To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.

A causal effect is defined as the distribution P (Y do(X), Z) where variables Y are observed, variables X are intervened upon (forced to values irrespective of their natural causes) and variables Z are conditioned on. Instead of placing various parametric restrictions based on background knowledge, we are interested in this paper in the question of identifiability: can the causal effect be uniquely determined from the distributions (data) we have and a graph representing our structural knowledge on the generating causal system. 1 In the most basic setting we are identifying causal effects from a single observational input distribution, corresponding to passively observed data. To solve such problems more generally than what is possible with the backdoor adjustment (Spirtes et al., 1993; Pearl, 2009; Greenland et al., 1999), Pearl (1995) introduced do-calculus, a set of three rules that together with probability theory enable the manipulation of interventional distributions. Shpitser and Pearl (2006a) and Huang and Valtorta (2006) showed that do-calculus is complete by presenting polynomial-time algorithms whose each step can be seen as a rule of do-calculus or as an operation based on basic probability theory. The algorithms have a high practical value because the rules of do-calculus do not by themselves provide an indication on the order in which they should be applied.

Data integrity is a property which a world state interpreted with a world model is consistent with the real operating environment. Even a formally verified safety claim of an autonomous system is prone to a malfunction caused by loss of data integrity. From a first-person viewpoint in a congested environment, some components of measurable part of the world state may become transiently deficient or unavailable because of the limited capability of sensor devices. If the system could get into a situation where the world state becomes suddenly unobservable, existing estimation methods may get unstable. These methods can hardly detect the loss of data integrity and produce an incorrect estimate without any notice. Our insight is that we can merge the original concept of observer theory with that of automated reasoning. Firstly, we propose a new way of unifying them into a problem of checking satisfiability of a formula that consists of predicates regarding the world model and decision variables regarding unmeasurable part of the world state. We can detect a loss of data integrity by checking if the problem is unsatisfiable. Secondly, we replace the idea of observability in control theory with identifiability with respect to a measure of tolerance and a world model. We show a procedure of estimating the world state with a bounded uncertainty specified by the measure of tolerance. Third, we show that a problem of sensor fusion, a problem of reasoning a world state of discrete and enumerated type, and a decision problem under uncertainty in the world state are formulated as an identifiability problem. The proposal presents a constructive basis for supporting the degree of confidence in the estimated world state.