TWiki> TFlex Web>Proposals>DeepGRs>DepStructureProperties (01 Sep 2008, MarkMcConville)

### Definition of dependency structures

A dependency structure over sentence S = w1 w2 ... wn is an ordered triple <N,f,R> where:

• N is a finite, non-empty set of nodes
• f is a (possibly partial) function from N to {i | Wi ∈ S}, i.e. every node is labelled with exactly one token from the sentence (but nothing stops the same token labelling more than one node)
• R is an irreflexive, rooted relation on N, i.e. no self-arcs, and there is exactly one element of N which is a source but not a target.

### Optional constraints

An 'acyclic' dependency structure over sentence S is a dependency structure <N,f,R> over S where:

• if there is a sequence of R-arcs from node n1 to node n2, then there is no sequence of R-arcs from n2 to n1.

A dependency 'tree' over sentence S is an acyclic dependency structure <N,f,R> over S where:

• every node is a target of at most one arc, i.e. no re-entrancy.

A 'projective' dependency tree over sentence S = w1 w2 ... wn is a dependency tree <N,f,R> over S, where:

• for all ni,j,k∈N, if niRnk and f(ni) < f(nj) < f(nk), then niR*nj, where R* is the reflexive transitive closure of R, i.e. no corssing dependencies.

A 'surjective' dependency structure over sentence S is a dependency structure <N,f,R> over S where:

• f is a surjective function, i.e. every token in S labels at LEAST one node in N.

An 'injective' dependency structure over sentence S is a dependency structure <N,f,R> over S where:

• f is an injective function, i.e. every token in S labels at MOST one node in N.

A 'bijective' dependency structure over sentence S is a surjective, injective dependency structure <N,f,R> over S, i.e. every token in S labels EXACTLY one node in N.

Most dependency parsers assume bijective, projective dependency trees. Recent work (based on the Prague Dependency Treebank and the Danish Dependency Treebank) has relaxed both the projective constraint (i.e. bijective dependency trees) and then the tree constraint (i.e. bijective acyclic dependency structures). There are good reasons for relaxing the surjective constraint (expletive pronouns) and the injective constraint (argument cluster coordination, noun modifier coordination).

proj tree non-proj tree acyclic non-tree non-acyclic 1 2 3 - - - - - - - - - - - - -

Note: it is also possible that we might want to relax the constraint that f is a function. Existence of multi-word expressions make the possibility of a 'total mapping' interesting.

### Definition of typed dependency structures

A 'typed' dependency structure over sentence S = w1 w2 ... wn and vocabulary Σ of dependency symbols is an ordered pair <N,f,R,g> where:

• <N,f,R> is a dependency structure over S
• g is a function from R to Σ, i.e. every arc is labelled with exactly one dependency symbol.

### Others

Functionality - for each word W and each dependency label T, there is no more than one dependency from W labelled with T

-- MarkMcConville - 14 Aug 2008

Topic revision: r6 - 01 Sep 2008 - 11:32:00 - MarkMcConville

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