Abstract
The apex graph grammars generate precisely the context-free graph languages of bounded degree, independently of whether one considers hyperedge replacement systems or (boundary or confluent) NLC or edNCE graph grammars. The main feature of apex graph grammars is that nodes cannot be “passed” from nonterminal to nonterminal. The proof is based on a normal form result for arbitrary hyperedge replacement systems that forbids “passing chains”. This generalizes Greibach Normal Form.
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References
[BauCou] Bauderon, M., Courcelle, B.: Graph expressions and graph rewritings. Math. Systems Theory20, 83–127 (1987)
[Ber] Berstel, J.: Transductions and context-free languages. Stuttgart: Teubner 1979
[Bra88] Brandenburg, F.J.: On polynomial time graph grammars. Proceedings STACS 88 (Lect. Notes Comput. Sci., vol. 294, pp. 227–236) Berlin, Heidelberg, New York: Springer 1988
[Bra91] Brandenburg, F.J.: The equivalence of boundary and confluent graph grammars on graph languages of bounded degree. In: Book, R.V. (ed.) Rewriting techniques and applications (Lect. Notes Comput. Sci., vol. 488, pp. 312–322) Berlin, Heidelberg, New York: Springer 1991
[Cou87] Courcelle, B.: An axiomatic definition of context-free rewriting and its application to NLC graph grammars. Theor. Comput. Sci.55, 141–181 (1987)
[Cou88] Courcelle, B.: On using context-free graph grammars for analyzing recursive definitions. In: Fuchi, K., Kott, L. (eds.) Programming of future generation computers II, pp. 83–122. Amsterdam: Elsevier 1988
[Cou90] Courcelle, B.: The monadic second-order logic of graphs. I: Recognizable sets of finite graphs. Inf. Comput.85, 12–75 (1990)
[Cou91] Courcelle, B.: On the structure of context-free sets of graphs generated by vertex replacement. Report 91-44, University of Bordeaux, France, December 1991.
[CouEng] Courcelle, B., Engelfriet, J.: A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars. Report 91-41, University of Bordeaux, France, June 1991
[EhrMaiRoz] Ehrenfeucht, A., Main, M.G., Rozenberg, G.: Restrictions on NLC grammars. Theor. Comput. Sci.31, 211–223 (1984)
[Eng89] Engelfriet, J.: Context-free NCE graph grammars. Proceedings FCT'89 (Lect. Notes Comput. Sci., vol. 380, pp. 148–161) Berlin, Heidelberg, New York: Springer 1989
[Eng91] Engelfriet, J.: A characterization of context-free NCE graph languages by monadic second-order logic on trees. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G., Rosenfeld, A. (eds.) Graph-grammars and their application to computer science (Lect. Notes Comput. Sci., vol. 532, pp. 311–327) Berlin, Heidelberg, New York: Springer 1991
[Eng92] Engelfriet, J.: A Greibach Normal Form for context-free graph grammars. In: Kuich, W. (ed.) ICALP'92 Proceedings (Lect. Notes Comput. Sci., vol. 623, pp. 138–149) Berlin, Heidelberg, New York: Springer 1992
[EngHey91a] Engelfriet, J., Heyker, L.M.: The string generating power of context-free hypergraph grammars. J. Comput. System Sci.43, 328–360 (1991)
[EngHey91b] Engelfriet, J., Heyker, L.M.: Hypergraph languages of bounded degree. Report 91-01, University of Leiden, January 1991.
[EngHey92] Engelfriet, J., Heyker, L.M.: Context-free hypergraph grammars have the same term-generating power as attribute grammars. Acta Inf.29, 161–210 (1992)
[EngLei] Engelfriet, J., Leih, G.: Linear graph grammars: power and complexity. Inf. Comput.81, 88–121 (1989)
[EngLeiRoz87] Engelfriet, J., Leih, G., Rozenberg, G.: Apex graph grammars. In: Ehrig, H., Nagl, M., Rozenberg, G., Rosenfeld, A. (eds.) Graph-grammars and their application to computer science (Lect. Notes Comput. Sci., vol. 291, pp. 167–185) Berlin, Heidelberg, New York: Springer 1987
[EngLeiRoz88] Engelfriet, J., Leih, G., Rozenberg, G.: Apex graph grammars and attribute grammars. Acta Inf.25, 537–571 (1988)
[EngLeiRoz91] Engelfriet, J., Leih, G., Rozenberg, G.: Nonterminal separation in graph grammars. Theor. Comput. Sci.82, 95–111 (1991)
[EngLeiWel] Engelfriet, J., Leih, G., Welzl, E.: Boundary graph grammars with dynamic edge relabeling. J. Comput. System Sci.40, 307–345 (1990)
[EngRoz90] Engelfriet, J., Rozenberg, G.: A comparison of boundary graph grammars and context-free hypergraph grammars. Inf. Comput.84, 163–206 (1990)
[EngRoz91] Engelfriet, J., Rozenberg, G.: Graph grammars based on node rewriting: an introduction to NLC graph grammars. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G., Rosenfeld, A. (eds.) Graph-grammars and their application to computer science (Lect. Notes Comput. Sci., vol. 532, pp. 12–23) Berlin, Heidelberg, New York: Springer 1991
[GinSpa66] Ginsburg, S., Spanier, E.H.: Finite-turn pushdown automata. SIAM J. Control4, 429–453 (1966).
[GinSpa68] Ginsburg, S., Spanier, E.H.: Derivation bounded languages. J. Comput. System Sci.2, 228–250 (1968)
[Gre] Greibach, S.A.: A new normal-form theorem for context-free phrase structure grammars. J. ACM12, 42–52 (1965)
[Hab] Habel, A.: Hyperedge replacement: grammars and languages (Lect. Notes Comput. Sci., vol. 643) Berlin, Heidelberg, New York: Springer 1992
[HabKre87a] Habel, A., Kreowski, H.-J.: Some structural aspects of hypergraph languages generated by hyperedge replacement. Proceedings STACS '87 (Lect. Notes Comput. Sci., vol. 247, pp. 207–219) Berlin Heidelberg, New York: Springer 1987
[HabKre87b] Habel, A., Kreowski, H.-J.: May we introduce to you: hyperedge replacement. In: Ehrig, H., Nagl, M., Rozenberg, G., Rosenfeld, A. (eds.) Graph-grammars and their application to computer science (Lect. Notes Comput. Sci., vol. 291, pp. 15–26) Berlin, Heidelberg, New York: Springer 1987
[HabKreVog] Habel, A., Kreowski, H.-J., Vogler, W.: Decidable boundedness problems for sets of graphs generated by hyperedge-replacement. Theor. Comput. Sci.89, 33–62 (1991)
[JanRozWel] Jassens, D., Rozenberg, G., Welzl, E.: The bounded degree problem for NLC grammars is decidable. J. Comput. System Sci.33, 415–422 (1986)
[Lau88a] Lautemann, C.: Decomposition trees: structured graph representation and efficient algorithms. In: Dauchet, M., Nivat, M. (eds.) CAAP '88 Proceedings (Lect. Notes Comput. Sci., vol. 299, pp. 28–39) Berlin, Heidelberg, New York: Springer 1988
[Lau88b] Lautemann, C.: Efficient algorithms on context-free graph languages. In: ICALP '88 Proceedings (Lect. Notes Comput. Sci., vol. 317, pp. 362–378) Berlin, Heidelberg, New York: Springer 1988
[Lau90] Lautemann, C.: Tree automata, tree decomposition and hyperedge replacement. Report 2/90, Johannes Gutenberg-Universität, Mainz, Germany, 1990
[LenWan]: Lengauer, T., Wanke, E.: Efficient decision procedures for graph properties on context-free graph languages. J. ACM40, 368–393 (1993)
[MonRos] Montanari, U., Rossi, F.: An efficient algorithm for the solution of hierarchical networks of constraints. In: Ehrig, H., Nagl, M., Rozenberg, G., Rosenfeld, A. (eds.) Graph-grammars and their application to computer science (Lect. Notes Comput. Sci., vol. 291, pp. 440–457) Berlin, Heidelberg, New York: Springer 1987
[Ros] Rosenkrantz, D.J.: Matrix equations and normal forms for context-free grammars. J. ACM14, 501–507 (1967)
[RozWel] Rozenberg, G., Welzl, E.: Boundary NLC graph grammars — basic definitions, normal forms, and complexity. Inf. Control.69, 136–167 (1986)
[Urb] Urbanek, F.J.: On Greibach normal form construction. Theor. Comput. Sci.40, 315–317 (1985)
[Vog] Vogler, W.: On hyperedge replacement and BNLC graph grammars. In: Graph-theoretic concepts in computer science WG '89 (Lect. Notes Comput. Sci., vol. 411, pp. 78–93) Berlin, Heidelberg, New York: Springer 1989
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Engelfriet, J., Heyker, L. & Leih, G. Context-free graph languages of bounded degree are generated by apex graph grammars. Acta Informatica 31, 341–378 (1994). https://doi.org/10.1007/BF01178511
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DOI: https://doi.org/10.1007/BF01178511