  The aim of the present paper is to generalise Sahlqvist correspondence theory
to the many-valued setting. For truth algebras that are perfect Heyting
algebras, we present the standard translation between many-valued modal
languages and suitably defined first-order and second-order correspondence
languages; and prove its correctness. We then define a general class of
inequalities, called Sahlqvist inequalities, to which algebraic methods can be
applied to obtain first-order local frame correspondents in the many-valued
setting. Difficulties that arise due to the lack of classical negation in the
many-valued setting are overcome by making use of signed generation trees. Next
we show that algorithmic approaches to obtain first-order local frame
correspondents can be generalise to the many-valued setting and present the
generalisation of the ALBA algorithm for this purpose. We conclude with a
comparison between classical (two-valued) and many-valued frame correspondents.
