Graph Rewriting: An Algebraic and Logic Approach

Publisher Summary This chapter presents three mathematical tools that can be used to describe graph-grammars and the sets generated by them. The chapter describes graph properties by logical formulas and presents a comparison of the powers of several logical languages. The chapter discusses first-order logic, second-order logic, and monadic second-order logic, together with a few variants and restrictions of these three languages. It presents algebraic techniques that are useful for defining and studying graph rewriting rules and context-free graph-grammars. The chapter also presents the use of category theory for specifying graph rewriting rules in a precise and concise way, and for properly defining the least or rather the initial solution of a system of graph equations. It presents some links between the context-free graph-grammars and monadic second-order logic. Some applications to the definition of sets of graphs by forbidden configurations and to the theory of NP-completeness are also presented in the chapter.

[1]  Seymour Ginsburg,et al.  Two Families of Languages Related to ALGOL , 1962, JACM.

[2]  Nobuji Saito,et al.  Linear-time computability of combinatorial problems on series-parallel graphs , 1982, JACM.

[3]  M. Rabin.  Automata on Infinite Objects and Church's Problem , 1972 .

[4]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[5]  Hans-Jörg Kreowski,et al.  Is parallelism already concurrency? Part 2: Non-sequential processes in graph grammars , 1986, Graph-Grammars and Their Application to Computer Science.

[6]  K. Wagner.  Über eine Eigenschaft der ebenen Komplexe , 1937 .

[7]  Detlef Seese,et al.  The Structure of Models of Decidable Monadic Theories of Graphs , 1991, Ann. Pure Appl. Log..

[8]  Annegret Habel,et al.  Graph-Theoretic Properties Compatible with Graph Derivations , 1988, WG.

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

[10]  Francesca Rossi,et al.  An Efficient Algorithm for the Solution of Hierarchical Networks of Constraints , 1986, Graph-Grammars and Their Application to Computer Science.

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

[12]  Paris C. Kanellakis,et al.  Elements of Relational Database Theory , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[13]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[14]  J. Engelfriet,et al.  IO and OI , 1975 .

[15]  Grzegorz Rozenberg,et al.  Neighbourhood-Uniform NLC Grammars , 1986, WG.

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

[17]  Hartmut Ehrig,et al.  Tutorial introduction to the algebraic approach of graph grammars , 1986, Graph-Grammars and Their Application to Computer Science.

[18]  Hartmut Schmeck,et al.  Algebraic Characterization of Reducible Flowcharts , 1983, J. Comput. Syst. Sci..

[19]  Bruno Courcelle,et al.  Equivalences and Transformations of Regular Systems-Applications to Recursive Program Schemes and Grammars , 1986, Theor. Comput. Sci..

[20]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[21]  Bruno Courcelle,et al.  On context-free sets of graphs and their monadic second-order theory , 1986, Graph-Grammars and Their Application to Computer Science.

[22]  Martin Wirsing,et al.  Algebraic Specification , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[23]  Clemens Lautemann,et al.  Efficient Algorithms on Context-Free Graph Grammars , 1988, ICALP.

[24]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

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

[26]  Ivan Hal Sudborough,et al.  Bandwidth constrained NP-Complete problems , 1981, STOC '81.

[27]  Hartmut Ehrig,et al.  Introduction to the Algebraic Theory of Graph Grammars (A Survey) , 1978, Graph-Grammars and Their Application to Computer Science and Biology.

[28]  Detlef Seese,et al.  Problems Easy for Tree-Decomposable Graphs (Extended Abstract) , 1988, ICALP.

[29]  Richard Kennaway.  On "On Graph Rewritings" , 1987, Theor. Comput. Sci..

[30]  David E. Muller,et al.  The Theory of Ends, Pushdown Automata, and Second-Order Logic , 1985, Theor. Comput. Sci..

[31]  Hartmut Ehrig,et al.  Graph-Grammars: An Algebraic Approach , 1973, SWAT.

[32]  Joseph A. Goguen,et al.  Initial Algebra Semantics and Continuous Algebras , 1977, J. ACM.

[33]  Oliver Vornberger,et al.  On Some Variants of the Bandwidth Minimization Problem , 1984, SIAM J. Comput..

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

[35]  Thomas Lengauer,et al.  Efficient Algorithms for Finding Minimum Spanning Forests of Hierarchically Defined Graphs , 1987, J. Algorithms.

[36]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

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

[38]  Annegret Habel,et al.  Characteristics of Graph Languages Generated by Edge Replacement , 1987, Theor. Comput. Sci..

[39]  H. Gaifman.  On Local and Non-Local Properties , 1982 .

[40]  Jesse B. Wright,et al.  Algebraic Automata and Context-Free Sets , 1967, Inf. Control..

[41]  Jean Berstel,et al.  Context-Free Languages , 1990, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[42]  Annegret Habel,et al.  Amalgamation of Graph Transformations: A Synchronization Mechanism , 1987, J. Comput. Syst. Sci..

[43]  Hans-Jörg Kreowski,et al.  Is parallelism already concurrency? Part 1: Derivations in graph grammars , 1986, Graph-Grammars and Their Application to Computer Science.

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

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

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

[47]  Joost Engelfriet,et al.  Fixed Point Languages, Equality Languages, and Representation of Recursively Enumerable Languages , 1980, JACM.

[48]  Michel Bauderon,et al.  On Systems of Equations Defining Infinite Graphs , 1988, WG.

[49]  Bruno Courcelle,et al.  A representation of graphs by algebraic expressions and its use for graph rewriting systems , 1986, Graph-Grammars and Their Application to Computer Science.

[50]  Hans L. Bodlaender,et al.  Dynamic Programming on Graphs with Bounded Treewidth , 1988, ICALP.

[51]  Joost Engelfriet,et al.  IO and OI. I , 1977, J. Comput. Syst. Sci..

[52]  Bruno Courcelle,et al.  Recursive Applicative Program Schemes , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[53]  P. Seymour,et al.  Some New Results on the Well-Quasi-Ordering of Graphs , 1984 .

[54]  Václav Koubek,et al.  Least Fixed Point of a Functor , 1979, J. Comput. Syst. Sci..

[55]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs: Definable Sets of Finite Graphs , 1988, WG.

[56]  Thomas Lengauer,et al.  Efficient Analysis of Graph Properties on Context-free Graph Languages (Extended Abstract) , 1988, ICALP.

[57]  H. Bodlaender.  Polynomial algorithms for chromatic index and graph isomorphism on partial k-trees , 1987 .

[58]  B. Courcelle.  Fundamental properties of infinite trees , 1983 .

[59]  Paul Molitor,et al.  On Network Algebras and Recursive Equations , 1986, Graph-Grammars and Their Application to Computer Science.

[60]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[61]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[62]  Jean-Claude Raoult,et al.  On graph rewritings , 1984, Bull. EATCS.

[63]  Manfred Nagl.  Bibliography on graph-rewriting systems (graph grammars) , 1982, Graph-Grammars and Their Application to Computer Science.