Monadic Second-Order Definable Graph Transductions: A Survey

Abstract Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multi-valued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or, more simply, definable transductions ) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of hyperedge replacement (HR) and the class of vertex replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees, independently of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions such that the image of every finite word is finite; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labelled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikh's theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs

[1]  Annegret Habel,et al.  May we introduce to you: hyperedge replacement , 1986, Graph-Grammars and Their Application to Computer Science.

[2]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[3]  Joost Engelfriet,et al.  The String Generating Power of Context-Free Hypergraph Grammars , 1991, J. Comput. Syst. Sci..

[4]  John Doner,et al.  Tree Acceptors and Some of Their Applications , 1970, J. Comput. Syst. Sci..

[5]  C. C. Elgot.  Decision problems of finite automata design and related arithmetics , 1961 .

[6]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[7]  Jean-Claude Raoult,et al.  A survey of tree transductions , 1992, Tree Automata and Languages.

[8]  Joost Engelfriet,et al.  A Characterization of Context-Free NCE Graph Languages by Monadic Second-Order Logic on Trees , 1990, Graph-Grammars and Their Application to Computer Science.

[9]  Joost Engelfriet,et al.  Graph Grammars Based on Node Rewriting: An Introduction to NLC Graph Grammars , 1990, Graph-Grammars and Their Application to Computer Science.

[10]  Grzegorz Rozenberg,et al.  Boundary NLC Graph Grammars-Basic Definitions, Normal Forms, and Complexity , 1986, Inf. Control..

[11]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs V: On Closing the Gap Between Definability and Recognizability , 1991, Theor. Comput. Sci..

[12]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs VII: Graphs as Relational Structures , 1992, Theor. Comput. Sci..

[13]  Bruno Courcelle,et al.  The Monadic Second order Logic of Graphs VI: on Several Representations of Graphs By Relational Structures , 1994, Discret. Appl. Math..

[14]  Sophie Tison,et al.  Decidability of the Confluence of Finite Ground Term Rewrite Systems and of Other Related Term Rewrite Systems , 1990, Inf. Comput..

[15]  Franz-Josef Brandenburg.  The Equivalence of Boundary and Confluent Graph Grammars on Graph Languages of Bounded Degree , 1991, RTA.

[16]  Bruno Courcelle.  Structural Properties of Context-Free Sets of Graphs Generated by Vertex Replacement , 1995, Inf. Comput..

[17]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[18]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[19]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

[20]  Bruno Courcelle,et al.  Graph grammars, monadic second-order logic and the theory of graph minors , 1991, Graph Structure Theory.

[21]  Klaus-Jörn Lange.  Context-Free Controlled ETOL Systems , 1983, ICALP.

[22]  Bruno Courcelle,et al.  An Axiomatic Definition of Context-Free Rewriting and its Application to NLC Graph Grammars , 1987, Theor. Comput. Sci..

[23]  Bruno Courcelle,et al.  A Logical Characterization of the Sets of Hypergraphs Defined by Hyperedge Replacement Grammars , 1995, Math. Syst. Theory.

[24]  Joost Engelfriet,et al.  Hypergraph Languages of Bounded Degree , 1994, J. Comput. Syst. Sci..

[25]  Annegret Habel,et al.  Hyperedge Replacement: Grammars and Languages , 1992, Lecture Notes in Computer Science.

[26]  Bruno Courcelle,et al.  The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues , 1992, RAIRO Theor. Informatics Appl..

[27]  Joost Engelfriet,et al.  Context-Free NCE Graph Grammars , 1989, FCT.

[28]  Bruno Courcelle,et al.  Monadic Second-Order Evaluations on Tree-Decomposable Graphs , 1991, Theor. Comput. Sci..

[29]  Joost Engelfriet,et al.  Tree transducers, L systems and two-way machines (Extended Abstract) , 1978, J. Comput. Syst. Sci..

[30]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[31]  Paul Seymour,et al.  Graph Structure Theory, Proceedings of a AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors held June 22 to July 5, 1991, at the University of Washington, Seattle, USA , 1993, Graph Structure Theory.

[32]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[33]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[34]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[35]  Grzegorz Rozenberg,et al.  A Survey of NLC Grammars , 1983, CAAP.

[36]  Joost Engelfriet,et al.  A Comparison of Boundary Graph Grammars and Context-Free Hypergraph Grammars , 1990, Inf. Comput..