  This paper presents a method for learning logical task specifications and
cost functions from demonstrations. Linear temporal logic (LTL) formulas are
widely used to express complex objectives and constraints for autonomous
systems. Yet, such specifications may be challenging to construct by hand.
Instead, we consider demonstrated task executions, whose temporal logic
structure and transition costs need to be inferred by an autonomous agent. We
employ a spectral learning approach to extract a weighted finite automaton
(WFA), approximating the unknown logic structure of the task. Thereafter, we
define a product between the WFA for high-level task guidance and a Labeled
Markov decision process (L-MDP) for low-level control and optimize a cost
function that matches the demonstrator's behavior. We demonstrate that our
method is capable of generalizing the execution of the inferred task
specification to new environment configurations.
