Context-free Tree Grammars Research Articles

Combinatory categorial grammars as generators of weighted forests

Combinatory Categorial Grammar (CCG) is an extension of categorial grammar that is well-established in computational linguistics. It is mildly context-sensitive, so it is efficiently parsable and reaches an expressiveness that is suitable for describing natural languages. Weighted CCG (wCCG) are introduced as a natural extension of CCG with weights taken from an arbitrary commutative semiring. Their expressive power is compared to other weighted formalisms with special emphasis on the weighted forests generated by wCCG since the ability to express the underlying syntactic structure of an input sentence is a vital feature of CCG in the area of natural language processing. Building on recent results for the expressivity in the unweighted setting, the corresponding results are derived for the weighted setting for any commutative semiring. More precisely, the weighted forests generatable by wCCG are also generatable by weighted simple monadic context-free tree grammar (wsCFTG). If the rule system is restricted to application rules and composition rules of first degree, then the generatable weighted forests are exactly the regular weighted forests. Finally, when only application rules are allowed, then a proper subset of the regular weighted forests is generatable.

The tree-generative capacity of combinatory categorial grammars

The generative capacity of combinatory categorial grammars (CCGs) as generators of tree languages is investigated. It is demonstrated that the tree languages generated by CCGs can also be generated by simple monadic context-free tree grammars. However, the important subclass of pure combinatory categorial grammars cannot even generate all regular tree languages. Additionally, the tree languages generated by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then these grammars can generate only a proper subset of the regular tree languages, whereas they can generate exactly the regular tree languages once first-degree composition rules are permitted.

Universal Tree Source Coding Using Grammar-Based Compression

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources.

Linear context-free tree languages and inverse homomorphisms

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. In fact, we prove a stronger result: we encode Dyck words into a linear context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-free tree languages are closed under inverse linear tree homomorphisms.

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.

On characterization of fuzzy tree pushdown automata

This paper introduces the concepts of fuzzy tree pushdown automata and fuzzy context-free tree grammars. At first, we show that any fuzzy context-free tree grammar can be converted into a corresponding definition of fuzzy Chomsky normal form for trees. Then, the relation between a fuzzy context-free tree grammar and a fuzzy tree pushdown automaton is investigated. In fact, we show that the class of languages accepted by fuzzy tree pushdown automata is identical to the one generated by fuzzy context-free tree grammars. Some examples are given to clarify the results.

Regular Approximation of Weighted Linear Context-Free Tree Languages

We show how to train a weighted regular tree grammar such that it best approximates a weighted linear context-free tree grammar concerning the Kullback–Leibler divergence between both grammars. Furthermore, we construct a regular tree grammar that approximates the tree language induced by a context-free tree grammar.

Constructing small tree grammars and small circuits for formulas

It is shown that every tree of size n over a fixed set of σ different ranked symbols can be produced by a so called straight-line linear context-free tree grammar of size O(nlogσ⁡n), which can be used as a compressed representation of the input tree. Two algorithms for computing such tree grammar are presented: One working in logarithmic space and the other working in linear time. As a consequence, it is shown that every arithmetical formula of size n, in which only m≤n different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size O(n⋅log⁡mlog⁡n) and depth O(log⁡n). This refines a classical result of Brent from 1974, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n).

Approximation of smallest linear tree grammar

A simple linear-time algorithm for constructing a linear context-free tree grammar of size O(rg+rglog⁡(n/rg)) for a given input tree T of size n is presented, where g is the size of a minimal linear context-free tree grammar for T, and r is the maximal rank of symbols in T (which is a constant in many applications). This is the first example of a grammar-based tree compression algorithm with a good, i.e. logarithmic in terms of the size of the input tree, approximation ratio. The analysis of the algorithm uses an extension of the recompression technique from strings to trees.

The generative power of delegation networks

A device that generates trees over a ranked alphabet, together with an interpretation of the symbols in that alphabet as functions or relations on some domain, generates subsets of that domain. This well-known concept of tree-based generator is essentially already present in the seminal paper by Mezei and Wright from 1967. A delegation network consists of a finite set of such generators that can “delegate” parts of the generation process to each other. It can be viewed as an (extended) IO context-free tree grammar with an interpretation. We investigate the theoretical properties of these networks and establish several characterizations of the generated subsets, in the style of Mezei and Wright. We also show that the hierarchy of tree language classes obtained by iterating the concept of delegation, is properly contained in the closure of the regular tree languages under nondeterministic macro tree transductions, but not contained in the IO-hierarchy.

Context-Free Tree Grammars are as Powerful as Context-Free Jungle Grammars

Jungles generalize trees by sharing subtrees and allowing garbage. It is shown that IO context-free tree grammars generate the same jungle languages as context-free jungle grammars. Also, they defi ...

Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Weir [34] proved a Chomsky-Schutzenberger-like representation theorem for the string languages of tree-adjoining grammars, where the Dyck language Dn in the Chomsky-Schutzenberger characterization is replaced by the intersection D2n ∩ g(D2n), where g is a certain bijection on the alphabet consisting of 2n pairs of brackets. This paper presents a generalization of this theorem to the string languages that are the yield images of the tree languages generated by simple (i.e., linear and non-deleting) context-free tree grammars. This result is obtained through a natural generalization of the original Chomsky-Schutzenberger theorem to the tree languages of simple context-free tree grammars. We use Rogers’s [24,23] notion of multi-dimensional trees to state this latter theorem in a very general, abstract form.

XML tree structure compression using RePair

XML tree structures can conveniently be represented using ordered unranked trees. Due to the repetitiveness of XML markup these trees can be compressed effectively using dictionary-based methods, such as minimal directed acyclic graphs (DAGs) or straight-line context-free (SLCF) tree grammars. While minimal SLCF tree grammars are in general smaller than minimal DAGs, they cannot be computed in polynomial time unless P=NP. Here, we present a new linear time algorithm for computing small SLCF tree grammars, called TreeRePair, and show that it greatly outperforms the best known previous algorithm BPLEX. TreeRePair is a generalization to trees of Larsson and Moffat's RePair string compression algorithm. SLCF tree grammars can be used as efficient memory representations of trees. Using TreeRePair, we are able to produce the smallest queryable memory representation of ordered trees that we are aware of. Our investigations over a large corpus of commonly used XML documents show that tree traversals over TreeRePair grammars are 14 times slower than over pointer structures and 5 times slower than over succinct trees, while memory consumption is only 1/43 and 1/6, respectively. With respect to file compression we are able to show that a Huffman-based coding of TreeRePair grammars gives compression ratios comparable to the best known XML file compressors.

Parameter reduction and automata evaluation for grammar-compressed trees

Trees can be conveniently compressed with linear straight-line context-free tree grammars. Such grammars generalize straight-line context-free string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression). It is shown that every linear straight-line context-free tree grammar can be transformed in polynomial time into a monadic (and linear) one. A tree grammar is monadic if each nonterminal uses at most one context parameter. Based on this result, polynomial time algorithms are presented for testing whether a given (i) nondeterministic tree automaton or (ii) nondeterministic tree automaton with sibling-constraints or (iii) nondeterministic tree walking automaton, accepts a tree represented by a linear straight-line context-free tree grammar. It is also shown that if tree grammars are nondeterministic or non-linear, then reducing their numbers of parameters cannot be done without an exponential blow-up in grammar size.

The Equivalence of Tree Adjoining Grammars and Monadic Linear Context-free Tree Grammars

The equivalence of leaf languages of tree adjoining grammars and monadic linear context-free grammars was shown about a decade ago. This paper presents a proof of the strong equivalence of these grammar formalisms. Non-strict tree adjoining grammars and monadic linear context-free grammars define the same class of tree languages. We also present a logical characterisation of this tree language class showing that a tree language is a member of this class iff it is the two-dimensional yield of an MSO-definable three-dimensional tree language.

Grammatical Verification for Mathematical Formula Recognition Based on Context-Free Tree Grammar

This paper proposes the use of a formal grammar for the verification of mathematical formulae for a practical mathematical OCR system. Like a C compiler detecting syntax errors in a source file, we want to have a verification mechanism to find errors in the output of mathematical OCR. A linear monadic context-free tree grammar (LM-CFTG) is employed as a formal framework to define “well-formed” mathematical formulae. A cubic time parsing algorithm for LM-CFTGs is presented. For the purpose of practical evaluation, a verification system for mathematical OCR is developed, and the effectiveness of the system is demonstrated by using the ground-truthed mathematical document database InftyCDB-1 and a misrecognition database newly constructed for this study.

Efficient memory representation of XML document trees

Implementations that load XML documents and give access to them via, e.g., the DOM, suffer from huge memory demands: the space needed to load an XML document is usually many times larger than the size of the document. A considerable amount of memory is needed to store the tree structure of the XML document. In this paper, a technique is presented that allows to represent the tree structure of an XML document in an efficient way. The representation exploits the high regularity in XML documents by compressing their tree structure; the latter means to detect and remove repetitions of tree patterns. Formally, context-free tree grammars that generate only a single tree are used for tree compression. The functionality of basic tree operations, like traversal along edges, is preserved under this compressed representation. This allows to directly execute queries (and in particular, bulk operations) without prior decompression. The complexity of certain computational problems like validation against XML types or testing equality is investigated for compressed input trees.

Application of the CKY Algorithm to Recognition of Tree Structures for Linear, Monadic Context-Free Tree Grammars

In this paper, a recognition algorithm for the class of tree languages generated by linear, monadic context-free tree grammars (LM-CFTGs) is proposed. LM-CFTGs define an important class of tree languages because LM-CFTGs are weakly equivalent to tree adjoining grammars (TAGs). The algorithm uses the CKY algorithm as a subprogram and recognizes whether an input tree can be derived from a given LM-CFTG in O(n4) time, where n is the number of nodes of the input tree.

Analogical Conception of Chomsky Normal Form and Greibach Normal Form for Linear, Monadic Context-Free Tree Grammars

This paper presents the analogical conception of Chomsky normal form and Greibach normal form for linear, monadic context-free tree grammars (LM-CFTGs). LM-CFTGs generate the same class of languages as four well-known mildly context-sensitive grammars. It will be shown that any LM-CFTG can be transformed into equivalent ones in both normal forms. As Chomsky normal form and Greibach normal form for context-free grammars (CFGs) play a very important role in the study of formal properties of CFGs, it is expected that the Chomsky-like normal form and the Greibach-like normal form for LM-CFTGs will provide deeper analyses of the class of languages generated by mildly context-sensitive grammars.

The complexity of tree automata and XPath on grammar-compressed trees

The complexity of various membership problems for tree automata on compressed trees is analyzed. Two compressed representations are considered: dags, which allow to share identical subtrees in a tree, and straight-line context-free tree grammars, which moreover allow to share identical intermediate parts in a tree. Several completeness results for the classes NL, P, and PSPACE are obtained. Finally, the complexity of the evaluation problem for (structural) XPath queries on trees that are compressed via straight-line context-free tree grammars is investigated.