We
study constant-free versions of the inclusion problem of pattern
languages and the satisfiability problem of word equations. The
inclusion problem of pattern languages is known to be undecidable for
both erasing and nonerasing pattern languages, but decidable for
constant-free erasing pattern languages. We prove that it is undecidable
for constant-free nonerasing pattern languages. The satisfiability
problem of word equations is known to be in PSPACE and NP-hard. We prove
that the nonperiodic satisfiability problem of constant-free word
equations is NP-hard. Additionally, we prove a polynomial-time reduction
from the satisfiability problem of word equations to the problem of
deciding whether a given constant-free equation has a solution morphism α
such that α(xy) ≠ α(yx) for given variables x and y.