  In this paper, we investigate the sequence estimation problem of
faster-than-Nyquist (FTN) signaling as a promising approach for increasing
spectral efficiency (SE) in future communication systems. In doing so, we
exploit the concept of Gaussian separability and propose two probabilistic data
association (PDA) algorithms with polynomial time complexity to detect binary
phase-shift keying (BPSK) FTN signaling. Simulation results show that the
proposed PDA algorithm outperforms the recently proposed SSSSE and SSSgb$K$SE
algorithms for all SE values with a modest increase in complexity. The PDA
algorithm approaches the performance of the semidefinite relaxation (SDRSE)
algorithm for SE values of $0.96$ bits/sec/Hz, and it is within the $0.5$ dB
signal-to-noise ratio (SNR) penalty at SE values of $1.10$ bits/sec/Hz for the
fixed values of $\beta = 0.3$.
