Efficient Algorithms for Recognizing Weighted Tree-Adjoining Languages
This work provides more efficient algorithms for computational linguists and formal language theorists, though it is incremental as it builds on existing formalisms with specific optimizations.
The paper tackled the problem of computing stringsums and allsums for weighted tree-adjoining languages by defining semiring-weighted versions of two-level formalisms and designing new algorithms, resulting in improved time and space efficiency for LIG and EPDA, with factors such as O(n|N|) time and O(|Γ|) space savings, and introduced the first algorithms for PAA.
The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdown-adjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiring-weighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of $\mathcal{O}(n|\mathcal{N}|)$ (where $n$ is the string length and $|\mathcal{N}|$ is the size of the nonterminal set) and more space-efficient by a factor of $\mathcal{O}(|Γ|)$ (where $|Γ|$ is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more space-efficient and time-efficient than the algorithm of Alonso et al. (2001) by factors of $\mathcal{O}(|Γ|^2)$ and $\mathcal{O}(|Γ|^3)$, respectively. Finally, we give the first PAA stringsum and allsum algorithms.