Incremental Construction of Minimal Tree Automata

We describe an algorithm that allows the incremental addition or removal of unranked ordered trees to a minimal frontier-to-root deterministic finite-state tree automaton (DTA). The algorithm takes a tree t and a minimal DTA A as input; it outputs a minimal DTA A′ which accepts the language L(A) accepted by A incremented (or decremented) with the tree t. The algorithm can be used to efficiently maintain dictionaries which store large collections of trees or tree fragments.     

Refinements to nearest-neighbor searching ink-dimensional trees

This note presents a simplification and generalization of an algorithm for searchingk-dimensional trees for nearest neighbors reported by Friedmanet al [3]. If the distance between records is measured usingL ₂ , the Euclidean norm, the data structure used by the algorithm to determine the bounds of the search space can be simplified to a single number. Moreover, because distance measurements inL ₂ are rotationally invariant, the algorithm can be generalized to allow a partition plane to have an arbitrary orientation, rather than insisting that it be perpendicular to a coordinate axis, as in the original algorithm. When ak-dimensional tree is built, this plane can be found from the principal eigenvector of the covariance matrix of the records to be partitioned. These techniques and others yield variants ofk-dimensional trees customized for specific applications.It is wrong to assume thatk-dimensional trees guarantee that a nearest-neighbor query completes in logarithmic expected time. For smallk, logarithmic behavior is observed on all but tiny trees. However, for largerk, logarithmic behavior is achievable only with extremely large numbers of records. Fork = 16, a search of ak-dimensional tree of 76,000 records examines almost every record.     

Construction of Aho Corasick automaton in linear time for integer alphabets

We present a new simple algorithm that constructs an Aho Corasick automaton for a set of patterns, P, of total length n, in O(n) time and space for integer alphabets. Processing a text of size m over an alphabet Σ with the automaton costs O(mlog|Σ|+k), where there are k occurrences of patterns in the text.A new, efficient implementation of nodes in the Aho Corasick automaton is introduced, which works for suffix trees as well.     

Embedding ofk-ary Complete Trees into Hypercubes with Uniform Load

The main result of this paper is an algorithm for embeddingk-ary complete trees into hypercubes with uniform load and asymptotically optimal dilation. The algorithm is fully scalable—the dimension of the hypercube can be chosen independent of the arity and height of the tree. The embedding enables to emulate optimally both Divide&Conquer computations onk-ary complete trees where only one level of nodes is active at a time and general computations based onk-ary complete trees where all tree nodes are active simultaneously. As a special case, we get an algorithm for embedding thek-ary complete tree of given height into its optimal hypercube with load 1 and asymptotically optimal dilation.     

Maintaining Regular Properties Dynamically in k-Terminal Graphs

The t -topology tree data structure is developed for maintaining t -ary trees dynamically. Each of a certain set of tree operations is shown to take O(log n) time, where n is the number of vertices in the trees. The t -topology trees are used to maintain any given regular property on members of the family of k -terminal graphs under a variety of dynamic graph operations. Received February 1994; revised February 1995.     

Formalizing and Analyzing the Needham-Schroeder Symmetric-Key Protocol by Rewriting

This paper reports on work in progress on using rewriting techniques for the specification and the verification of communication protocols. As in Genet and Klay's approach to formalizing protocols, a rewrite system describes the steps of the protocol and an intruder's ability of decomposing and decrypting messages, and a tree automaton encodes the initial set of communication requests and an intruder's initial knowledge. In a previous work we have defined a rewriting strategy that, given a term t that represents a property of the protocol to be proved, suitably expands and reduces t using the rules in and the transitions in to derive whether or not t is recognized by an intruder. In this paper we present a formalization of the Needham-Schroeder symmetric-key protocol and use the rewriting strategy for deriving two well-known authentication attacks.     

Creating Decision Trees from Rules using RBDT‐1

Most of the methods that generate decision trees for a specific problem use the examples of data instances in the decision tree–generation process. This article proposes a method called RBDT‐1—rule‐based decision tree—for learning a decision tree from a set of decision rules that cover the data instances rather than from the data instances themselves. The goal is to create on demand a short and accurate decision tree from a stable or dynamically changing set of rules. The rules could be generated by an expert, by an inductive rule learning program that induces decision rules from the examples of decision instances such as AQ‐type rule induction programs, or extracted from a tree generated by another method, such as the ID3 or C4.5. In terms of tree complexity (number of nodes and leaves in the decision tree), RBDT‐1 compares favorably with AQDT‐1 and AQDT‐2, which are methods that create decision trees from rules. RBDT‐1 also compares favorably with ID3 while it is as effective as C4.5 where both (ID3 and C4.5) are well‐known methods that generate decision trees from data examples. Experiments show that the classification accuracies of the decision trees produced by all methods under comparison are indistinguishable.     

格值交替树自动机

交替（树）自动机因其本身关于取补运算的简洁性及其与非确定型（树）自动机的等价性,成为自动机与模型检测领域研究的一个新方向.在格值交替自动机与经典交替树自动机概念的基础上,引入格值交替树自动机的概念,并研究了格值交替树自动机的代数封闭性和表达能力.首先,证明了对格值交替树自动机的转移函数取对偶运算,终止权重取补之后所得自动机与原自动机接受语言互补这一结论.其次,证明了格值交替树自动机关于交、并运算的封闭性.最后,讨论了格值交替树自动机和格值树自动机、格值非确定型自动机的表达能力;证明了格值交替树自动机与格值树自动机的等价性,并给出了二者相互转化的算法及其复杂度分析;同时,提供了用格值非确定型自动机来模拟格值交替树自动机的方法.     

模糊树自动机语言的封闭性

给出了[Σ-]代数、[Σ-]树、模糊[Σ-]树自动机、模糊[Σ-]树自动机行为的定义。引入了模糊树自动机语言的并、交、连接和Kleene闭包运算,证明了在这些运算下模糊树自动机语言的封闭性。     

使用事件自动机规约的C语言有界模型检测

提出使用事件自动机对 C 程序的安全属性进行规约,并给出了基于有界模型检测的形式化验证方法。事件自动机可以规约程序中基于事件的安全属性,且可以描述无限状态的安全属性。事件自动机将属性规约与C程序本身隔离,不会改变程序的结构。在事件自动机的基础上,提出了自动机可达树的概念。结合自动机可达树和有界模型检测技术,给出将事件自动机和C程序转化为可满足性模理论SMT(satisfiability modulo theory)模型的算法。最后,使用SMT求解器对生成的SMT模型求解,并根据求解结果给出反例路径分析算法。实例分析和实验结果表明,该方法可以有效验证软件系统中针对事件的属性规约。     

Descendants of a recognizable tree language for sets of linear monadic term rewrite rules

For a tree language L and a set S of term rewrite rules over Σ, the descendant of L for S is the set S^∗(L) of trees reachable from a tree in L by rewriting in S. For a recognizable tree language L, we study the set D(L) of descendants of L for all sets of linear monadic term rewrite rules over Σ. We show that D(L) is finite. For each tree automaton A over Σ, we can effectively construct a set {R₁,…,Rk} of linear monadic term rewrite systems over Σ such that and for any 1?i<j?k, .     

Reformulation of Global Constraints Based on Constraints Checkers

This article deals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in n, where n is the number of variables of the corresponding global constraint. By reducing the automaton to a conjunction of signature and transition constraints we show how to systematically obtain an automaton reformulation. Under some restrictions on the signature and transition constraints, this reformulation maintains arc-consistency. An implementation based on some constraints as well as on the metaprogramming facilities of SICStus Prolog is available. For a restricted class of automata we provide an automaton reformulation for the relaxed case, where the violation cost is the minimum number of variables to unassign in order to get back to a solution.     

A <Emphasis Type="Italic">k</Emphasis>-norm pruning algorithm for decision tree classifiers based on error rate estimation

Decision trees are well-known and established models for classification and regression. In this paper, we focus on the estimation and the minimization of the misclassification rate of decision tree classifiers. We apply Lidstone’s Law of Succession for the estimation of the class probabilities and error rates. In our work, we take into account not only the expected values of the error rate, which has been the norm in existing research, but also the corresponding reliability (measured by standard deviations) of the error rate. Based on this estimation, we propose an efficient pruning algorithm, called k-norm pruning, that has a clear theoretical interpretation, is easily implemented, and does not require a validation set. Our experiments show that our proposed pruning algorithm produces accurate trees quickly, and compares very favorably with two other well-known pruning algorithms, CCP of CART and EBP of C4.5. Editor: Hendrik Blockeel.     

Abstract Regular Tree Model Checking

Regular (tree) model checking (RMC) is a promising generic method for formal verification of infinite-state systems. It encodes configurations of systems as words or trees over a suitable alphabet, possibly infinite sets of configurations as finite word or tree automata, and operations of the systems being examined as finite word or tree transducers. The reachability set is then computed by a repeated application of the transducers on the automata representing the currently known set of reachable configurations. In order to facilitate termination of RMC, various acceleration schemas have been proposed. One of them is a combination of RMC with the abstract-check-refine paradigm yielding the so-called abstract regular model checking (ARMC). ARMC has originally been proposed for word automata and transducers only and thus for dealing with systems with linear (or easily linearisable) structure. In this paper, we propose a generalisation of ARMC to the case of dealing with trees which arise naturally in a lot of modelling and verification contexts. In particular, we first propose abstractions of tree automata based on collapsing their states having an equal language of trees up to some bounded height. Then, we propose an abstraction based on collapsing states having a non-empty intersection (and thus “satisfying”) the same bottom-up tree “predicate” languages. Finally, we show on several examples that the methods we propose give us very encouraging verification results.     

Parameterized complexity of basic decision problems for tree automata

There are many decision problems in automata theory (including membership, emptiness, inclusion and universality problems) that are NP-hard for some classes of tree automata (TA). The study of their parameterized complexity allows us to find new bounds of their nonpolynomial time algorithmic behaviours. We present results of such a study for classical TA, rigid tree automata, TA with global equality and disequality and t-DAG automata. As parameters we consider the number of states, the cardinality of the signature, the size of the term or the t-dag and the size of the automaton.     

Inductive inference of logic programs based on algebraic semantics

In this paper we present a new inductive inference algorithm for a class of logic programs, calledlinear monadic logic programs. It has several unique features not found in Shapiro’s Model Inference System. It has been proved that a set of trees isrational if and only if it is computed by a linear monadic logic program, and that the rational set of trees is recognized by a tree automaton. Based on these facts, we can reduce the problem of inductive inference of linear monadic logic programs to the problem of inductive inference of tree automata. Further several efficient inference algorithms for finite automata have been developed. We extend them to an inference algorithm for tree automata and use it to get an efficient inductive inference algorithm for linear monadic logic programs. The correctness, time complexity and several comparisons of our algorithm with Model Inference System are shown.     

A multidimensional approach to syntactic pattern recognition

This paper describes a syntactic method for representing the primitive parts of a pattern as nodes of a type of directed graph. A linear representation of the digraph can then be presented to a regular unordered tree automaton for classification. Regular unordered tree automata can be simulated by deterministic pushdown automata, so this procedure can be implemented easily. Regular u-tree automata and the corresponding generative systems, regular u-tree grammars are formally defined. Several results are shown which are applicable to all syntactic pattern recognition schemes involving the use of primitives.     

Dividedk-d trees

A variant ofk-d trees, thedivided k-d tree, is described that has some important advantages over ordinaryk-d trees. The dividedk-d tree is fully dynamic and allows for the insertion and deletion of points inO(logn) worst-case time. Moreover, dividedk-d trees allow for split and concatenate operations. Different types of queries can be performed with equal or almost equal efficiency as on ordinaryk-d trees. Both two- and multidimensional dividedk-d trees are studied.     

Mutation systems

We propose mutation systems as a model of the evolution of a string subject to the effects of mutations and a fitness function. One fundamental question about such a system is whether knowing the rules for mutations and fitness, we can predict whether it is possible for one string to evolve into another. To explore this issue, we define a specific kind of mutation system with point mutations and a fitness function based on conserved strongly k-testable string patterns. We show that for any k greater than 1, such systems can simulate computation by both finite state machines (FSMs) and asynchronous cellular automata. The cellular automaton simulation shows that in this framework, universal computation is possible and the question of whether one string can evolve into another is undecidable. We also analyse the efficiency of the FSM simulation assuming random point mutations.