Transduction from trees to graphs through folding

Parsing Unranked Tree Languages, Folded Once

“…This section recalls relevant formal language theory and fixes notation. Central definitions have been lifted from the introduction of regular tree foldings in [18,19].…”

“…Just like OPDGs, regular tree-foldings are suitable for modelling AMRs, but unlike OPDGs, they can also accommodate cyclic graphs and more varied types of node-sharing. On the down-side, the parsing problem is more difficult: If the relative order among the edges attaching to a node is preserved by the folding, then parsing can be performed in polynomial time in the size of the input graph [19]. If, however, this order is relaxed so that the output graph is considered to be unordered, then the parsing problem is NP-complete [19].…”

“…On the down-side, the parsing problem is more difficult: If the relative order among the edges attaching to a node is preserved by the folding, then parsing can be performed in polynomial time in the size of the input graph [19]. If, however, this order is relaxed so that the output graph is considered to be unordered, then the parsing problem is NP-complete [19]. In the tree in Subfigure (a), round nodes with and without annotation denote a label in Σ × ∆ and Σ, respectively, and square nodes denote a label in ∆ = {α, β, γ, δ, ϵ}.…”

“…The proof of NP-completeness in [19] reduces the 3-PARTITION problem into a problem of parsing a regular tree language under two applications of the folding operation. However, the case where only one folding operation is used was left as an open problem.…”

“…However, the case where only one folding operation is used was left as an open problem. The original construction in [19] does not work, because it requires two fold operations to hide information on how to to partition a multiset into blocks, such that all have three elements and the same sum M. In this paper, we show that the parsing problem for unranked regular tree languages folded once under an order-cancelling fold semantics (see [19] for the contrasting order-preserving case) is solvable in non-uniform polynomial time, i.e., in polynomial time when considering the language as fixed.…”