The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

The monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first-order closure of existential monadic second-order logic. We also verify that these graph properties even belong to the complexity class NLOG, which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy.     

Model checking LTL with regular valuations for pushdown systems

Recent works have proposed pushdown systems as a tool for analyzing programs with (recursive) procedures, and the model-checking problem for LTL has received special attention. However, all these works impose a strong restriction on the possible valuations of atomic propositions: whether a configuration of the pushdown system satisfies an atomic proposition or not can only depend on the current control state of the pushdown automaton and on its topmost stack symbol. In this paper we consider LTL with regular valuations: the set of configurations satisfying an atomic proposition can be an arbitrary regular language. The model-checking problem is solved via two different techniques, with an eye on efficiency. The resulting algorithms are polynomial in certain measures of the problem which are usually small, but can be exponential in the size of the problem instance. However, we show that this exponential blowup is inevitable. The extension to regular valuations allows to model problems in different areas; for instance, we show an application to the analysis of systems with checkpoints. We claim that our model-checking algorithms provide a general, unifying and efficient framework for solving them.     

Fixed-Point Logics and Solitaire Games

The model-checking games associated with fixed-point logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study Solitaire-LFP, a fragment of least fixed-point logic, whose evaluation games are nested soltaire games. This means that on each strongly connected component of the game, only one player can make nontrivial moves. Winning sets of nested solitaire games can be computed efficiently. The model-checking problem for Solitaire-LFP is Pspace-complete in general and Ptime-complete for formulae of bounded width. On finite structures (but not on infinite ones), Solitaire-LFP is equivalent to transitive closure logic. We also consider the solitaire fragment of guarded fixed-point logics. Due to the restricted quantification pattern of these logics, the associated games are small and therefore admit more efficient model-checking algorithms.     

A decision tree approach to graph and subgraph isomorphism detection

A new approach to the problem of graph and subgraph isomorphism detection from an input graph to a database of model graphs is proposed in this paper. It is based on a preprocessing step in which the model graphs are used to create a decision tree. At run time, subgraph isomorphisms are detected by means of decision tree traversal. If we neglect the time needed for preprocessing, the computational complexity of the new graph algorithm is only polynomial in the number of input graph vertices. In particular, it is independent of the number of model graphs and the number of edges in any of the graphs. However, the decision tree is of exponential size. Several pruning techniques which aim at reducing the size of the decision tree are presented. A computational complexity analysis of the new method is given and its behavior is studied in a number of practical experiments with randomly generated graphs.     

带复杂数据结构的模型检测工具

模型检测是近二十几年来最成功的自动验证技术之一，而模型检测工具的开发是将模型检测和实际相结合的关键．为了有效地对涉及到复杂数据类型的并发传值系统进行模型检测，总结了以扩展的带赋值符号迁移图和模态图分别作为并发系统和逻辑公式的语义模型来实现模型检测工具的工作，特别是将复杂数据结构引入传值进程定义语言和带赋值符号迁移图．同时结合实际例子说明模型检测工具的有效性．     

基于Petri网的模型检测研究

模型检测是关于系统属性验证的算法和方法.它通常采用状态空间搜索的方法来检测一个给定的计算模型是否满足某个用时序逻辑公式表示的特定属性.系统模型的状态空间的爆炸问题是模型检测所面临的主要问题,其主要原因是系统自身的并发特性和状态变迁的语义交织对基于Petri网的模型检测理论和验证技术进行了较为详细的研究,着重探讨了基于Petri网状态可达图的偏序简化和偏序语义技术、基于自动机的模型检测算法、基于Petri网的状态聚合法以及基于系统对称性的参数化和符号模型检测技术，并给出了研究思路以及未来所要进行的重点研究工作.模型检测技术已在通信协议和硬件系统的验证等领域得到成功应用，并且随着各种状态空间简化技术和模型检测算法的不断优化，其在其他应用领域也展示出广泛的应用前景.     

Model-checking games for fixpoint logics with partial order models

In this paper, we introduce model-checking games that allow local second-order power on sets of independent transitions in the underlying partial order models where the games are played. Since the interleaving semantics of such models is not considered, some problems that may arise when using interleaving representations are avoided and new decidability results for partial order models of concurrency are achieved. The games are shown to be sound and complete, and therefore determined. While in the interleaving case they coincide with the local model-checking games for the μ-calculus, in a partial order setting they verify properties of a number of fixpoint modal logics that can specify, in concurrent systems with partial order semantics, several properties not expressible with the μ-calculus. The games underpin a novel decision procedure for model-checking all temporal properties of a class of infinite and regular event structures, thus improving, in terms of temporal expressive power, previous results in the literature.     

Computing invariants in graphs of small bandwidth

Many graph invariants (chromatic number, rook polynomial, Tutte polynomial, etc.) are known to be computable for general graphs in exponential time only. Algorithms for their computation usually depend on special properties of the invariants and are not extendable to slightly different problems.Some general framework (called composition method) for the construction of algorithms for the solution graph of related problems has been developed during the last two years. In this paper we will demonstrate the power of this method by automating and applying it to a class of well-known problems covering e.g. the chromatic and the Tutte polynomial. The obtained algorithms are comparable to existing hand optimized ones in running time and memory usage.     

Linear-Time Algorithms for Partial \boldmathk -Tree Complements

It is known that a graph decision problem can be solved in linear time over partial k -trees if the problem can be defined in Monadic Second-order (or MS) logic. MS logic allows quantification of vertex and edge subsets, with respect to which logical sentences can encode many different conditions that an input graph must satisfy. It is not always clear, however, which graph problems can be expressed in such a way. In this paper we consider problems stated as logical conditions on subsets of the vertices and nonedges of the input graph. If such a problem can be defined in MS logic (i.e., in terms of the vertices and edges of the input graph), then there is a linear-time algorithm to solve the problem over partial k -trees. This algorithm also provides a solution to some problem over the graph-theoretic complements of partial k -trees. We study several examples of these ``complement-problems.' We introduce a variation of MS logic in which, if a graph-problem can be defined over the class of partial k -tree complements, then there is a linear-time algorithm to solve that problem over partial k -tree complements, and (equivalently) a linear-time algorithm to solve its complement-problem over partial k -trees.     

离散时段演算的符号模型验证

模型验证是对有限状态系统的一种形式化确认方法，近几年，模型验证方法已逐步扩展到实时系统应用中，为解决实时系统的模型验证问题，本文采用离散时段演算人实时系统规格说明的形式语言，用时间自动机作为实时系统的实现模型，对模型验证问题进行了细致的分析，并提出了一种具有实际应用价值的方法－商技术，该方法可以在避免当多个时间自动机并行组合时可能产生的状态空间组合爆炸问题，同时还可以简化整个模型验证问题。     

Bounded Degree Interval Sandwich Problems

The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter. Received October 2, 1997; revised April 1, 1998.     

State-variable planning under structural restrictions: algorithms and complexity

Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying state-transition graph. The exponential size of this graph, though, makes such restrictions costly to test. Hence, we propose an intermediate approach, using a state-variable model for planning and defining restrictions on the separate state-transition graphs for each state variable. We identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions. The algorithm has been implemented and it outperforms Graphplan on a number of test instances. In addition, we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions. This complexity map considers both the optimal and non-optimal plan generation problem.     

Outlier detection for simple default theories

It was noted recently that the framework of default logics can be exploited for detecting outliers. Outliers are observations expressed by sets of literals that feature unexpected properties. These observations are not explicitly provided in input (as it happens with abduction) but, rather, they are hidden in the given knowledge base. Unfortunately, in the two related formalisms for specifying defaults — Reiter's default logic and extended disjunctive logic programs — the most general outlier detection problems turn out to lie at the third level of the polynomial hierarchy. In this note, we analyze the complexity of outlier detection for two very simple classes of default theories, namely NU and DNU, for which the entailment problem is solvable in polynomial time. We show that, for these classes, checking for the existence of an outlier is anyway intractable. This result contributes to further showing the inherent intractability of outlier detection in default reasoning.     

Generalized Model-Checking over Locally Tree-Decomposable Classes

It has been proved in Frick and Grohe that properties of graphs or other relational structures that are definable in first-order logic can be decided in linear time when the input structures are restricted to come from a {\em locally tree-decomposable} class of structures. Important examples of such classes are the class of planar graphs and classes of graphs of bounded degree. In this paper we consider more general computational problems than decision problems, which are induced by formulas with free variables. In this generalized setting we investigate the construction (find a satisfying assignment), listing (all satisfying assignments) and counting (compute the number of satisfying assignments) problems for formulas of first-order logic. We show that each of these problems can be solved in linear time on locally tree-decomposable classes of structures. For instance, we devise an algorithm that, given a planar graph $\cal G$ and a first-order logic formula $\phi(\barx)$, computes a $\bara \in G$ such that $\cal G \models \phi(\bara)$ in time $f(||\phi||) \cdot ||\cal G||$ for some computable function $f$ (the construction case). Accordingly, we obtain linear time algorithms for the counting and listing cases.     

GrouseFlocks: steerable exploration of graph hierarchy space

Several previous systems allow users to interactively explore a large input graph through cuts of a superimposed hierarchy. This hierarchy is often created using clustering algorithms or topological features present in the graph. However, many graphs have domain-specific attributes associated with the nodes and edges which could be used to create many possible hierarchies providing unique views of the input graph. GrouseFlocks is a system for the exploration of this graph hierarchy space. By allowing users to see several different possible hierarchies on the same graph, it allows users to investigate hierarchy space instead of a single, fixed hierarchy. GrouseFlocks provides a simple set of operations so that users can create and modify their graph hierarchies based on selections. These selections can be made manually or based on patterns in the attribute data provided with the graph. It provides feedback to the user within seconds, allowing interactive exploration of this space.     

A fast identification algorithm for systems with delayed inputs

A fast identification algorithm is proposed for systems with delayed inputs. It is based on a non-asymptotic distributional estimation technique initiated in the framework of systems without delay. Such a technique leads to simple realisation schemes, involving integrators, multipliers and piecewise polynomial or exponential time functions. Thus, it allows for a real-time implementation. In order to introduce a generalisation to systems with input delay, three simple examples are presented here. The first illustration is a first-order model with delayed input and noise. Then, a second-order system driven through a transmission line is considered. A third example shows a possible link between simultaneous identification and generalised eigenvalue problems.     

Hierarchical Multilabel Classification with Optimal Path Prediction

We consider multilabel classification problems where the labels are arranged hierarchically in a tree or directed acyclic graph (DAG). In this context, it is of much interest to select a well-connected subset of nodes which best preserve the label dependencies according to the learned models. Top-down or bottom-up procedures for labelling the nodes in the hierarchy have recently been proposed, but they rely largely on pairwise interactions, thus susceptible to get stuck in local optima. In this paper, we remedy this problem by directly finding a small number of label paths that can cover the desired subgraph in a tree/DAG. To estimate the high-dimensional label vector, we adopt the advantages of partial least squares techniques which perform simultaneous projections of the feature and label space, while constructing sound linear models between them. We then show that the optimal label prediction problem with hierarchy constraints can be reasonably transformed into the optimal path prediction problem with the structured sparsity penalties. The introduction of path selection models further allows us to leverage the efficient network flow solvers with polynomial time complexity. The experimental results validate the promising performance of the proposed algorithm in comparison to the state-of-the-art algorithms on both tree- and DAG-structured data sets.     

A study on multi-dimensional products of graphs and hybrid logics

In this work, we address some issues related to products of graphs and products of modal logics. Our main contribution is the presentation of a necessary and sufficient condition for a countable and connected graph to be a product, using a property called intransitivity. We then proceed to describe this property in a logical language. First, we show that intransitivity is not modally definable and also that no necessary and sufficient condition for a graph to be a product can be modally definable. Then, we exhibit a formula in a hybrid language that describes intransitivity. With this, we get a logical characterization of products of graphs of arbitrary dimensions. We then use this characterization to obtain two other interesting results. First, we determine that it is possible to test in polynomial time, using a model-checking algorithm, whether a finite connected graph is a product. This test has cubic complexity in the size of the graph and quadratic complexity in its number of dimensions. Finally, we use this characterization of countable connected products to provide sound and complete axiomatic systems for a large class of products of modal logics. This class contains the logics defined by product frames obtained from Kripke frames that satisfy connectivity, transitivity and symmetry plus any additional property that can be defined by a pure hybrid formula. Most sound and complete axiomatic systems presented in the literature are for products of a pair of modal logics, while we are able, using hybrid logics, to provide sound and complete axiomatizations for many products of arbitrary dimensions.     

A Polynomial Approach to the Constructive Induction of Structural Knowledge

The representation formalism as well as the representation language is of great importance for the success of machine learning. The representation formalism should be expressive, efficient, useful, and applicable. First-order logic needs to be restricted in order to be efficient for inductive and deductive reasoning. In the field of knowledge representation, term subsumption formalisms have been developed which are efficient and expressive. In this article, a learning algorithm, KLUSTER, is described that represents concept definitions in this formalism. KLUSTER enhances the representation language if this is necessary for the discrimination of concepts. Hence, KLUSTER is a constructive induction program. KLUSTER builds the most specific generalization and a most general discrimination in polynomial time. It embeds these concept learning problems into the overall task of learning a hierarchy of concepts.