  Given the emergent reasoning abilities of large language models, information
retrieval is becoming more complex. Rather than just retrieve a document,
modern information retrieval systems advertise that they can synthesize an
answer based on potentially many different documents, conflicting data sources,
and using reasoning. But, different kinds of questions have different answers,
and different answers have different complexities. In this paper, we introduce
a novel framework for analyzing the complexity of a question answer based on
the natural deduction calculus as presented in Prawitz (1965). Our framework is
novel both in that no one to our knowledge has used this logic as a basis for
complexity classes, and also in that no other existing complexity classes to
these have been delineated using any analogous methods either. We identify
three decidable fragments in particular called the forward, query and planning
fragments, and we compare this to what would be needed to do proofs for the
complete first-order calculus, for which theorem-proving is long known to be
undecidable.
