Logical Interpolation and Projection onto State in the Duration Calculus

The classical interpolation theorem of Craig (cf. e.g. [ChK73]) states that if φ ⇒ ψ is a valid first order predicate logic formula, then there exists a first order formula θ built using only nonlogical symbols occurring in both φ and ψ and, possibly, equality, such that the formulas φ ⇒ θ and θ ⇒ ψ are valid too. A formula θ with this property is called an interpolant between φ and ψ. Similar statements apply to a variety of non-classical and modal logics. In [Gue01] it was shown that abstract time interval temporal logics admit a different kind of interpolation theorems. In these theorems the parts φ and ψ are written using a shared vocabulary of rigid symbols and disjoint copies of the same vocabulary of flexible symbols. Instead of having shared flexible symbols between φ and ψ, it is required that the pairs of corresponding symbols from the two copies of the flexible symbol vocabulary occurring in φ and ψ evaluate to the same predicates, functions and constants within specified intervals of time. Given that, it is shown that interpolants between φ and ψ can be restricted to specify properties of the considered intervals of time only too. Let the languages built using the two copies of the flexible symbol vocabulary mentioned above be called L1 and L2, respectively. Let, given a formula α from L1, the result of replacing its flexible symbols except the flexible constant l (see the definition of ITL in Section 1) by their counterparts from L2 be denoted by α . Let Φ be a finite set of formulas from L1. Let φ and ψ be in L1 too. Let c0, c1 and c2 be rigid constants, which are shared by L1 and L2. Then in the case of ITL the interpolated formula has the form