  We develop foundations for computing Craig interpolants and similar
intermediates of two given formulas with first-order theorem provers that
construct clausal tableaux. Provers that can be understood in this way include
efficient machine-oriented systems based on calculi of two families:
goal-oriented like model elimination and the connection method, and bottom-up
like the hyper tableau calculus. The presented method for Craig-Lyndon
interpolation involves a lifting step where terms are replaced by quantified
variables, similar as known for resolution-based interpolation, but applied to
a differently characterized ground formula and proven correct more abstractly
on the basis of Herbrand's theorem, independently of a particular calculus.
Access interpolation is a recent form of interpolation for database query
reformulation that applies to first-order formulas with relativized quantifiers
and constrains the quantification patterns of predicate occurrences. It has
been previously investigated in the framework of Smullyan's non-clausal
tableaux. Here, in essence, we simulate these with the more machine-oriented
clausal tableaux through structural constraints that can be ensured either
directly by bottom-up tableau construction methods or, for closed clausal
tableaux constructed with arbitrary calculi, by postprocessing with
restructuring transformations.
