Regular Sets of Descendants by Leftmost Strategy

For a constructor-based rewrite system R, a regular set of ground terms E, and assuming some additional restrictions, we build a finite tree automaton that recognizes the descendants of E, i.e. the terms issued from E by rewriting, according to leftmost strategy.     

Structural Properties of Context-Free Sets of Graphs Generated by Vertex Replacement

We establish that a Vertex Replacement set of graphs, i.e., a set of graphs generated by a C-edNCE or, equivalently, by a separated handle rewriting graph grammar is Hyperedge Replacement, i.e., is generated by a hyperedge replacement graph grammar, iff its graphs do not contain arbitrary large complete bipartite graphs K_{n, n} as subgraphs. Another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices. These properties are decidable by means of an appropriate extension of the theorem by Parikh that characterizes the commutative images of context-free languages. We extend these results to hypergraphs.     

一种Active XML模式重写算法

基于树自动机理论,研究了Active xML(简记为AXML)模式重写问题,提出了一种多项式时间的AXML模式重写判定算法,并对算法进行了实现.实验结果证明了所提算法用于判定AXML模式重写的优越性.     

Tree Spanners for Bipartite Graphs and Probe Interval Graphs

A tree t-spanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree t-spanner problem asks whether a graph admits a tree t-spanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree t-spanner problem is NP-complete even for chordal bipartite graphs for t ≥ 5, and every bipartite ATE-free graph has a tree 3-spanner, which can be found in linear time. The previous best known results were NP-completeness for general bipartite graphs, and that every convex graph has a tree 3-spanner. We next focus on the tree t-spanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7-spanner admissible, and a tree 7-spanner can be constructed in (m log n) time.     

On the reachability of a version of graph-rewriting system

In this paper, we consider a problem on the reachability of a version of graph-rewriting system. It deals with 3-regular graphs with states for the vertices. They differ from ordinary graphs so that a cyclic order of the edges is assigned on each vertex. Graphs are rewritten with a rule set of graph rewriting. For any two such connected graphs with at least four vertices of distinct states, we show that there exists a rule set that rewrites one to the other.     

Termination Proofs for String Rewriting Systems via Inverse Match-Bounds

Annotating a letter by a number, one can record information about its history during a rewrite derivation. In each rewrite step, numbers in the reduct are updated depending on the redex numbering. A string rewriting system is called match-bounded if there is a global upper bound to these numbers. Match-boundedness is known to be a strong sufficient criterion for both termination and preservation of regular languages. We show that the string rewriting systems whose inverse (left and right hand sides exchanged) is match-bounded, also have exceptional properties, but slightly different ones. Inverse match-bounded systems need not terminate; they effectively preserve context-free languages; their sets of normalizable strings and their sets of immortal strings are effectively regular. These languages can be used to decide the normalization, the uniform normalization, the termination and the uniform termination problem for inverse match-bounded systems. We also prove that the termination problem is decidable in linear time, and that a certain strong reachability problem is decidable, thereby solving two open problems of McNaughton’s. Like match-bounds, inverse match-bounds entail linear derivational complexity on the set of terminating strings.     

Active XML文档安全重写判定算法

研究了AXML文档安全重写判定问题,即判定给定AXML文档通过触发其包含的服务调用生成的文档集合是否能够全部重写为符合目标模式的文档实例.基于树自动机理论,定义了用于抽象AXML文档的树自动机--ATA机(AXML treeautomata),ATA机等价于给定AXML文档通过触发其包含的服务调用所能生成的文档集合.基于ATA机,提出一个AXML文档安全重写判定算法,表明了该算法的正确性及有效性.     

Recognition of directed acyclic graphs by spanning tree automata

In this paper, we study tree automata for directed acyclic graphs (DAGs). We define the movement of a tree automaton on a DAG so that a DAG is accepted by a tree automaton if and only if the DAG has a spanning tree accepted by the tree automaton. We call this automaton a spanning tree automaton. The NP-completeness of the membership problem of DAGs for spanning tree automata is shown. However, if inputs are restricted to series–parallel graphs or generalized series–parallel graphs, it is shown that the membership problem for spanning tree automata is solvable in linear time.     

Linearizing Term Rewriting Systems Using Test Sets

Often non left-linear rules in term rewriting systems can be replaced by a finite set of left-linear ones without changing the set of irreducible ground terms. Using appropriate test sets, we can always decide if this is possible and, in case it is, effectively perform such a transformation. We thus can also decide if the set of irreducible ground terms is a regular tree language.     

Non-Linear Rewrite Closure and Weak Normalization

A rewrite closure is an extension of a term rewrite system with new rules, usually deduced by transitivity. Rewrite closures have the nice property that all rewrite derivations can be transformed into derivations of a simple form. This property has been useful for proving decidability results in term rewriting. Unfortunately, when the term rewrite system is not linear, the construction of a rewrite closure is quite challenging. In this paper, we construct a rewrite closure for term rewrite systems that satisfy two properties: the right-hand side term in each rewrite rule contains no repeated variable (right-linear) and contains no variable occurring at depth greater than one (right-shallow). The left-hand side term is unrestricted, and in particular, it may be non-linear. As a consequence of the rewrite closure construction, we are able to prove decidability of the weak normalization problem for right-linear right-shallow term rewrite systems. Proving this result also requires tree automata theory. We use the fact that right-shallow right-linear term rewrite systems are regularity preserving. Moreover, their set of normal forms can be represented with a tree automaton with disequality constraints, and emptiness of this kind of automata, as well as its generalization to reduction automata, is decidable. A preliminary version of this work was presented at LICS 2009 (Creus 2009).     

Decidability for Left-Linear Growing Term Rewriting Systems

A term rewriting system is called growing if each variable occurring on both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemard's result can be extended to left-linear growing rewriting systems that may have right-nonlinear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for right-ground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems.     

Decidable First-Order Theories of One-Step Rewriting in Trace Monoids

We prove that the first-order theory of the one-step rewriting relation associated with a trace rewriting system is decidable but in general not elementary. This extends known results on semi-Thue systems but our proofs use new methods; these new methods yield the decidability of local properties expressed in first-order logic augmented by modulo-counting quantifiers. Using the main decidability result, we define several subclasses of trace rewriting systems for which the confluence problem is decidable.     

A Monadic Second-Order Definition of the Structure of Convex Hypergraphs

We consider finite hypergraphs with hyperedges defined as sets of vertices of unbounded cardinality. Each such hypergraph has a unique modular decomposition, which is a tree, the nodes of which correspond to certain subhypergraphs (induced by certain sets of vertices called strong modules) of the considered hypergraph. One can define this decomposition by monadic second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the hyperedges form intervals. Our main result says that the unique linear order witnessing the convexity of a prime hypergraph (i.e., of one, the modular decomposition of which is trivial) can be defined in MS logic. As a consequence, we obtain that if a set of bipartite graphs that correspond (in the usual way) to convex hypergraphs has a decidable monadic second-order theory (which means that one can decide whether a given MS formula is satisfied in some graph of the set) then it has bounded clique-width. This yields a new case of validity of a conjecture which is still open.     

From Chemical Rules to Term Rewriting

In this paper, rule-based programming is explored in the field of automated generation of chemical reaction mechanisms. We explore a class of graphs and a graph rewriting relation where vertices are preserved and only edges are changed. We show how to represent cyclic labeled graphs by decorated labeled trees or forests, then how to transform trees into terms. A graph rewriting relation is defined, then simulated by a tree rewriting relation, which can be in turn simulated by a rewriting relation on equivalence classes of terms. As a consequence, this kind of graph rewriting can be implemented using term rewriting. This study is motivated by the design of the GasEl system for the generation of kinetics reactions mechanisms. In GasEl, chemical reactions correspond to graph rewrite rules and are implemented by conditional rewriting rules in ELAN. The control of their application is done through the ELAN strategy language.     

Tree spanners on chordal graphs: complexity and algorithms

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The T t-S problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil (SIAM J. Discrete Math. 8 (1995) 359–387) by showing that, for any t4, T t-S is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t+2 (if t is odd). Then we point out that every chordal graph of diameter at most t−1 (respectively, t−2) admits a tree t-spanner whenever t2 is even (respectively, t3 is odd), and such a tree spanner can be constructed in linear time.

The complexity status of T 3-S still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide T 3-S efficiently.     

Learning rules for graph transformations by induction from examples

The input to the described program, in learning mode, consists of examples of starting graph and result graph pairs. The starting graph is transformable into the result graph by adding or deleting certain edges and vertices. The essential common features of the starting graphs are stored together with specifications of the edges and vertices to be deleted or added. This latter information is obtained by mapping each starting graph onto the corresponding result graph. On subsequent input of similar starting graphs without a result graph, the program, in performance mode, recognizes the characterizing set of features in the starting graph and can perform the proper transformation on the starting graph to obtain the corresponding result graph. The program also adds the production to its source code so that after recompilation it is permanently endowed with the new production. If any feature which lacks the property "ordinary" is discovered in the starting graph and only one example has been given, then there is feedback to the user including a request for more examples to ascertain whether the extraordinary property is a necessary part of the situation.     

Widening techniques for regular tree model checking

We consider the framework of regular tree model checking where sets of configurations of a system are represented by regular tree languages and its dynamics is modeled by a term rewriting system (or a regular tree transducer). We focus on the computation of the reachability set R*(L) where R is a regular tree transducer and L is a regular tree language. The construction of this set is not possible in general. Therefore, we present a general acceleration technique, called regular tree widening which allows to speed up the convergence of iterative fixpoint computations in regular tree model checking. This technique can be applied uniformly to various kinds of transformations. We show the application of our framework to different analysis contexts: verification of parameterized tree networks and data-flow analysis of multithreaded programs. Parametrized networks are modeled by relabeling tree transducers, and multithreaded programs are modeled by term rewriting rules encoding transformations on control structures. We prove that our widening technique can emulate many existing algorithms for special classes of transformations and we show that it can deal with transformations beyond the scope of these algorithms.     

Finite complete rewriting systems and the complexity of the word problem

Summary It is well known that the word problem for a finite complete rewriting system is decidable. Here it is shown that in general this result cannot be improved. This is done by proving that each sufficiently rich complexity class can be realized by the word problem for a finite complete rewriting system. Further, there is a gap between the complexity of the word problem for a finite complete rewriting system and the complexity of the least upper bound for the lengths of the chains generated by this rewriting system, and this gap can get arbitrarily large. Thus, the lengths of these chains do not give any information about the complexity of the word problem. Finally, it is shown that the property of allowing a finite complete rewriting system is not an invariant of finite monoid presentations.Some of the results presented here are from the doctoral dissertation of the first author, which was supervised by Prof. H. Brakhage at the University of Kaiserslautern     

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

For directed and undirected graphs, we study how to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness. In particular, we provide a dynamic programming algorithm for graphs of bounded treewidth and a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding non-existence results when replacing vertex cover number by treewidth or feedback vertex set number.     

Emptiness and Finiteness for Tree Automata with Global Reflexive Disequality Constraints

In recent years, several extensions of tree automata have been considered. Most of them are related with the capability of testing equality or disequality of certain subterms of the term evaluated by the automaton. In particular, tree automata with global constraints are able to test equality and disequality of subterms depending on the state to which they are evaluated. The emptiness problem is known decidable for this kind of automata, but with a non-elementary time complexity, and the finiteness problem remains unknown. In this paper, we consider the particular case of tree automata with global constraints when the constraint is a conjunction of disequalities between states, and the disequality predicate is forced to be reflexive. This restriction is significant in the context of XML definitions with monadic key constraints. We prove that emptiness and finiteness are decidable in triple exponential time for this kind of automata.