  In this paper we provide for parsing with respect to grammars expressed in a
general TFS-based formalism, a restriction of ALE. Our motivation being the
design of an abstract (WAM-like) machine for the formalism, we consider parsing
as a computational process and use it as an operational semantics to guide the
design of the control structures for the abstract machine.
  We emphasize the notion of abstract typed feature structures (AFSs) that
encode the essential information of TFSs and define unification over AFSs
rather than over TFSs. We then introduce an explicit construct of multi-rooted
feature structures (MRSs) that naturally extend TFSs and use them to represent
phrasal signs as well as grammar rules. We also employ abstractions of MRSs and
give the mathematical foundations needed for manipulating them. We then present
a simple bottom-up chart parser as a model for computation: grammars written in
the TFS-based formalism are executed by the parser. Finally, we show that the
parser is correct.
