Reflexive Transitive Closure Research Articles

Теоретичні основи програмної релятивізації у технологічних системах програмування

The research claims the need for paradigmatic changes in the technological foundations of programming, which involve a transition from an individual-subjective programming paradigm to an intersubjective one. The proposed intersubjective paradigm is based on the understanding of programming as an activity conditioned by program likeness, and subject-oriented complementarity of its active and passive forms. To implement such a transition, the objectification of active-passive complementarity is considered, which is the main prerequisite for the real modernization of the understanding of programming as a reflexive-transitive closure of the programming analogy generated by the subject. The Determinant for the objectification of the active role of the subject in the context of technological programming environments is a concept — essence—simile, presented in the form of one or another specification. Passive constituent conditioning is objectified by similitude as a whole consequence of assimilation, namely by a program. To achieve this goal, a tool for logical-mathematical specifications of semantic-syntactic aspects of programming is proposed. This tool was tested on examples that demonstrated the general features of the application of the technological programming environment to the generation of subject-oriented technological programming systems and their use. In particular, the use of this tool allows to create more efficient technological programming systems that have higher quality and flexibility in performing tasks. The obtained results confirm the need for paradigmatic changes in the technological foundations of programming and are one of the possible options for solving the problem of paradigmatic changes and further technologization of programming, and also demonstrate the general features of the application of a technological programming environment for the generation of subject-oriented technological programming systems and their use.

Free Kleene algebras with domain

First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are ‘regular’ sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.

Relational characterisations of paths

Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in many areas of mathematics and computing, researchers usually fall back to point-wise reasoning when it comes to arguments about paths in a graph. We present a purely algebraic way to specify different kinds of paths in Kleene relation algebras, which are relation algebras equipped with an operation for reflexive transitive closure. We study the relationship between paths with a designated root vertex and paths without such a vertex. Since we stay in first-order logic this development helps with mechanising proofs. To demonstrate the applicability of the algebraic framework we verify the correctness of three basic graph algorithms. All results of this paper are formally verified using the interactive proof assistant Isabelle/HOL.

An expressive and tractable rule-based description language

ABSTRACT is a rule language introduced by Nguyen, Nguyen, and Szałas for the Semantic Web. It has a PTime data complexity and is more expressive than Horn- . These rule languages, however, do not allow PDL-like role constructors (i.e., union, sequential composition, the reflexive-transitive closure and the test operator). Using the transitive closure of a role is not the same as using a transitive role that is a superrole of . Furthermore, the test operator together with the other PDL-like role constructors would allow users to express sophisticated concepts. In this paper, we extend with PDL-like role constructors to obtain the rule language . We modify the algorithm given by Nguyen, Nguyen, and Szałas (2015) to obtain another one that checks satisfiability of a knowledge base in instead of . The resulting algorithm also has a PTime data complexity and is therefore tractable.

Intersection of the reflexive transitive closures of two rewrite relations induced by term rewriting systems

We show that it is undecidable whether the intersection of the reflexive transitive closures of two rewrite relations induced by term rewriting systems is equal to the reflexive transitive closure of a rewrite relation induced by a term rewriting system.

Using relation-algebraic means and tool support for investigating and computing bipartitions

Using Dedekind categories as an algebraic structure for (binary) set-theoretic relations without complements, we present purely algebraic definitions of “to be bipartite” and “to possess no odd cycles” and prove that both notions coincide in case that the reflexive–transitive closure of the relation in question is symmetric. This generalises D. Kőnig's well-known theorem from undirected graphs to abstract relations and, hence, to models such as L-relations that are different from set-theoretic relations. One direction of this generalisation holds for general relations. The other direction requires the symmetry of the reflexive–transitive closure as premise and is shown by specifying a bipartition for the given relation in form of a pair of disjoint relational vectors. For set-theoretic relations this immediately leads to a relational program for computing a bipartition. We also investigate the structure of a certain set of bipartitions that is generated by a given bipartition, explore the set of all bipartitions of a bipartite relation with respect to minimality and maximality of elements in view of the component-wise inclusion, show a relationship between bipartitions and bichromatic partitions and also discuss how the algebraic proofs can be mechanised using theorem proving tools.

Algorithms for Kleene algebra with converse

The equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik and Stefanescu in 1995. In particular, they obtained its decidability by using a particular automata construction.We show that deciding this equational theory is PSpace-complete, by providing a PSpace algorithm (the problem is easily shown to be PSpace-hard). We obtain other algorithms that are time-efficient in practice, despite not being PSpace.Our results use an alternative automata construction, inspired by the one from Bloom, Ésik and Stefanescu. We relate those two constructions by exhibiting a bisimulation between the resulting deterministic automata, and by showing how our construction results in more sharing between states, thus producing smaller automata.

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.

A Model of Layered Architectures

Architectural styles and patterns play an important role in software engineering. One of the most known ones is the layered architecture style. However, this style is usually only stated informally, which may cause problems such as ambiguity, wrong conclusions, and difficulty when checking the conformance of a system to the style. We address these problems by providing a formal, denotational semantics of the layered architecture style. Mainly, we present a sufficiently abstract and rigorous description of layered architectures. Loosely speaking, a layered architecture consists of a hierarchy of layers, in which services communicate via ports. A layer is modeled as a relation between used and provided services, and layer composition is defined by means of relational composition. Furthermore, we provide a formal definition for the notions of syntactic and semantic dependency between the layers. We show that these dependencies are not comparable in general. Moreover, we identify sufficient conditions under which, in an intuitive sense which we make precise in our treatment, the semantic dependency implies, is implied by, or even coincides with the reflexive-transitive closure of the syntactic dependency. Our results provide a technology-independent characterization of the layered architecture style, which may be used by software architects to ensure that a system is indeed built according to that style.

Combining relation algebra and data refinement to develop rectangle-based functional programs for reflexive–transitive closures

We show how to systematically derive simple purely functional algorithms for computing the reflexive–transitive closure of directed graphs. Directed graphs can be represented as binary relations and we develop our algorithms based on a relation-algebraic description of reflexive–transitive closures. This description employs the notion of rectangles and instantiating the resulting algorithm with different kinds of rectangles leads to different algorithms for computing reflexive–transitive closures. Using data refinement, we then develop simple Haskell programs for two specific choices of rectangles and show that one of them has cubic running time like Warshall's standard algorithm. Finally, we apply our approach to other standard operations of relation algebra and present graph theoretic applications of our developments.

On the Join Dependency Relation in Multinomial Lattices

Aiming to understand equivalence relations that model concurrent computation, we investigate congruences of multinomial lattices \(\mathcal{L}(v)\) introduced by Bennett and Birkhoff (Algebra Univers. 32(1):115–144, 1994). Our investigation gives rise to an explicit description of the join dependency relation between two join irreducible elements and of its reflexive transitive closure. The explicit description emphasizes several properties and makes it possible to separate the equational theories of multinomial lattices by their dimensions. In their covering of non modular varieties Jipsen and Rose (Varieties of lattices, Lecture Notes in Mathematics, vol. 1533, Springer, Berlin, 1992) define a sequence of equations \(SD_{n}(\land)\), for n ≥ 0. Our main result sounds as follows: if\(v = (v_{1},\ldots,v_{n}) \in \mathbb{N}^{n}\)andvi > 0 for i = 1,...,n, then the multinomial lattice\(\mathcal{L}(v)\)satisfies\(SD_{n-1}(\land)\)and fails\(SD_{n-2}(\land)\)

On Equations for Union-Free Regular Languages

In this paper we consider the variety UF generated by all algebras of binary relations equipped with the operations of composition, reflexive-transitive closure, and the empty set and the identity relation as constants. This variety coincides with the variety generated by the union-free reducts of Kleene algebras of languages and its free objects are formed by union-free regular languages, that is, regular languages represented by regular expressions having no occurrence of +. We show that the variety UF is not finitely based. The situation does not change if we consider the variety UF∨ generated by the above algebras of binary relations equipped with the conversion operation.

The variety of Kleene algebras with conversion is not finitely based

Given an arbitrary set A, one obtains the full Kleene algebra of binary relations over A by considering the operations of union, composition, reflexive-transitive closure, conversion, and the empty set and the identity relation as constants. Such algebras generate the variety of Kleene algebras (with conversion). As a result of a general analysis of identities satisfied by varieties having an involution operation, we prove that the variety of Kleene algebras with conversion has no finite equational axiomatization. In our argument we make use of the fact that the variety of Kleene algebras without conversion is not finitely based and that, relatively to this variety, the variety of Kleene algebras with conversion is finitely axiomatized.

The Price of Universality

We investigate the effect on the complexity of adding the universal modality and the reflexive transitive closure modality to modal logics. From the examples in the literature, one might conjecture that adding the reflexive transitive closure modality is at least as hard as adding the universal modality, and that adding either of these modalities to a multi-modal logic where the modalities do not interact can only increase the complexity to EXPTIME-complete. We show that the first conjecture holds under reasonable assumptions and that, except for a number of special cases which we fully characterize, the hardness part of the second conjecture is true. However, the upper bound part of the second conjecture fails miserably: we show that there exists a uni-modal, decidable, finitely axiomatizable, and canonical logic for which adding the universal modality causes undecidability and for which adding the reflexive transitive closure modality causes high undecidability.

A time- and cost-optimal algorithm for interlocking sets-with applications

Given a family I of intervals, two intervals in I interlock if they overlap but neither of them strictly contains the other. A set of intervals in which every two are related in the reflexive transitive closure of the interlock relation is referred to as an interlocking set. The task is determining the maximal interlocking sets of I arises in numerous applications, including traffic control, robot arm manipulation, segmentation of range images, routing, automated surveillance systems, recognizing polygonal configurations, and code generation for parallel machines. Our first contribution is to show that any sequential algorithm that computes the maximal interlocking sets of a family of n intervals must take /spl Omega/(n log n) time in the algebraic tree model. Next, we show that any parallel algorithm for this problem must take /spl Omega/(log n) time in the CREW model even if an infinite number of processors and memory cells are available. We then go on to show that both the sequential and the parallel lower bounds are tight by providing matching algorithms running, respectively, in /spl Theta/(n log n) sequential time and in /spl Theta/(log n) time using n processors in the CREW model. At the same time, if the endpoints of the intervals are specified in sorted order, our sequential algorithm runs in O(n) time, improving the best previously known result. It is interesting to note that even if the endpoints are sorted, /spl Omega/(log n) is a time lower bound for solving the problem in the CREW model, regardless of the amount of resources available. As an application of our algorithm for interlocking sets, we obtain a time- and cost-optimal solution to a restricted version of the single row routing problem. The best previously known result for routing a set of n nets without street crossovers runs in O(log n loglog n) time using n processors in the CRCW model. By contrast, our algorithm runs in /spl Theta/(log n) time using n/log n processors in the CREW model, being both time- and cost-optimal.

Notes on equational theories of relations

We describe explicitly the free algebras in the equational class generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure and neutral elements 0 (empty relation) and 1 (identity relation). We show the corresponding equational theory is decidable by reducing the problem to a question about regular sets. Similar results are given for two related equational theories.

Equational properties of Kleene algebras of relations with conversion

We describe a set of equational axioms for the variety generated by all algebras of binary relations with operations of union, composition, conversion and reflexive-transitive closure and neutral elements 0 (the empty relation) and 1 (the identity relation).

PARALLEL DOMINATOR COMPUTATION ON A RAP

ABSTRACT Two constant time algorithms, which are based on the reflexive transitive closure of a directed graph, are proposed for computing dominators and dominator tree of a flow graph respectively. The parallel computation model used is a reconfigurable array of processors. A reconfigurable array of processors is defined to be an array of processors connected to a reconfigurable bus system whose configuration can be dynamically changed. Other applications that are based on the proposed algorithms are also solved in a constant time respectively. These problems include finding the back edges in a flow graph, recognizing the acyclic directed graph, recognizing the reducible flow graph, finding the natural loops, and finding the inner loop.

A criterion for proving noetherianity of a relation

Let -+ be a binary relation (or rewriting rule) on a set Y, and denote by +* its reflexive-transitive closure. Recall that --+ is said to be noetherian if and only if there is no infinitely long sequences x1, x2, . . . in .IT such that xi~Xi+ 1 holds for every index i; + is conjluent (locally confluent) if and only if for all x, y, z in X, x+*y and x+* z (x-y and .x-+z) imply that there exists X satisfying both y+*X and z+*.U. Finally, + is said to be complete whenever it is both confluent and noetherian. We are looking for criteria implying completeness. According to a tradition that goes back at least to [S], “the” natural way for proving noetherianity of + consists in constructing for the elements of 9” a notion of “degree” with values in N (or, more generally, in a well-order) in such a way that degree strictly decreuses along -+-paths. Once noetherianity is known, local confluence implies global one, providing the classical criterion described by the following proposition.

Dynamic algebras: Examples, constructions, applications

Dynamic algebras combine the classes of Boolean (B ∨ ′ 0) and regular (R ∪; *) algebras into a single finitely axiomatized variety (B R ◊) resembling an R-module with “scalar” multiplication ◊. The basic result is that * is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of prepositional dynamic logic, and yet another notion of regular algebra.