Transitive Closure Operator Research Articles

On undecidability of unary non-nested PFP-operators for one successor function theory

We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.

Logics for Epistemic Actions: Completeness, Decidability, Expressivity

We build and study dynamic versions of epistemic logic. We study languages parameterized by an action signature that allows one to express epistemic actions such as (truthful) public announcements, completely private announcements to groups of agents, and more. The language 𝓛(Σ) is modeled on dynamic logic. Its sentence-building operations include modalities for the execution of programs, and for knowledge and common knowledge. Its program-building operations include action execution, composition, repetition, and choice. We consider two fragments of 𝓛(Σ). In 𝓛1(Σ), we drop action repetition; in 𝓛0(Σ), we also drop common knowledge. We present the syntax and semantics of these languages and sound proof systems for the validities in them. We prove the strong completeness of a logical system for 𝓛0(Σ) and the weak completeness of one for 𝓛1(Σ). We show the finite model property and, hence, decidability of 𝓛1(Σ). We translate 𝓛1(Σ) into PDL, obtaining a second proof of decidability. We prove results on expressive power, comparing 𝓛1(Σ) with modal logic together with transitive closure operators. We prove that a logical language with operators for private announcements is more expressive than one for public announcements.

Fuzzy Cluster Analysis: Pseudometrics and Fuzzy Clusters

Introduction. Clustering problems arise in various spheres of human activity. In cases where there are no initial data sufficient for statistical analysis or information obtained from experts is used, fuzzy models are proposed that take into account different types of uncertainty and more argumentatively reflect real situations that model systems of different purposes. Particular attention is drawn to invariance in problems with different types of data measured in different scales according to the classification of S. Stevens. It is known that when solving cluster analysis problems using the transitive closure operation with respect to the equivalence that is obtained, such connections between objects as similarity and dissimilarity are changed. Therefore, it is necessary to take into account the problem of adequacy when developing models and algorithms for solving problems of fuzzy cluster analysis. The purpose of the paper is an analyzing the problem of adequacy of the results of fuzzy cluster analysis on the introduction of metrics and pseudometrics on fuzzy sets in the presence of several qualitative and quantitative characteristics of objects. Propose an approach that ensures the adequacy of pseudometrics, that is, provides invariance with respect to permissible transformations of the values of fuzzy features, and also ensures the division of objects into equivalence classes without distorting the distance between them. Results. Axiomatic definitions of a fuzzy cluster and a fuzzy α level cluster are proposed, which are introduced as fuzzy sets of elements similar to certain elements of a given set, if the condition is met: the dissimilarity ratio must be an invariant pseudometric. This condition is ensured by the use of the linguistic correlation coefficient when calculating fuzzy relations of similarity and dissimilarity. Based on the definition of a fuzzy cluster of α level and threshold conorm, the distance between fuzzy clusters of α level is determined. Conclusions. The proposed approach can be the basis for the development of algorithms for solving cluster analysis problems. This provides a meaningful interpretation of the obtained clusters, and the possibility of clarifying the results in further studies of their structure. Keywords: fuzzy set, conorm, metric, pseudometric, fuzzy similarity relation, fuzzy cluster.

Non-well-founded Proof Theory of Transitive Closure Logic

Supporting inductive reasoning is an essential component is any framework of use in computer science. To do so, the logical framework must extend that of first-order logic. Transitive closure logic is a known extension of first-order logic that is particularly straightforward to automate. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of a single transitive closure operator has the advantage of uniformly capturing all finitary inductive definitions. To further improve the reasoning techniques for transitive closure logic, we here present an infinitary proof system for it, which is an infinite descent –style counterpart to the existing (explicit induction) proof system for the logic. We show that the infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e., cyclic ) proofs provides the basis for an effective system for automating inductive reasoning.

On computational complexity of successor theory with unary transitive closure

The present article concerns the theory of the successor function with the unary transitive closure (TC) operator. This theory is equivalent to the TC-theory of discrete linear order with respect to TC-definability. We prove that any decision algorithm for this theory has at least hyperexponential computational complexity. The latter is much higher than the complexity of the same theories without the TC-operator.

Minimization of Tree Patterns

Many of today’s graph query languages are based on graph pattern matching. We investigate optimization of tree-shaped patterns that have transitive closure operators. Such patterns not only appear in the context of graph databases but also were originally studied for querying tree-structured data, where they can perform child, descendant, node label, and wildcard tests. The minimization problem aims at reducing the number of nodes in patterns and goes back to the early 2000s. We provide an example showing that, in contrast to earlier claims, tree patterns cannot be minimized by deleting nodes only. The example resolves the M = ? NR problem, which asks if a tree pattern is minimal if and only if it is nonredundant. The example can be adapted to prove that minimization is Σ P 2 -complete, which resolves another question that was open since the early research on the problem. The latter result shows that, unless NP = Π P 2 , more general approaches for minimizing tree patterns are also bound to fail in general.

On the Coherence of Probabilistic Relational Formalisms.

There are several formalisms that enhance Bayesian networks by including relations amongst individuals as modeling primitives. For instance, Probabilistic Relational Models (PRMs) use diagrams and relational databases to represent repetitive Bayesian networks, while Relational Bayesian Networks (RBNs) employ first-order probability formulas with the same purpose. We examine the coherence checking problem for those formalisms; that is, the problem of guaranteeing that any grounding of a well-formed set of sentences does produce a valid Bayesian network. This is a novel version of de Finetti’s problem of coherence checking for probabilistic assessments. We show how to reduce the coherence checking problem in relational Bayesian networks to a validity problem in first-order logic augmented with a transitive closure operator and how to combine this logic-based approach with faster, but incomplete algorithms.

Bisimulations on Data Graphs

Bisimulation provides structural conditions to characterize indistinguishability from an external observer between nodes on labeled graphs. It is a fundamental notion used in many areas, such as verification, graph-structured databases, and constraint satisfaction. However, several current applications use graphs where nodes also contain data (the so called "data graphs"), and where observers can test for equality or inequality of data values (e.g., asking the attribute 'name' of a node to be different from that of all its neighbors). The present work constitutes a first investigation of "data aware" bisimulations on data graphs. We study the problem of computing such bisimulations, based on the observational indistinguishability for XPath ---a language that extends modal logics like PDL with tests for data equality--- with and without transitive closure operators. We show that in general the problem is PSpace-complete, but identify several restrictions that yield better complexity bounds (coNP, PTime) by controlling suitable parameters of the problem, namely the amount of non-locality allowed, and the class of models considered (graphs, DAGs, trees). In particular, this analysis yields a hierarchy of tractable fragments.

Adaptive Ensembling of Semi-Supervised Clustering Solutions

Conventional semi-supervised clustering approaches have several shortcomings, such as (1) not fully utilizing all useful must-link and cannot-link constraints, (2) not considering how to deal with high dimensional data with noise, and (3) not fully addressing the need to use an adaptive process to further improve the performance of the algorithm. In this paper, we first propose the transitive closure based constraint propagation approach, which makes use of the transitive closure operator and the affinity propagation to address the first limitation. Then, the random subspace based semi-supervised clustering ensemble framework with a set of proposed confidence factors is designed to address the second limitation and provide more stable, robust, and accurate results. Next, the adaptive semi-supervised clustering ensemble framework is proposed to address the third limitation, which adopts a newly designed adaptive process to search for the optimal subspace set. Finally, we adopt a set of nonparametric tests to compare different semi-supervised clustering ensemble approaches over multiple datasets. The experimental results on 20 real high dimensional cancer datasets with noisy genes and 10 datasets from UCI datasets and KEEL datasets show that (1) The proposed approaches work well on most of the real-world datasets. (2) It outperforms other state-of-the-art approaches on 12 out of 20 cancer datasets, and 8 out of 10 UCI machine learning datasets.

On decidability of the theory with the transitive closure operator

In this paper we consider a theory of integers with the successor function, divisibility predicates, equality and a transitive closure operator. The order relation can be expressed in this theory using the transitive closure operator.

Fuzzy-Clustering Based Cost Modeling of Disassembly Planning for EOL Products

Cost model is a key issue in the disassembly process planning, because the optimized disassembly sequence is determined by ranking several possible disassembly operations. In this study, a fuzzy cost model for disassembly processes was developed based on the fuzzy clustering method. The objective was to solve the problems in the practical applications of currently used quantitative models of disassembly cost model, which is based on the change times of the tools operation or the disassembly time. The sample data were obtained through the disassembly tests of the typical EOL (end-of-life) products. Following this , the transitive closure operations were performed after standardization and normalization. Dynamic clustering was carried out on the basis of above results, and appropriate clustering results were selected to construct the membership function of fuzzy costs. This study also proposed a method of proportional interpolation to expand the directly built membership function to the uncovered discourse domain, resulting in more practical fuzzy cost models. Finally, the disassembly process of a general reducer was adopted as an example to verify the feasibility of the above method.

The middle ground-ancestral logic

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic (FOL). However, there are good reasons to avoid using full second-order logic (SOL) for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory (which is much more relevant for the task of mechanizing mathematics). Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof.

Characterizing weighted MSO for trees by branching transitive closure logics

We introduce the branching transitive closure operator on progressing weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching transitive closure or (ii) one existential second-order quantifier followed by one universal first-order quantifier; in both cases the operator is applied to step-formulas over (a) Boolean first-order logic enriched by modulo counting or (b) Boolean monadic-second order logic.

Synthesizing iterators from abstraction functions

A technique for synthesizing iterators from declarative abstraction functions written in a relational logic specification language is described. The logic includes a transitive closure operator that makes it convenient for expressing reachability queries on linked data structures. Some optimizations, including tuple elimination, iterator flattening, and traversal state reduction, are used to improve performance of the generated iterators. A case study demonstrates that most of the iterators in the widely used JDK Collections classes can be replaced with code synthesized from declarative abstraction functions. These synthesized iterators perform competitively with the hand-written originals. In a user study the synthesized iterators always passed more test cases than the hand-written ones, were almost always as efficient, usually took less programmer effort, and were the qualitative preference of all participants who provided free-form comments.

The Descriptive Complexity of the Deterministic Exponential Time Hierarchy

In Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP).

The Undecidability of Iterated Modal Relativization

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ11–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 Σ01–complete. These results go via reduction to problems concerning domino systems.

On logics with two variables

This paper is a survey and systematic presentation of decidability and complexity issues for modal and non-modal two-variable logics. A classical result due to Mortimer says that the two-variable fragment of first-order logic, denoted FO 2, has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2. Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points, etc. Examples of such logics are computation tree logic CTL, the modal μ-calculus L μ , or popular description logics used in artificial intelligence. Although the additional features are usually not first-order constructs, the resulting logics can still be seen as two-variable logics that are embedded in suitable extensions of FO 2. Typically, the applications call for an analysis of the satisfiability and model checking problems of the logics employed. The decidability and complexity issues for modal and non-modal two-variables logics have been studied quite intensively in the last years. It has turned out that the satisfiability problems for two-variable logics with full first-order quantification are usually much harder (and indeed highly undecidable in many cases) than the satisfiability problems for corresponding modal logics. On the other side, the situation is different for model checking problems. The model checking problem of a modal logic has essentially the same complexity as the model checking problem of the corresponding two variable logic with full quantification.

Undecidability results on two-variable logics

It is a classical result of Mortimer that $L^2$ , first-order logic with two variables, is decidable for satisfiability. We show that going beyond $L^2$ by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Hartig (or equicardinality) quantifier¶– a choice construct known as Hilbert's $\epsilon$ -operator. In fact all these extensions of $L^2$ prove to be undecidable both for satisfiability, and for satisfiability in finite structures. Moreover most of them are hard for $\Sigma^1_1$ , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic.

Parallel transitive closure computation in relational databases

The transitive closure operation is an important extension to relational algebra. Because of its high computation cost, it is of great interest to design efficient parallel algorithms for computing the transitive closure in relational database systems. In this paper, we present a new algorithm to compute transitive closures on SIMD meshes based on relational algebra operations. Double-hash distribution is used to avoid rehashing new tuples for the next join phase. There presently exists no extra step for the redistribution of these tuples. Possible redundant computation between different join phases has been prevented without using global operations. As only regular linear communication occurs on the mesh, and the workload is fully distributed, a speedup of O( n × n) has been achieved, where n × n is the size of mesh. Therefore, this algorithm is an optimal parallel version of the transitive closure algorithms based on relational algebra operations on SIMD meshes.

Hierarchies in transitive closure logic, stratified Datalog and infinitary logic

We establish a general hierarchy theorem for quantifier classes in the infinitary logic L ∞ ω ω on finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog.