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 several types of graph grammars have been introduced and a rich theory of graph grammars and their languages has been developed. However, there are no graph automata corresponding to the major classes of graph grammars. There is no duality between generative and recognizing devices, as it is known for the Chomsky hierarchy of formal languages. We introduce graph automata as devices for the recognition of graph languages. A graph automaton consists of a finite state control, a finite set of instructions, and a collection of heads or guards. It traverses and reads an input graph in a systematic way and performs a graph search directed by the instructions. As our main results we show that nite 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. These results can be specialized to hold for connected graph languages and graph languages of bounded degree. We generalize some automata theoretic properties from string to graph automata and re-establish the best possible complexity results of some decision problems in a straightforward manner