See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2806886 On Internal Presentation of Regular Graphs Article · July 1999 Source: CiteSeer CITATIONS READS 0 13 1 author: Didier Caucal Université Gustave Eiffel 47 PUBLICATIONS 1,015 CITATIONS SEE PROFILE All content following this page was uploaded by Didier Caucal on 05 February 2015. The user has requested enhancement of the downloaded file.
On In ternal Presen tation of Regular Graphs 1 2 Didier Caucal and T eo dor Knapik 1 IRISA�CNRS , Campus de Beaulieu 35042 Rennes , F rance e�mail: caucal@irisa.fr 2 IREMIA , Univ ersit� de la R�union, BP 7151, 97715 Saint Denis Messageries Cedex 9, R�union e�mail: knapik@univ�reunion.fr Abstract. The study of in�nite graphs has p oten tial applications in the sp ec- i�cation and v eri�cation of in�nite systems and in the transformation of suc h systems. Pre�x�recognizable graphs and regular graphs are of particular in terest in this area since their monadic second�order theories are decidable. Although the latter form a prop er sub class of the former, no c haracterization of regular graphs within the class of pre�x�recognizable ones has b een kno wn, except for a graph�theoretic one of [2]. W e pro vide here three suc h new c haracterizations. In particular, a decidable, language�theoretic, necessary and su�cien t condition for the regularit y of an y pre�x�recognizable graph is established. Our pro ofs yield a construction of a deterministic h yp eredge�replacemen t grammar for an y pre�x�recognizable graph that is regular. In tro duction Graphs are one of the most general structures in computer science. While the study of �nite graphs has a long history , in�nite graphs ha v e only recen tly at- tracted the atten tion of computer scien tists. As a matter of fact, in�nite graphs are indisp ensable for mo deling large or in�nite systems. In order to put in prac- tice v eri�cation of suc h systems, in v estigation of the logical and algorithmic prop erties of in�nite graphs needs to b e pursued. Another direction where in- �nite graphs ha v e promising applications is the elimination of redundancy in syn tactic ob jects, suc h as e.g. recursiv e program sc hemes (see [12] for a surv ey). T o b e studied b y a computer scien tist, an e�ectiv e presen tation is required for in�nite graphs. Let us men tion v arious presen tations in tro duced in the lit- erature esp ecially in connection with logic. One of the �rst suc h presen tations uses pushdo wn automata [22]. Presen ted graphs are transition graphs of push- do wn automata where the in ternal con�guration form the set of v ertices. A quite di�eren t approac h ma y b e found in [11] where presen tations consist of systems of equations o v er h yp eredge replacemen t (HR) op erations. The graph presen ted b y a system is the least solution of the system. This ma y also b e un- dersto o d as an ! -iteration of a deterministic graph grammar (see e.g. [7]). The class of graphs ha ving this presen tation is called equational in [11] and regu- lar in [7]. A more general class of graphs is obtained [1] when, instead of HR op erations, v ertex replacemen t (VR) op erations [14, 15] are used in systems of equations. This leads to an alternativ e presen tation of the class of graphs �rst de�ned [9] and called recognizable. (T o b e more precise, w e call these graphs pre�x�recognizable.) Since b oth classes of graphs ma y b e presen ted b y systems
2 of equations, w e prefer to sp eak of regular (resp. pre�x�recognizable) graphs 1 instead of equational. F or the class of pre�x�recognizable graphs, t w o kinds of presen tations are in tro duced in [9]. The �rst one is based on the comp osition of t w o language�theoretic op erations that act on paths of the in�nite binary tree: in v erse rational substitutions, and rational restrictions. The second one consists of a kind of extension of recognizable relations whic h is further used for sp ecifying the edge relation b et w een v ertices. In the presen t pap er, w e call the latter pre�x�recognizable relations. An extension of this approac h to (not necessarily simple) h yp ergraphs is in tro duced in [25]. Finally , let us men tion t w o additional graph presen tations. In [6] a graph presen tation consists of a �nite string�rewriting system together with a rational subset of the set of irreducible w ords. The graph is de�ned similarly to the Ca yley graph of a group presen ta- tion. Last but not least, an in teresting idea is dev elop ed in [17] where graphs are presen ted b y systems of equations with higher�order recursion. Most of the presen tations men tioned ab o v e are related to the class of pre�x� recognizable graphs or to its prop er sub class of regular graphs. Th us the tran- sition graphs of pushdo wn automata are precisely the ro oted regular graphs of �nite degree [7], and an alternativ e presen tation of this class is the one of [6] although more general classes of graphs ma y b e obtained within the latter approac h b y appropriate restrictions on string�rewriting systems used in the presen tations [19, 20]. On the other hand, the VR�systems of equations consid- ered in [1] presen t pre�x�recognizable graphs or their extensions to h yp ergraphs. W e conjecture that the approac h of [25] leads to the same class of h yp ergraphs. On the con trary , the presen tations of [17] lead to a class of graphs that seems to b e more general than the pre�x�recognizable graphs, but this h yp othesis needs to b e in v estigated. The classes of graphs men tioned ab o v e ha v e b een disco v ered mainly in logical in v estigations. Th us the decidabilit y of the monadic second�order theory has 2 b een established in [22] for regular ro oted graphs of �nite degree, in [11] for regular graphs, and in [9] for pre�x�recognizable graphs. As sho wn in [18], the class of graphs describ ed there has decidable CTL [10] and S1S [4], and the decidabilit y of the mo dal � �calculus [21] and ev en of the monadic second�order logic is conjectured. V ery di�eren t motiv ations app ear in [16]. The author of that pap er is in- terested in connections b et w een problems on �nite graphs and their in�nite equiv alen ts. Since an y recursiv e presen tation mak es sense with resp ect to this goal, the author stipulates the use of either terminating T uring mac hines that recognize the edge relation or some other equiv alen t means. Among the presen tations review ed ab o v e t w o kinds ma y b e distinguished: those where the graph is de�ned up to isomorphism and those where the v ertices are explicitly named. W e shall call the former external and the latter in ternal. According to this classi�cation, presen tations of [6, 7, 16, 22, 25] are in ternal and those of [1, 11, 17] are external. Concerning [9], one of the t w o presen tations of 1 In order to a v oid a confusion, the terms of HR�equational and VR�equational are used in [2]. 2 In fact, the primary motiv ation comes here from the com binatorial semigroup theory .
Recommend
More recommend
