  Often times, we specify tasks for a robot using temporal language that can
also span different levels of abstraction. The example command ``go to the
kitchen before going to the second floor'' contains spatial abstraction, given
that ``floor'' consists of individual rooms that can also be referred to in
isolation ("kitchen", for example). There is also a temporal ordering of
events, defined by the word "before". Previous works have used Linear Temporal
Logic (LTL) to interpret temporal language (such as "before"), and Abstract
Markov Decision Processes (AMDPs) to interpret hierarchical abstractions (such
as "kitchen" and "second floor"), separately. To handle both types of commands
at once, we introduce the Abstract Product Markov Decision Process (AP-MDP), a
novel approach capable of representing non-Markovian reward functions at
different levels of abstractions. The AP-MDP framework translates LTL into its
corresponding automata, creates a product Markov Decision Process (MDP) of the
LTL specification and the environment MDP, and decomposes the problem into
subproblems to enable efficient planning with abstractions. AP-MDP performs
faster than a non-hierarchical method of solving LTL problems in over 95% of
tasks, and this number only increases as the size of the environment domain
increases. We also present a neural sequence-to-sequence model trained to
translate language commands into LTL expression, and a new corpus of
non-Markovian language commands spanning different levels of abstraction. We
test our framework with the collected language commands on a drone,
demonstrating that our approach enables a robot to efficiently solve temporal
commands at different levels of abstraction.
