Hyperedge Replacement Grammars Research Articles

Hypergraph Lambek grammars

In the graph grammar theory, there is a natural generalization of the context-free grammar approach to graphs called hyperedge replacement grammar (HRG). The latter is based on the hyperedge replacement procedure, and it preserves the main principles and properties of context-free grammars (e.g. the pumping lemma). In the string case, there is an alternative approach to describing formal languages, which was introduced by J. Lambek; it is based on a substructural logic called the Lambek calculus. In this paper, we generalize this calculus and this grammar approach to hypergraphs, which results in defining the hypergraph Lambek calculus (HL) and hypergraph Lambek grammars (HL-grammars). We show how to embed the Lambek calculus in HL justifying that the presented generalization is appropriate. After that, we study several structural properties of HL and turn to the question of expressive power of its grammars. It appears that HL-grammars are more powerful than HRGs while having the same algorithmic complexity with them; in particular, we show that intersection of a language generated by an HRG and of a language generated by an HL-grammar can be generated by some HL-grammar as well. Consequently, HL-grammars might be considered as enhancement of HRGs.

Grammars Based on a Logic of Hypergraph Languages

The hyperedge replacement grammar (HRG) formalism is a natural and well-known generalization of context-free grammars. HRGs inherit a number of properties of context-free grammars, e.g. the pumping lemma. This lemma turns out to be a strong restriction in the hypergraph case: it implies that languages of unbounded connectivity cannot be generated by HRGs. We introduce a formalism that turns out to be more powerful than HRGs while having the same algorithmic complexity (NP-complete). Namely, we introduce hypergraph Lambek grammars; they are based on the hypergraph Lambek calculus, which may be considered as a logic of hypergraph languages. We explain the underlying principles of hypergraph Lambek grammars, establish their basic properties, and show some languages of unbounded connectivity that can be generated by them (e.g. the language of all graphs, the language of all bipartite graphs, the language of all regular graphs).

Membership Problem with Adjacency Matrix

In this article, proposed a algorithm to solve the membership problem in Hyperedge Replacement Grammars (HRG). Given a hypergraph H with labeled nodes rooted and directed hyperedges, the problem consists in determining if H 2 L(G), where G is in HRG, this is to say, if H is in the language generated by G, for this the analysis is done directly in the Adjacency Matrix of the hypergraph H. For the algorithm proposed, also presents the proof of its correctness.

Weak Greibach Normal Form for Hyperedge Replacement Grammars

It is known that hyperedge replacement grammars are similar to string context-free grammars in the sense of definitions and properties. Therefore, we expect that there is a generalization of the well-known Greibach normal form from string grammars to hypergraph grammars. Such generalized normal forms are presented in several papers; however, they do not cover a large class of hypergraph languages (e.g. languages consisting of star graphs). In this paper, we introduce a weak Greibach normal form, whose definition corresponds to the lexicalized normal form for string grammars, and prove that every context-free hypergraph language (with nonsubstantial exceptions) can be generated by a grammar in this normal form. The proof presented in this paper generalizes a corresponding one for string grammars with a few more technicalities.

Uniform parsing for hyperedge replacement grammars

It is well known that hyperedge-replacement grammars can generate NP-complete graph languages even under seemingly harsh restrictions. This means that the parsing problem is difficult even in the non-uniform setting, in which the grammar is considered to be fixed rather than being part of the input. Little is known about restrictions under which truly uniform polynomial parsing is possible. In this paper we propose a low-degree polynomial-time algorithm that solves the uniform parsing problem for a restricted type of hyperedge-replacement grammars which we expect to be of interest for practical applications.

Speeding up Generalized PSR Parsers by Memoization Techniques

Predictive shift-reduce (PSR) parsing for hyperedge replacement (HR) grammars is very efficient, but restricted to a subclass of unambiguous HR grammars. To overcome this restriction, we have recently extended PSR parsing to generalized PSR (GPSR) parsing along the lines of Tomita-style generalized LR parsing. Unfortunately, GPSR parsers turned out to be too inefficient without manual tuning. This paper proposes to use memoization techniques to speed up GPSR parsers without any need of manual tuning, and which has been realized within the graph parser distiller Grappa. We present running time measurements for some example languages; they show a significant speed up by some orders of magnitude when parsing valid graphs. But memoization techniques do not help when parsing invalid graphs or if all parses of an ambiguous input graph shall be determined.

Ordered Tree Decomposition for HRG Rule Extraction

We present algorithms for extracting Hyperedge Replacement Grammar (HRG) rules from a graph along with a vertex order. Our algorithms are based on finding a tree decomposition of smallest width, relative to the vertex order, and then extracting one rule for each node in this structure. The assumption of a fixed order for the vertices of the input graph makes it possible to solve the problem in polynomial time, in contrast to the fact that the problem of finding optimal tree decompositions for a graph is NP-hard. We also present polynomial-time algorithms for parsing based on our HRGs, where the input is a vertex sequence and the output is a graph structure. The intended application of our algorithms is grammar extraction and parsing for semantic representation of natural language. We apply our algorithms to data annotated with Abstract Meaning Representations and report on the characteristics of the resulting grammars.

Formalization and correctness of predictive shift-reduce parsers for graph grammars based on hyperedge replacement

Hyperedge replacement (HR) grammars can generate NP-complete graph languages, which makes parsing hard even for fixed HR languages. Therefore, we study predictive shift-reduce (PSR) parsing that yields efficient parsers for a subclass of HR grammars, by generalizing the concepts of SLR(1) string parsing to graphs. We formalize the construction of PSR parsers and show that it is correct. PSR parsers run in linear space and time, and are more efficient than the predictive top-down (PTD ) parsers recently developed by the authors.

Learning Hyperedge Replacement Grammars for Graph Generation

The discovery and analysis of network patterns are central to the scientific enterprise. In the present work, we developed and evaluated a new approach that learns the building blocks of graphs that can be used to understand and generate new realistic graphs. Our key insight is that a graph's clique tree encodes robust and precise information. We show that a Hyperedge Replacement Grammar (HRG) can be extracted from the clique tree, and we develop a fixed-size graph generation algorithm that can be used to produce new graphs of a specified size. In experiments on large real-world graphs, we show that graphs generated from the HRG approach exhibit a diverse range of properties that are similar to those found in the original networks. In addition to graph properties like degree or eigenvector centrality, what a graph "looks like" ultimately depends on small details in local graph substructures that are difficult to define at a global level. We show that the HRG model can also preserve these local substructures when generating new graphs.

Juggrnaut: using graph grammars for abstracting unbounded heap structures

This paper presents a novel abstraction framework for heap data structures. It employs graph grammars, more precisely context-free hyperedge replacement grammars. We will show that this is a very natural formalism for modelling dynamic data structures in an intuitive way. Our approach aims at extending finite-state verification techniques to handle pointer-manipulating programs operating on complex dynamic data structures that are potentially unbounded in their size. The theoretical foundations of our approach and its correctness are the main focus of this paper. In addition, we present a prototypical tool entitled Juggrnaut that realizes our approach and show encouraging experimental verification results for three case studies: a doubly-linked list reversal, the flattening of binary trees, and the Deutsch---Schorr---Waite tree traversal algorithm.

Contextual hyperedge replacement

Contextual hyperedge-replacement grammars (contextual grammars, for short) are an extension of hyperedge replacement grammars. They have recently been proposed as a grammatical method for capturing the structure of object-oriented programs, thus serving as an alternative to the use of meta-models like uml class diagrams in model-driven software design. In this paper, we study the properties of contextual grammars. Even though these grammars are not context-free, one can show that they inherit several of the nice properties of hyperedge replacement grammars. In particular, they possess useful normal forms and their membership problem is in NP.

Test generation from recursive tile systems

SUMMARYThis paper explores the generation of conformance test cases for recursive tile systems (RTSs) in the framework of the classical ioco testing theory. The RTS model allows the description of reactive systems with recursion and is very similar to other models like pushdown automata, hyperedge replacement grammars or recursive state machines. Test generation for this kind of infinite state labelled transition systems is seldom explored in the literature. The first part presents an off‐line test generation algorithm for weighted RTSs, a determinizable sub‐class of RTSs, and the second one an on‐line test generation algorithm for the full RTS model. Both algorithms use test purposes to guide test selection through targeted behaviours. Additionally, essential properties relating verdicts produced by generated test cases with both the soundness, with respect to the specification, and the precision, with respect to a test purpose, are proved. Copyright © 2014 John Wiley & Sons, Ltd.

Juggrnaut: Graph Grammar Abstraction for Unbounded Heap Structures

We present a novel abstraction framework for heap data structures that uses graph grammars, more precisely context-free hyperedge replacement grammars, as an intuitive formalism for efficiently modeling dynamic data structures. It aims at extending finite-state verification techniques to handle pointer-manipulating programs operating on complex dynamic data structures that are potentially unbounded in their size. We demonstrate how our framework can be employed for analysis and verification purposes by instantiating it for binary trees, and by applying this instantiation to the well-known Deutsch-Schorr-Waite traversal algorithm. Our approach is supported by a prototype tool, enabling the quick verification of essential program properties such as heap invariants, completeness, and termination.

Second-Order Abstract Categorial Grammars as Hyperedge Replacement Grammars

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148---155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83---127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207---219, 1987) are two natural ways of generalizing "context-free" grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191---229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15---29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars.

An Internal Presentation of Regular Graphs by Prefix-Recognizable Graphs

The study of infinite graphs has potential applications in the specification and verification of infinite systems and in the transformation of such systems. Prefix-recognizable graphs and regular graphs are of particular interest in this area since their monadic second-order theories are decidable. Although the latter form a proper subclass of the former, no characterization of regular graphs within the class of prefix-recognizable ones has been known, except for a graph-theoretic one in [2]. We provide here three such new characterizations. In particular, a decidable, language-theoretic, necessary and sufficient condition for the regularity of any prefix-recognizable graph is established. Our proofs yield a construction of a deterministic hyperedge-replacement grammar for any prefix-recognizable graph that is regular.

Cooperating Distributed Hyperedge Replacement Grammars

The concept of cooperation and distribution as known from the area of grammar systems is introduced to graph grammars; more precisely, to hyperedge replacement (for short, HR) grammars. This can be seen as a modest approach to controlled HR. Two different derivation modes are considered, the so-called t-mode and the =k-mode. For the t-mode it is shown that the class of hypergraph languages obtained is equal to the class of ET0L HR languages which is a natural generalization of the class of ET0L languages to hypergraphs.

The obstructions of a minor-closed set of graphs defined by a context-free grammar

We establish that the finite set of obstructions of a minor-closed set of graphs given by a hyperedge replacement grammar can be effectively constructed. Our proof uses an auxiliary result stating that the system of equations associated with a proper hyperedge replacement grammar has a unique solution.

Context-free graph grammars and concatenation of graphs

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

(UN-)DECIDABILITY OF GEOMETRIC PROPERTIES OF PICTURES GENERATED BY COLLAGE GRAMMARS

Collage grammars are based on hyperedge replacement in a geometric environment and provide context-free syntactic devices for the generation of picture languages. An intriguing question is which geometric properties of the generated pictures can be decided by inspecting the generating collage grammars. The results presented in this paper address the question in two respects. (1) The decidability results known for hyperedge replacement grammars generating graph languages, based on the notion of compatibility, can be carried over to collage grammars. Unfortunately, compatible properties seem rare in the geometric setting. In this paper three concrete ones are presented. (2) In some other cases being only a minor extensions of situations for which compatibility is obtained, we can prove undecidability results.

A logical characterization of the sets of hypergraphs defined by Hyperedge Replacement grammars

A logical characterization of the sets of hypergraphs defined by Hyperedge Replacement grammars