  Logic Programming is a Turing complete language. As a consequence, designing
algorithms that decide termination and non-termination of programs or decide
inductive/coinductive soundness of formulae is a challenging task. For example,
the existing state-of-the-art algorithms can only semi-decide coinductive
soundness of queries in logic programming for regular formulae. Another, less
famous, but equally fundamental and important undecidable property is
productivity. If a derivation is infinite and coinductively sound, we may ask
whether the computed answer it determines actually computes an infinite
formula. If it does, the infinite computation is productive. This intuition was
first expressed under the name of computations at infinity in the 80s. In
modern days of the Internet and stream processing, its importance lies in
connection to infinite data structure processing.
  Recently, an algorithm was presented that semi-decides a weaker property --
of productivity of logic programs. A logic program is productive if it can give
rise to productive derivations. In this paper we strengthen these recent
results. We propose a method that semi-decides productivity of individual
derivations for regular formulae. Thus we at last give an algorithmic
counterpart to the notion of productivity of derivations in logic programming.
This is the first algorithmic solution to the problem since it was raised more
than 30 years ago. We also present an implementation of this algorithm.
