Thank you for your continued engagement with our paper! We improve the formalisation of deixis below, making it more environment-agnostic.
To do so, we expand the observation to be an $m$-dimensional $R^n$ tensor, which can represent any possible properties of the observation.
## Formalisation of deixis
Let $O$ represent an abstract observation that an agent perceives from its environment, $O \in R^{n \times n \times n ... n = n^m}$. The $m$ dimensions can represent the spatial, temporal, or other positions. For example $R^{n^3}$ could represent a 3-D observatrion.
Let $O_p$ be the reference point in $O$, which coordinates are represented by an $n$-tuple of real numbers $(x_1,x_2 ... x_m)$ and $O_t$ be the target point in $O$, with its coordinates represented by an $n$-tuple of real numbers $(y_1, y_2 ... y_m)$.
Then, the relative distance function $d(O_p,O_t)$ returns an $n$-tuple of real numbers $(z_1,z_2 … z_m)$, such that $z_i = x_i - y_i$. This relative distance function allows for unambiguous identification of the target object $O_t$, given that the position of $O_p$ is known.
A deictic expression can be defined as a mapping of the value of $d(O_p,O_t)$, the reference point $O_p$, and their context $O$, to a specific linguistic or symbolic expression that describes the relationship between $O_p$ and $O_t$. This mapping can be represented as:
$(O,d(O_p,O_t),O_p) \rightarrow \text{Expression}(O,d(O_p,O_t),O_p)$
where the resulting expression $\text{Expression}(O,d(O_p,O_t),O_p)$ is a description of the reference point $O_p$ and its relative distances to the target point $O_t$, given the context $O$.
### Examples
Consider two objects, $O_p$ and $O_t$, located at Cartesian coordinates $(x_1, x_2)$ and $(y_1, y_2)$, respectively. The relative distance function $d(O_p, O_t) = (z_1, z_2)$ captures the differences in both dimensions.
In this context, suppose the expression "2 to the left and 3 up" is used. This would correspond to $z_1 = -2$ (indicating 2 to the left) and $z_2 = 3$ (indicating 3 up) relative to the reference point $O_p$. However, without knowing the exact position of $O_p$, this information alone is insufficient to identify $O_t$.
To resolve this, the reference point $O_p$ must either be explicitly included in the deictic expression or implicitly understood by all interlocutors. For example, if it is agreed that the reference point is at $(5, 4)$, where $(5,4)$ could be an abstraction of any object, the expression "2 to the left and 3 up" clearly locates $O_t$ at $(3, 7)$. Without such an agreement, a full deictic expression is necessary, such as "2 to the left and 3 up from $(5, 4)$".
Depending on the context $O$ and the need for clarity between agents, some information provided by the relative distance function $d$ may be omitted. For example, if the target object $O_t$ is the only item of a given type in the environment to the left of the reference point $O_p$, simply stating "to the left" might be sufficient without specifying the exact distances or the $y$ direction. Similarly, if the agents are conversing about objects on a flat surface, the height difference may be irrelevant and ignored, simplifying the expression further.
The examples above could also be rewritten to showcase the ability to specify any other deictic basis, by simply changing the underlying meaning of a given dimension. The expression "5 minutes after 10:00" specifies a temporal deixis by explicitly locating an event $O_t$ relative to a reference time $O_p$. In a 3D spatial context, an expression like "4 to the left, 2 forward, and 1 down" captures the exact relative position of an object $O\_t$ in Euclidean space.
This formalization will be included in the revised version of our paper.