Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences

We prove that the property of being closed (resp., rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., privileged, rich, trapezoidal, balanced) factor of length n is automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence.