  Data-driven systems are gathering increasing amounts of data from users, and
sensitive user data requires privacy protections. In some cases, the data
gathered is non-numerical or symbolic, and conventional approaches to privacy,
e.g., adding noise, do not apply, though such systems still require privacy
protections. Accordingly, we present a novel differential privacy framework for
protecting trajectories generated by symbolic systems. These trajectories can
be represented as words or strings over a finite alphabet. We develop new
differential privacy mechanisms that approximate a sensitive word using a
random word that is likely to be near it. An offline mechanism is implemented
efficiently using a Modified Hamming Distance Automaton to generate whole
privatized output words over a finite time horizon. Then, an online mechanism
is implemented by taking in a sensitive symbol and generating a randomized
output symbol at each timestep. This work is extended to Markov chains to
generate differentially private state sequences that a given Markov chain could
have produced. Statistical accuracy bounds are developed to quantify the
accuracy of these mechanisms, and numerical results validate the accuracy of
these techniques for strings of English words.
