Equality Predicate Research Articles

Decidability of the Satisfiability Problem for Boolean Set Theory with the Unordered Cartesian Product Operator

We give a positive solution to the decidability problem for the fragment of set theory, dubbed BST ⊗, consisting of quantifier-free formulae involving the Boolean set operators of union, intersection, and set difference, along with the unordered Cartesian product operator ⊗ (where \(s \otimes t := \big \lbrace \lbrace u,v\rbrace \,\texttt {|}\:u \in s \wedge v \in t \big \rbrace\) ), and the equality predicate, but no membership. Specifically, we provide nondeterministic exponential decision procedures for both the ordinary and the finite satisfiability problems for BST ⊗. We expect that these decision procedures can be adapted for the standard Cartesian product and, with added technicalities, to the cases involving membership, providing a solution to a longstanding problem in computable set theory.

Решетка $E$-замкнутых классов мультифункций ранга 2

Multifunctions are discrete functions defined on a finite set and returning as their values all subsets of the considered set. The paper considers the classification of multifunctions defined on a two-element set with respect to the E-closure operator. E-closed sets of multifunctions are sets that are closed under superposition, the closure operator with branching by the equality predicate, the identification of variables, and the addition of dummy variables. The concept of separating sets was introduced using a greedy algorithm for the set covering problem, and 22 classes of separating sets were obtained. It is shown that the classification under consideration leads to a finite set of closed classes. The work describes all 359 E-closed classes of multifunctions, among which there are 138 pairs of dual classes and 83 self-dual classes. For each class consisting only of multifunctions, its generating system is indicated.

The fluted fragment with transitive relations

The fluted fragment is a fragment of first-order logic (without equality) in which, roughly speaking, the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that this fragment has the finite model property. We consider extensions of the fluted fragment with various numbers of transitive relations, as well as the equality predicate. In the presence of one transitive relation (together with equality), the finite model property is lost; nevertheless, we show that the satisfiability and finite satisfiability problems for this extension remain decidable. We also show that the corresponding problems in the presence of two transitive relations (with equality) or three transitive relations (without equality) are undecidable, even for the two-variable sub-fragment.

АЛГОРИТМЫ РЕШЕНИЯ СИСТЕМ УРАВНЕНИЙ НАД РАЗЛИЧНЫМИ КЛАССАМИ КОНЕЧНЫХ ГРАФОВ

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n > 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.

О решетке <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><msub><mi>S</mi><mi>I</mi></msub></math>-замкнутых классов мультифункций ранга 2

The paper considers multifunctions on a two-element set with superposition and the equality predicate branching operator. The superposition operator is based on the intersection of sets. The main purpose of the work is to describe all closed classes with respect to the considered operators. The equality predicate branching operator allows the task to be reduced to a description of all closed classes generated by 2-variable multifunctions. Using this, it is shown that the lattice of classes closed with respect to the considered operators contains 237 elements. A generating set is specified for each closed class. The result obtained in the paper extends the known result for all closed classes of partial functions on a two-element set.

First-order swap structures semantics for some logics of formal inconsistency

Abstract The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (i.e. logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous approaches to quantified LFIs presented in the literature. The case of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called $\textbf{QLFI1}_\circ $ is also studied, which is equivalent to the quantified version of da Costa and D’Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and $\textbf{QLFI1}_\circ $ with a standard equality predicate is also considered.

Elementary classes of graphs in a language without equality predicate

In this paper we describe equivalence classes generated by finite graphs in the graph theory language without equality predicate. These classes coincide with classes of so called hard extensions of graphs.

E-closed Sets of Hyperfunctions on Two-Element Se

Hyperfunctions are functions that are defined on a finite set and return all non-empty subsets of the considered set as their values. This paper deals with the classification of hyperfunctions on a two-element set. We consider the composition and the closure operator with the equality predicate branching (E-operator). E-closed sets of hyperfunctions are sets that are obtained using the operations of adding dummy variables, identifying variables, composition, and E-operator. It is shown that the considered classification leads to a finite set of closed classes. The paper presents all 78 E-closed classes of hyperfunctions, among which there are 28 pairs of dual classes and 22 self-dual classes. The inclusion diagram of the E-closed classes is constructed, and for each class, its generating system is obtained

Top-k spatial distance joins

Top-k joins have been extensively studied when numerical valued attributes are joined on an equality predicate. Other types of join attributes and predicates have received little to no attention. In this paper, we consider spatial objects that are assigned a score (e.g., a ranking). Give two collections R, S of such objects and a spatial distance threshold 𝜖, we introduce the top-k spatial distance join (k-SDJoin) to identify the k pairs of objects, which have the highest combined score (based on an aggregate function γ) among all object pairs in R × S with a spatial distance at most 𝜖. State-the-of-art methods for relational top-k joins can be adapted for k-SDJoin, but their focus is on minimizing the number of objects accessed from the inputs; however, when spatial objects are joined, the computational cost can easily become the bottleneck. In view of this, we propose a novel evaluation algorithm, which greatly reduces the computational cost, without compromising the access cost. The main idea is to access and efficiently join blocks of objects from each collection, using appropriate bounds to avoid computing the entire spatial 𝜖-distance join. As the performance of our solution heavily relies on the size of the input blocks, we devise an approach for automated block size tuning enhanced by a novel generic model for estimating the number of objects to be accessed from each input. Contrary to previous efforts, our model employs cheap-to-compute statistics and requires no prior knowledge of data distribution. Our extensive experimental analysis demonstrates the efficiency of our algorithm compared to methods based on existing literature that prioritize either the ranking or the spatial join component of k-SDJoin queries.

Computational Psycholinguistics

Computational Psycholinguistics

Классификация мультиопераций ранга 2 относительно E-предполных множеств

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is 129. All types of bases are described and it is proved that the maximum cardinality of a basis is 4.

Критерий полноты для оператора замыкания с разветвлением по предикату равенства на множестве мультиопераций ранга 2

Критерий полноты для оператора замыкания с разветвлением по предикату равенства на множестве мультиопераций ранга 2

Translation of first order formulas into ground formulas via a completion theory

A translation technique is presented which transforms a class of First Order Logic formulas, called Restricted formulas, into ground formulas. For the formulas in this class the range of quantified variables is restricted by Domain formulas.If we have a complete knowledge of the predicates involved in the Domain formulas their extensions can be evaluated with the Relational Algebra and these extensions are used to transform universal (respectively existential) quantifiers into finite conjunctions (respectively disjunctions).It is assumed that the complete knowledge is represented by Completion Axioms and Unique Name Axioms à la Reiter. These axioms involve the equality predicate. However, the translation allows to remove the equality in the ground formulas and for a large class of formulas their consequences are the same as the initial First Order formulas. This result open the door for the design of efficient deduction techniques.

Математические основы теории познания на основе экспериментов

В статье рассматриваются математические предпосылки для разработки теории познания, использующей информацию о реальных экспериментах, проводимых над реальными объектами с помощью формально-технологических аналогов машин Тьюринга. Такие аналоги, получившие название «универсальных синтезаторов-анализаторов объектов», теоретически позволяют выполнять синтез и анализ самых разнообразных конструкций (в том числе объектов, полученных путем соединения конечного числа исходных более мелких объектов, так называемых элементов базы) по различным алгоритмам в рамках некоторых ограничений, накладываемых алгоритмическими системами, названными формальными технологиями. Эти алгоритмические системы оказываются весьма близкими по своей формальной структуре и сути к структурам алгебраических систем Мальцева. Формальная близость таких алгоритмических систем позволяет, во-первых, выдвинуть гипотезу об алгоритмической основе практически всех окружающих нас понятных или непонятных (пока) нам физических процессов, что в какой-то степени объясняет широкую применимость математики к объяснению самых различных особенностей окружающего нас мира; во-вторых, она же позволяет сформулировать и доказать ряд теорем (называемых утверждениями), касающихся особенностей и ключевых характеристик алгоритмов познания в одномерных, двухмерных и трехмерных средах для различных формальных технологических систем, включая так называемую «теорему об эффективности накопленных знаний». Теорема (утверждение) оказывается применимой к очень широкому классу технологий, использующих в качестве операции анализа предикат равенства двух объектов. В статье приводятся и доказываются утверждения о существовании алгоритмов познания с различными наборами технологических операций типа синтеза и декомпозиции, а также с различными наборами операций анализа, включая предикат равенства, операцию «случайного стационарного отображения» (механизм действия которой неизвестен, а потому близок к концепции оракулов в машинах Тьюринга), операции определения формы объектов и др. Приводится структура соответствующего автоматически действующего устройства, названного «познавателем».

Order-sorted equality enrichments modulo axioms

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation E↦E∼ that extends an algebraic specification E to a specification E∼ having an equationally-defined equality predicate? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications E that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have a subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity–commutativity axioms, so that the constructors are free modulo B. We prove that the transformation E↦E∼ preserves all the just-mentioned properties of E. The transformation has been automated in Maude using reflection and is used as a component in many Maude formal tools.

Lower complexity bounds for lifted inference

AbstractOne of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such “lifted inference” techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show the feasibility of lifted inference on certain syntactically defined classes of models. Lower complexity bounds that imply some limitations for the feasibility of lifted inference on more expressive model classes were established earlier in Jaeger (2000; Jaeger, M. 2000. On the complexity of inference about probabilistic relational models. Artificial Intelligence 117, 297–308). However, it is not immediate that these results also apply to the type of modeling languages that currently receive the most attention, i.e., weighted, quantifier-free formulas. In this paper we extend these earlier results, and show that under the assumption that NETIME≠ETIME, there is no polynomial lifted inference algorithm for knowledge bases of weighted, quantifier-, and function-free formulas. Further strengthening earlier results, this is also shown to hold for approximate inference and for knowledge bases not containing the equality predicate.

Efficient and Secure File Deduplication in Cloud Storage

Outsourcing to a cloud storage brings forth new challenges for the efficient utilization of computing resources as well as simultaneously maintaining privacy and security for the outsourced data. Data deduplication refers to a technique that eliminates redundant data on the storage and the network, and is considered to be one of the most-promising technologies that offers efficient resource utilization in the cloud computing. In terms of data security, however, deduplication obstructs applying encryption on the outsourced data and even causes a side channel through which information can be leaked. Achieving both efficient resource utilization and data security still remains open. This paper addresses this challenging issue and proposes a novel solution that enables data deduplication while also providing the required data security and privacy. We achieve this goal by constructing and utilizing equality predicate encryption schemes which allow to know only equivalence relations between encrypted data. We also utilize a hybrid approach for data deduplication to prevent information leakage due to the side channel. The performance and security analyses indicate that the proposed scheme is efficient to securely manage the outsourced data in the cloud computing.

Proof-Guided Test Selection from First-Order Specifications with Equality

This paper deals with test case selection from axiomatic specifications whose axioms are quantifier-free first-order formulas with equality. We first prove the existence of an ideal exhaustive test set to start the selection from. We then propose an extension of the test selection method called axiom unfolding, originally defined for algebraic specifications, to quantifier-free first-order specifications with equality. This method basically consists of a case analysis of the property under test (the test purpose) according to the specification axioms. It is based on a proof search for the different instances of the test purpose. Since the calculus is sound and complete, this allows us to provide a full coverage of this property. The generalisation we propose allows to deal with any kind of predicate (not only equality) and with any form of axiom and test purpose (not only equations or Horn clauses). Moreover, it improves our previous works with efficiently dealing with the equality predicate, thanks to the paramodulation rule.

Generating Minimum Transitivity Constraints in P-time for Deciding Equality Logic

In a CAV'05 paper [Meir, O. and O. Strichman, Yet another decision procedure for equality logic, in: K. Etessami and S. Rajamani, editors, Proc. 17th Intl. Conference on Computer Aided Verification (CAV'05), Lect. Notes in Comp. Sci. 3576 (2005), pp. 307–320] we introduced a new decision procedure for Equality Logic: each equality predicate is encoded with a Boolean variable, and then a set of transitivity constraints are added to compensate for the loss of transitivity of equality. The constraints are derived by analyzing Contradictory Cycles: cycles in the equality graph with exactly one disequality. Such a cycle is called constrained under a formula ϕ if ϕ is not satisfied with an assignment of true to all equality edges and false to the disequality edge. While we proved in [Meir, O. and O. Strichman, Yet another decision procedure for equality logic, in: K. Etessami and S. Rajamani, editors, Proc. 17th Intl. Conference on Computer Aided Verification (CAV'05), Lect. Notes in Comp. Sci. 3576 (2005), pp. 307–320] that it is sufficient to constrain all simple contradictory cycles, we left open the question of how to find the necessary constraints in polynomial time. Instead, we showed two possible compromises: an exponential algorithm, or, alternatively, a polynomial approximation that constrains all contradictory cycles rather than only the simple ones. In this article we show a polynomial algorithm that constrains only the simple contradictory cycles.

The closure operator with the equality predicate branching on the set of partial Boolean functions

The closure operator with the equality predicate branching on the set of partial Boolean functions