Emptiness Problems of eNCE Graph Languages

Abstract

We consider the complexity of the emptiness problem for various classes of graph languages de ned 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 con uent and boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, NLcomplete for deterministic con uent, deterministic boundary, and deterministic linear eNCE graph grammars. The exponential time algorithm for deciding emptiness of con uent eNCE graph grammars is based on an exponential time transformation of a con uent eNCE graph grammar into a nonblocking con uent eNCE graph grammar generating the same language.

Node Replacements in Embedding Normal Form

“…In an earlier version of this paper [19], we have shown that even every confluent eNCE graph grammar can be transformed into an equivalent neighborhood-preserving grammar, see also [11,Theorem 1.3.33]. Similarly, it has been shown in [18] that every confluent eNCE graph grammar can be transformed into an equivalent nonblocking grammar, see also [11,Theorem 1.3.21].…”

“…It is well known that the confluent, nonblocking, and unrestricted eNCE graph languages form a proper hierarchy [3,8]. Furthermore, every confluent eNCE graph grammar can be transformed into an equivalent nonblocking grammar [18], see also [11,Theorem 1.3.21].…”

“…Many problems are even undecidable for general eNCE graph grammars and still PSPACE-hard for linear eNCE graph grammars, see [12,18,20]. However, if the grammars satisfy certain properties then their languages are sometimes more efficiently analyzable.…”

“…The nonblocking property can also considerably simplify the analysis of the language of a given graph grammar. For example, given an eNCE graph grammar, the problem whether or not its language is empty or finite is in general undecidable [18], but can be solved efficiently in the same NODE REPLACEMENTS way as for context-free string grammars if the grammar is nonblocking. We get a similar effect if we analyze the language of a grammar with respect to a graph property.…”

Regular Description of Context-free Graph Languages

“…It is shown in [SW1] that for every C-edNCE grammar an equivalent nonblocking C-edNCE grammar can be constructed. This fact will not be used, but will be a consequence of our proofs (cf.…”