A Kleene Theorem for a Class of Planar Acyclic Graphs

Graph automata

“…Subsets of smag(Σ ) are called pattern languages. Our patterns are analogous with the unsorted abstract dags of [18,19,5]. For another formalization see also [15].…”

“…An element a of a magmoid M is said to be connected if it cannot be factorized as a = a 1 a 2 for a 1 , a 2 ∈ M − {e 0 }. It is easily seen that val −1 Σ (CON(Σ )) is just the set of all connected patterns of smag(Σ ∪ D) which is shown to be recognizable in [5]. However according to Proposition 11 and Proposition 3 this graph language cannot be recognized by a pattern automaton with graph structure.…”

“…Pattern automata have originated in [10], see also [21], and their basic properties are examined in [18,19] and [5] under the name of pdag automata. A pattern automaton is a structure…”

“…[18,19]). The automata considered by Bossut, Dauchet and Warin, work exactly in the same way except that they use pdags with an unbounded number of roots and that they control the sequences of initial and final states by rational languages of state words (cf [5]). They have shown that the class of automaton definable languages of pdags is the smallest class containing finite sets of pdags and closed under union, (iterated) serial composition and (iterated) nondeterministic parallel composition.…”

Weighted DAG Automata for Semantic Graphs

“…Without going into further detail, we mention here some additional publications by diverse authors on automata recognizing DAGs or graphs: Bossut, Dauchet, and Warin (1988); Kaminski and Pinter (1992); Potthoff, Seibert, and Thomas (1994); Bossut, Dauchet, and Warin (1995). Furthermore, there exists considerable work within the XML community on evaluating tree automata and logical queries on compressed representations of trees, which are DAGs (see, e.g., Frick, Grohe, and Koch 2003;Lohrey and Maneth 2006).…”