eNCE Graph Research Articles

Finite graph automata for linear and boundary graph languages

Graph grammars can be regarded as a generalization of context-free grammars from strings to graphs. Over the past 30 years a rich theory of graph grammars and their languages has been developed. However, there are no graph automata. There is no duality between generative and recognizing devices, as it is known for the Chomsky hierarchy of formal languages. Here we introduce graph automata as devices for the recognition of sets of undirected node labelled graphs. A graph automaton consists of a finite state control, a finite set of instructions, and a collection of heads or guards. It reads an input graph in a systematic way and performs a graph search directed by the instructions. As our main results we show that finite graph automata recognize exactly the set of graph languages generated by linear NCE graph grammars and that alternating finite graph automata recognize exactly the languages of boundary graph grammars. Finally, we generalize some automata theoretic properties from string to graph automata, integrate the connectivity of graphs into graph automata, and explain why graph automata cannot be generalized to deal with dynamic edge relabellings and eNCE graph languages.

Node Replacements in Embedding Normal Form

We introduce the embedding normal form for eNCE graph grammars to simplify the analysis of eNCE graph languages. Using this normal form we show that for a large class of eNCE graph grammars useless edges and blocking edges can be removed. This class of eNCE graph grammars can be characterized in terms of weak notions of order independency.

The Complexity of the Kn,n-Problem for Node Replacement Graph Languages

The bounded Kn,n-problem is the question whether or not a graph language of a given graph grammar contains arbitrarily large complete bipartite subgraphs Kn,n. In this paper, we investigate the complexity of this problem for all relevant classes of node replacement graph grammers. Our main result states that the bounded Kn,n-problem is NL-complete for reduced nonblocking eNCE graph grammars and for reduced linear NCE graph grammars. As an application, our results settle the complexity of the problems whether or not the graph language of a given confluent, boundary, or linear graph grammar has bounded tree-width and whether or not it is an HR graph language.

Separation Results for Separated Apex NLC and NCE Graph Languages

Relations among various types of node replacement graph languages are known in the literature but there has remained a gap in the hierarchy of these graph languages. The present paper fills this gap by proving a separation result for k-separated apex graph languages and by extending a known result for k-separated graph languages. This yields, together with other known relations, a complete hierarchy of NLC, NCE, eNCE graph languages and their k-separated, linear, and/or apex subclasses.

Efficient recognition algorithms for boundary and linear eNCE graph languages

It is well known that every boundary (linear) eNCE graph language that is connected and degree-bounded by a constant is in LOGCFL (NLOG). The present paper proves an upper bound of the complexity of boundary (linear) eNCE graph languages whose component and degree bounds are functions in the size of the graph. As a corollary of this analysis, the known LOGCFL and NLOG results stated above hold even if “connected” is replaced by “component-bounded by log n.”

The Bounded Degree Problem for eNCE Graph Grammars

The complexity of the bounded degree problem is analyzed for graph languages generated by eNCE graph grammars. In particular, the bounded degree problem is shown to be undecidable for eNCE graph grammars, DEXPTIME-complete for confluent/boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, and P-complete for non-blocking eNCE graph grammars. In our main theorem we show that the bounded degree problem is NL-complete for reduced non-blocking eNCE graph grammars. Many of the shown results carry over to other types of graph grammars.

Emptiness Problems of eNCE Graph Languages

We consider the complexity of the emptiness problem for various classes of graph languages defined by eNCE (edge label neighborhood controlled embedding) graph grammars. In particular, we show that the emptiness problem is undecidable for general eNCE graph grammars, DEXPTIME-complete for confluent and boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, NL-complete for deterministic confluent, deterministic boundary, and deterministic linear eNCE graph grammars. The exponential time algorithm for deciding emptiness of confluent eNCE graph grammars is based on an exponential time transformation of a confluent eNCE graph grammar into a nonblocking confluent eNCE graph grammar generating the same language.

Separating k-separated eNCE graph languages

An eNCE graph grammar is k-separated ( k⩾1) if the distance between any two nonterminal nodes in any of its sentential forms is at least k. Let SEP k denote the class of graph languages generated by k-separated grammars. Then, SEP 1 (SEP 2) is the class of eNCE (boundary eNCE) graph languages, and so SEP 2⊊SEP 1. Recently, Engelfriet (1991) showed that SEP 3⊊SEP 2 and conjectured that, in fact, SEP k+1 ⊊SEP k for each k⩾ 1. We prove this conjecture affirmatively.

Nonterminal separation in graph grammars

The notion of nonterminal separation in eNCE graph grammars is introduced: such a graph grammar is k-separated (with k⩾1) if the distance between any two nonterminal nodes in any of its sentential forms is at least k (2-separated eNCE grammars are known as boundary eNCE graph grammars). There exists a 2-separated eNCE graph language that is not 3-separated. Apex eNCE graph grammars can be arbitrarily separated: every apex eNCE graph language can be generated by a k-separated apex eNCE grammar, for every k.

Boundary graph grammars with dynamic edge relabeling

Most NLC-like graph grammars generate node-labeled graphs. As one of the exceptions, eNCE graph grammars generate graphs with edge labels as well. We investigate this type of graph grammar and show that the use of edge labels (together with the NCE feature) is responsible for some new properties. Especially boundary eNCE (B-eNCE) grammars are considered. First, although eNCE grammars have the context-sensitive feature of “blocking edges,” we show that B-eNCE grammars do not. Second, we show the existence of a Chomsky normal form and a Greibach normal form for B-eNCE grammars. Third, the boundary eNCE languages are characterized in terms of regular tree and string languages. Fourth, we prove that the class of (boundary) eNCE languages properly contains the closure of the class of (boundary) NLC languages under node relabelings. Analogous results are shown for linear eNCE grammars.