Recursive querying of neural networks via weighted structures

Expressive querying of machine learning models - viewed as a form of intentional data - enables their verification and interpretation using declarative languages, thereby making learned representations of data more accessible. Motivated by the querying of feedforward neural networks, we investigate logics for weighted structures. In the absence of a bound on neural network depth, such logics must incorporate recursion; thereto we revisit the functional fixpoint mechanism proposed by Gr"adel and Gurevich. We adopt it in a Datalog-like syntax; we extend normal forms for fixpoint logics to weighted structures; and show an equivalent "loose" fixpoint mechanism that allows values of inductively defined weight functions to be overwritten. We propose a "scalar" restriction of functional fixpoint logic, of polynomial-time data complexity, and show it can express all PTIME model-agnostic queries over reduced networks with polynomially bounded weights. In contrast, we show that very simple model-agnostic queries are already NP-complete. Finally, we consider transformations of weighted structures by iterated transductions.