Completeness

Completeness is a more interesting property because it depends crucially on the lookup relation, and on the chosen indexing scheme. Incompleteness can arise either when the lookup step does not add all the clauses that are needed for a proof, or if the combination of indices prevents the creation of an item that would be needed to prove the goal.

Without lookup and indexing (i.e., if all clauses of the program are items in the chart), the completeness argument would be even simpler than that for regular Earley deduction, since the addition of items is not even restricted by the prediction step.

3.4.2.1 Restrictions on the Lookup Operation

We show that the lookup step preserves completeness for the case of hpsgwith sequence union. We make the simplified assumption that anhpsggrammar con-sists of clauses of the following form (ignoring the cp/2 goal for the moment), where S stands for the predicate sign/1, L for lexical sign/1, and P for phrasal sign/1. We assume binary branching for this example.

S ←L.

S ←P.

P ←S∧S.

(3.13) Given this abstract program, all branches of a proof tree have the following form.

S(P S)^∗L (3.14) It is easy to see that all leaves of any proof tree are lexical signs (L). Therefore, a specification of the lookup relation is possible that adds only lexical signs to the chart if the goal is to prove that the input is a sign, as in the following abstract specification of the lookup relation.

lookup(S, L). (3.15)

Addition of thecp/2goals is not problematic since it is not proved by bottom-up methods, but rather by normal top-down resolution for which all needed clauses are present.

We must now further refine the above argument for the case where only a subset of the lexical signs are added to the chart by showing that the lexical signs which are not added as items cannot be leaves of the proof tree.

In the parsing case, the argument builds on the fact that a non-erasing opera-tion is used for the combinaopera-tion of strings. As a consequence, no lexical sign can be part of the proof tree that does not cover a word of the input string.¹⁹ Therefore, words that are not in the input string need not be added as lexical items to the chart.

Completeness for the parsing case

If every reading of every word in the input string and every non-unit clause of the program are added to the chart, then completeness of parsing is guaranteed.

For the case of generation, the same argument can be made on the basis of the semantic content of the goal, of the lexical signs, and of the compositional operation for the combination of semantic contents of the constituents. In case of a semantic monotonicity requirement, any semantic content of a leaf of the proof tree becomes part of the semantic content of nodes that dominate it. Therefore, any lexical sign whose semantic content is not a component of the semantic content

19If a grammar makes use of empty categories, these must also be added to the chart since they can be part of any input string. All grammar rules which introduce lexical entries must also be added to the chart.

of the goal need not be added as items to the chart.

Completeness for semantically monotonic generation

If every lexical entry for every component of the input logical form and every non-unit clause of the program are added to the chart, then completeness of generation with semantically monotonic grammars is guaranteed.

In the case of semantically non-monotonic grammars, the completeness depends crucially on the fact that the lookup step adds all the lexical entries that are needed for a proof of the goal, even if their semantics is not a component of the logical form to be generated. This is ensured by a precompilation of the lookup relation by making use of the non-chain rules and of lexicalised chain rules of the grammar, as illustrated in section 3.3.2.

Completeness for semantically non-monotonic generation

If every lexical entry for every decomposition (via non-chain rules and lexicalised chain rules) of the input logical form and every non-unit clause of the program are added to the chart, then completeness of generation with semantically non-monotonic grammars is guaranteed.

We conclude that a specification of the lookup relation preserves completeness if it adds all items that can occur as leaves in the proof tree for a given goal.

3.4.2.2 Restrictions on Indexing

Indexing serves the purpose of preventing the creation of items that cannot be part of the proof of a given goal. In order to show completeness, it must be shown that the indexing operation does not rule out any items that could be part of the proof.

In section 3.2.6, we have argued that indexing is redundant in the sense that it only represents information that is an instantiation of the program clauses for a particular parsing or generation instance. The clearest example of this is the case where thephonvalue in a chart parser with a context-free backbone is rep-resented in the clause part as well as in the index part of each item. In this case, the combination of indices will only fail if the combination of the clauses by the fundamental rule also fails for that particular problem instance.

[[index(C)]]⊇[[C]] (3.16)

[[R_i(index(C₁),index(C₂))]]⊇[[R(C₁, C₂)]] (3.17)

Hence, if the requirements postulated in section 3.2.6 on the choice of indices in (3.16), and on the operation for combining indices in (3.17) are fulfilled, indexing will only block derivation steps that do not contribute to a proof of the particular problem instance, and the use of indexing does not affect completeness.