In this system the input for the parser is not a simple list of words, as we have assumed up to now, but rather a word-graph: a directed, acyclic graph where the states are points in time, and the edges are labelled with word hypotheses and their corresponding acoustic score. Thus, such word-graphs are acyclic weighted finite-state automata.
In  a framework for processing ill-formed input is described in which certain common errors are modelled as (weighted) finite-state transducers. The composition of an input sentence with these transducers produces a (weighted) finite state automaton which is then input for the parser. In such an approach the need to generalize from input strings to input finite-state automata is also clear.
The generalization from strings to weighted acyclic finite-state automata introduces essentially two complications. Firstly, we cannot use string indices anymore. Secondly we need to keep track of the acoustic scores of the words used in a certain derivation.
In the head-corner parser, this leads to an alternative to the predicate smaller_equal/2. Rather than a simple integer comparison, we now need to check that a derivation from P0 to P can be extended to a derivation from E0 to E by checking that there are paths in the word-graph from E0 to P0 and from P to E.
The predicate connection/2 is true if there is a path in the word-graph from the first argument to the second argument. It is assumed that state names are integers; to rule out cyclic word-graphs we also require that for all transitions from P0 to P it is the case that P0 < P. Transitions in the word-graph are represented by clauses of the form wordgraph:trans(P0,Sym,P,Score) which indicate that there is a transition from state P0 to P with symbol Sym and acoustic score Score. The connection predicate can be specified simply as the reflexive and transitive closure of the transition relation between states:
A somewhat different approach that may turn out to be more efficient is to use the ordinary comparison operator that we used in the original definition of the head-corner parser. The possible extra cost of allowing impossible partial analyses is worthwhile if the more precise check would be more expensive. If for typical input word-graphs the number of transitions per state is high (such that almost all pairs of states are connected), then this may be an option.