time, as well as between proof length and the complexity of decision procedures of logical theories. (shrink)

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and (...) also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. (shrink)