Trajectory Space, Situations, Fluents Sample Clauses

Trajectory Space, Situations, Fluents. ‌ In general, a system can behave in a non-deterministic manner and therefore have multiple possible trajectories through the state space. If we know all possible trajectories of the system, we know everything that the state space can express about the system.1 In GEM we identify a system Sys with the set of all its possible trajectories in the SOTA space which we call the system’s trajectory space Ξ. Writing T for the time domain, S for the state space, and F [T → S] for the set of all functions from T to S, the trajectory space is simply Ξ = F [T → S]. Then, a system is a subset of the trajectory space, Sys ⊆ Ξ. The state of the affairs concept of SOTA can therefore also be expressed in an enriched way to account for such trajectories: for each trajectory ξ of the system, and at each point in time t the state 1This does not hold in the probabilistic case where the probability distribution of trajectories gives additional information. of affairs is the value Sys(ξ, t), which is a point of the state space S which we assume to be a product of sets with the (finite or infinite) index set I: Sys(ξ, t) = ξ(t) = ⟨si⟩i∈I ∈ S if ξ ∈ Sys. ⊗ × ⊗ ⊆ ⊆ ⊗ × → In GEM we structure the state space as the result of an interaction between the ensemble and its environment. We formalize this using the notion of a combination operator: let Ξens and Ξenv be the trajectory spaces of the ensemble and environment, respectively2, and let : Ξens Ξenv Ξ be a partial map that is a surjection onto Sys, i.e., there exist Sysens Ξens and Sysenv Ξenv such that Sysens Sysenv = Sys. In this case we obtain a trajectory of the system for compatible pairs of ensemble and environment trajectories in Sysens Sysenv . We therefore regard the system as the result of combining ensemble Sysens and environment Sysenv using the operator . For example, in GEM we can structure a model of the robot ensemble as follows. First we define the state space Srobot as the Cartesian product of the robot’s coordinates (R2) and its current state {Exploring, Resting, Trapped}, and the trajectory space Ξrobot as usual Srobot = R2 × {Exploring, Resting, Trapped} Ξrobot = F [T → Srobot ]. ⊗ ⊗ →