# Generic Cognition

## Cognitive Interpretations

Consider an embodied agent whose internal state $$y \in Y$$, sensor inputs $$s \in S$$ and actuator outputs $$a \in A$$ vary over time.

We will define a cognitive interpretation $$C$$ of the agent as a function which maps the agent’s state $$y$$ to a belief $$b \in B$$, a desire $$d \in D$$ and a fundamental set of immutable assumptions $$\phi \in \Phi$$. We will require that the assumptions do not vary with the agent’s state. $C: Y \to (B \times D \times \Phi) \textrm{ where } \\ \forall y_1, y_2 \in Y {\ \cdot \ }\bigg[ C(y_1) = (b_1, d_1, \phi_1) \ \land \ C(y_2) = (b_2, d_2, \phi_2) \bigg] \implies \phi_1 = \phi_2$

Given a cognitive interpretation, we can project the agent’s physical internal state into a space of cognitive variables. Consequently, we can observe a “cognitive dynamics”: the changes in the agent’s (attributed) beliefs and desires caused by sensory input (and responsible for actuator output).

## Theories Of Rationality

By itself, the cognitive interpretation $$C$$ is meaningless, since there are no semantics attached to the attributed cognitive content of the agent’s mind. It acquires meaning only when coupled with a theory of rationality $$R$$: a normative ideal of how beliefs and desires should be related to sensations and actions.

We will define a theory of rationality $$R$$ as a function describing, for an agent with cognitive state $$(b, d, \phi) \in B \times D \times \Phi$$, faced with sensory input $$s \in S$$, what the rational options are for the agent’s next beliefs, desires and actions. $R: (B \times D \times \Phi \times S) \to (2^B \times 2^D \times 2^A)$ ...where $$2^X$$ denotes the powerset of $$X$$. Note that more than one belief, desire and / or action may be evaluated as rational by the theory; this is common when the effects of different actions are indistinguishable, and for generality we allow it to be the case for beliefs and desires as well.

## Subsumption

We will say that one theory of rationality $$R_1: (B_1 \times D_1 \times \Phi_1 \times S) \to (2^{B_1} \times 2^{D_1} \times 2^A)$$ is subsumed by another $$R_2: (B_2 \times D_2 \times \Phi_2 \times S) \to (2^{B_2} \times 2^{D_2} \times 2^A)$$ if and only if $$R_2$$ can compute $$R_1$$ under a transformation of the individual variables $$b, d, \phi$$. We write this $$R_1 \preceq R_2$$:

\begin{aligned} & R_1 \preceq R_2 \iff \\ & \forall b \in B_1, \ d \in D_1, \ \phi \in \Phi_1, \ s \in S \\ & \exists f_B: B_1 \to B_2, \ f_D: D_1 \to D_2, \ f_\Phi: \Phi_1 \to \Phi_2 {\ \cdot \ }\\ & R_1(b, d, \phi, s) = R_2\left(f_B(b), f_D(d), f_\phi(\phi), s\right) \end{aligned}

Note that the equality in the final expression implies set identity in the components of $$R_1$$ and $$R_2$$’s values. It should be obvious that $$\preceq$$ constitutes a partial order relation, and hence defines an equivalence relation $$R_1 \sim R_2$$ in the case that $$R_1 \preceq R_2 \preceq R_1$$.

# Bayesian Inference

One very natural way of representing beliefs $$b$$ and $$\phi$$ mathematically is as subjective probability functions. In this case, we may restrict our theories of rationality to those which are consistent with Bayesian inference.

Assume that a belief $$b$$ actually corresponds to a probability function representing a belief about the instantaneous state $$z_t$$ of a “phenomenal world” variable