Structural identifiability of Order-of-Processing models: When do different cognitive architectures generate identical response time distributions

The Order-of-Processing (OP) method, introduced by Goldstein and Fisher in 1991, provides a powerful framework for deriving response-time distributions from stochastic network models of cognitive architecture. While the forward problem—predicting response-time distributions from a hypothesised network structure and process parameters—is well understood, the inverse problem remains unexamined: given an observed response-time distribution, when can researchers uniquely recover the underlying rate parameters, and when can they not? This work studies a restricted path-based subclass of OP models and develops a theory of structural identifiability for that subclass using moment-generating functions, Laplace-transform arguments, and diagram symmetries. The work shows that, under stated regularity and no-cancellation conditions, the multiset of rate parameters is generically recoverable from the reduced rational transform, whereas assignment of those rates to specific process labels can fail whenever the OP diagram contains nontrivial permutation symmetries. The work formalises the symmetry group of an OP diagram, show how its action induces observationally equivalent parameter assignments, and distinguish this within-framework non-identifiability from the broader serial–parallel mimicry problem. Applications to visual-search architectures illustrate symmetric, asymmetric, and partially symmetric cases and clarify the inferential scope of OPM-based analyses. The results provide practical guidance on when additional constraints or experimentally induced asymmetries are needed before fitted OPM parameters can be interpreted psychologically.