Modeling Graph Languages with Grammars Extracted via Tree Decompositions

Work on probabilistic models of natural language tends to focus on strings and trees, but there is increasing interest in more general graph-shaped structures since they seem to be better suited for representing natural language semantics, ontologies, or other varieties of knowledge structures. However, while there are relatively simple approaches to defining generative models over strings and trees, it has proven more challenging for more general graphs. This paper describes a natural generalization of the n-gram to graphs, making use of Hyperedge Replacement Grammars to define generative models of graph languages.9 page(s

Definability equals recognizability for graphs of bounded treewidth

We prove a conjecture of Courcelle, which states that a graph property is definable in MSO with modular counting predicates on graphs of constant treewidth if, and only if it is recognizable in the following sense: constant-width tree decompositions of graphs satisfying the property can be recognized by tree automata. While the forward implication is a classic fact known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS 201

The category of MSO transductions

MSO transductions are binary relations between structures which are defined using monadic second-order logic. MSO transductions form a category, since they are closed under composition. We show that many notions from language theory, such as recognizability or tree decompositions, can be defined in an abstract way that only refers to MSO transductions and their compositions

Treewidth, Pathwidth and Cospan Decompositions

We will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata

Algebras for Tree Decomposable Graphs

Complex problems can be sometimes solved efficiently via recursive decomposition strategies. In this line, the tree decomposition approach equips problems modelled as graphs with tree-like parsing structures. Following Milner’s flowgraph algebra, in a previous paper two of the authors introduced a strong network algebra to represent open graphs (up to isomorphism), so that homomorphic properties of open graphs can be computed via structural recursion. This paper extends this graphical-algebraic foundation to tree decomposable graphs. The correspondence is shown: (i) on the algebraic side by a loose network algebra, which relaxes the restriction reordering and scope extension axioms of the strong one; and (ii) on the graphical side by Milner’s binding bigraphs, and elementary tree decompositions. Conveniently, an interpreted loose algebra gives the evaluation complexity of each graph decomposition. As a key contribution, we apply our results to dynamic programming (DP). The initial statement of the problem is transformed into a term (this is the secondary optimisation problem of DP). Noting that when the scope extension axiom is applied to reduce the scope of the restriction, then also the complexity is reduced (or not changed), only so-called canonical terms (in the loose algebra) are considered. Then, the canonical term is evaluated obtaining a solution which is locally optimal for complexity. Finding a global optimum remains an NP-hard problem

Weighted DAG Automata for Semantic Graphs

Graphs have a variety of uses in natural language processing, particularly as representations of linguistic meaning. A deficit in this area of research is a formal framework for creating, combining, and using models involving graphs that parallels the frameworks of finite automata for strings and finite tree automata for trees. A possible starting point for such a framework is the formalism of directed acyclic graph (DAG) automata, defined by Kamimura and Slutzki and extended by Quernheim and Knight. In this article, we study the latter in depth, demonstrating several new results, including a practical recognition algorithm that can be used for inference and learning with models defined on DAG automata. We also propose an extension to graphs with unbounded node degree and show that our results carry over to the extended formalism

Multiple Context-Free Tree Grammars: Lexicalization and Characterization

Multiple (simple) context-free tree grammars are investigated, where "simple" means "linear and nondeleting". Every multiple context-free tree grammar that is finitely ambiguous can be lexicalized; i.e., it can be transformed into an equivalent one (generating the same tree language) in which each rule of the grammar contains a lexical symbol. Due to this transformation, the rank of the nonterminals increases at most by 1, and the multiplicity (or fan-out) of the grammar increases at most by the maximal rank of the lexical symbols; in particular, the multiplicity does not increase when all lexical symbols have rank 0. Multiple context-free tree grammars have the same tree generating power as multi-component tree adjoining grammars (provided the latter can use a root-marker). Moreover, every multi-component tree adjoining grammar that is finitely ambiguous can be lexicalized. Multiple context-free tree grammars have the same string generating power as multiple context-free (string) grammars and polynomial time parsing algorithms. A tree language can be generated by a multiple context-free tree grammar if and only if it is the image of a regular tree language under a deterministic finite-copying macro tree transducer. Multiple context-free tree grammars can be used as a synchronous translation device.Comment: 78 pages, 13 figure