Rule Heads Research Articles

SimRE: Simple contrastive learning with soft logical rule for knowledge graph embedding

Knowledge graphs serve as a pivotal framework for the structured representation of information regarding entities and relations. However, in the real world, these knowledge graphs are often incomplete and harboring missing facts. Knowledge graph completion (KGC) has emerged as a central research focus, entailing the automated prediction of these missing facts and garnering substantial scholarly attention in recent years. Text-based knowledge graph embedding methods have demonstrated considerable potential for tackling the challenges associated with KGC by employing pre-trained language models. However, their limitation lies in the lack of logical features, which constrains the efficacy of capturing intricate patterns within knowledge graphs. This paper proposed SimRE, a straightforward contrastive learning framework augmented with soft logic rules. SimRE introduces a self-supervised framework that leverages the input rule bodies to predict the corresponding rule heads through a contrastive objective. We introduced two rule sampling techniques to enhance the efficiency and accuracy of the model: in-batch rule negatives and pre-batch rule negatives. SimRE employs a simple method for integrating logical features with the text-based model. The experimental results on benchmark datasets demonstrate that the proposed approach outperforms state-of-the-art methods.

Some Transnational Aspects of Environmental Disputes in Kazakhstan and Its New Environmental Code

Environmental tort litigation is widely seen as an appropriate mechanism to redress wrongs caused by pollution, oil spills and other kinds of environmental degradation. Victims and NGOs try to find ways to sue parent companies in their country of incorporation, which is usually different from the country where the harmful event occurred. Central Asia is a region where part of the population suffers from similar situations. Kazakhstan, for instance, is very rich in minerals and foreign and domestic extractive companies often damage the soil and the air quality of towns and communities in the vicinity of their operations. The practical disappearance of the Aral Sea, partly due to the inappropriate use of water by Uzbekistan, is another example. Environmental regulation and administrative fines imposed by the countries concerned leaves the prosecution of reckless industrial practices in the hands of the Government, in countries were collusion between Governmental cliques and extractive industries is well known and where courts are often poorly trained or far from independent. Cases of transnational environmental damage are also possible due to the large border that Central Asian countries share with each other, as well as with other industrial giants like Russia and China. In such a scenario, what are the chances of using tort law and civil litigation for victims and other stakeholders, especially in view of the new Environmental Code of 2021? Central Asian countries basically belong to the civil law tradition and are greatly influenced by Russian legislation. This means that tort law and conflict of laws are usually codified in civil codes and in codes of civil procedure. Heads of jurisdiction and applicable law rules are not that different from those of other civil law countries. However, those rules often display certain “nationalist” features in that nationals or residents of the country of the forum usually have easier access to court against foreign defendants, at least on paper. Key words: Environment, Code, Litigation, Kazakhstan.

Strong Equivalence of Logic Programs with Ordered Disjunction: A Logical Perspective

AbstractLogic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing preferential disjunctions in the heads of program rules. The initial semantics of LPODs, although simple and quite intuitive, is not purely model-theoretic. As a result, certain properties of programs appear non-trivial to formalize in purely logical terms. For example, the current characterization of strong equivalence for LPODs, does not coincide with logical equivalence in some specific logic. This comes in sharp contrast with the well-known characterization of strong equivalence for classical logic programs, which coincides with logical equivalence in the logic of here-and-there. In this paper we obtain a purely logical characterization of strong equivalence for LPODs as logical equivalence in a four-valued logic. Moreover, we provide a new proof of the coNP-completeness of strong equivalence for LPODs, which has an interest in its own right since it relies on the special structure of such programs. Our results are based on the recent logical semantics of LPODs, a fact which we believe indicates that this new semantics may prove to be a useful tool in the further study of LPODs.

WORD OUTLOOK IN THE POEM «KUTADGU BILIG» BY JUSUP BALASAGYN

This article is devoted to research J. Balasagyn’s poem «Kutadgu Bilig» which is devoted to the study of philosophical views, understanding of the world, worldview, i.e. epistemology. The materialism of J. Balasagyn is closely connected with his spontaneous-dialectical views. He wrote that all the phenomena of nature grow and are constantly in motion in accordance with that change, that the old thing dies and gives way to the new, that human life is not eternal, that man is born again, grows old and is forced to die. J. Balasagyn used his ideas about materialism and spontaneous dialectics not only in the study of natural phenomena, but also in the study of social life. J. Balasagyn considered Kuntuduu to be a symbol of legitimacy. According to him, justice and morality are one of the most important qualities of a ruler. If the head of state rules fairly on the basis of the law, justice will surely prevail.

Loop restricted existential rules and first-order rewritability for query answering

Abstract In ontology-based data access (OBDA), the classical database is enhanced with an ontology in the form of logical assertions generating new intensional knowledge. A powerful form of such logical assertions is the tuple-generating dependencies (TGDs), also called existential rules, where Horn rules are extended by allowing existential quantifiers to appear in the rule heads. In this paper, we introduce a new language called loop restricted (LR) TGDs (existential rules), which are TGDs with certain restrictions on the loops embedded in the underlying rule set. We study the complexity of this new language. We show that the conjunctive query answering (CQA) under the LR TGDs is decidable. In particular, we prove that this language satisfies the so-called bounded derivation-depth property (BDDP), which implies that the CQA is first-order rewritable, and its data complexity is in Ac$^{0}$. We also prove that the combined complexity of the CQA is 2-ExpTime complete, while the language membership is Pspace complete. Then we extend the LR TGDs language to the generalized loop restricted (GLR) TGDs language and prove that this class of TGDs still remains to be first-order rewritable and properly contains most of other first-order rewritable TGDs classes discovered in the literature so far.

A Logical Characterization of the Preferred Models of Logic Programs with Ordered Disjunction

AbstractLogic programs with ordered disjunction (LPODs) extend classical logic programs with the capability of expressing alternatives with decreasing degrees of preference in the heads of program rules. Despite the fact that the operational meaning of ordered disjunction is clear, there exists an important open issue regarding its semantics. In particular, there does not exist a purely model-theoretic approach for determining the most preferred models of an LPOD. At present, the selection of the most preferred models is performed using a technique that is not based exclusively on the models of the program and in certain cases produces counterintuitive results. We provide a novel, model-theoretic semantics for LPODs, which uses an additional truth value in order to identify the most preferred models of a program. We demonstrate that the proposed approach overcomes the shortcomings of the traditional semantics of LPODs. Moreover, the new approach can be used to define the semantics of a natural class of logic programs that can have both ordered and classical disjunctions in the heads of clauses. This allows programs that can express not only strict levels of preferences but also alternatives that are equally preferred.

Study on Analysis and Improvement of Head of Ri Election Rules & Regulations -Focusing on Eup/Myeon Regional Village Codes of Jeju Special Self-Governing Province-

Study on Analysis and Improvement of Head of Ri Election Rules & Regulations -Focusing on Eup/Myeon Regional Village Codes of Jeju Special Self-Governing Province-

Characterizing Boundedness in Chase Variants

AbstractExistential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in the context of ontology-mediated query answering. Forward chaining, also known as the chase, is a fundamental tool for computing universal models of knowledge bases, which consist of existential rules and facts. Several chase variants have been defined, which differ on the way they handle redundancies. A set of existential rules is bounded if it ensures the existence of a bound on the depth of the chase, independently from any set of facts. Deciding if a set of rules is bounded is an undecidable problem for all chase variants. Nevertheless, when computing universal models, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound remains unknown or even very large. Hence, we investigate the decidability of the k-boundedness problem, which asks whether the depth of the chase for a given set of rules is bounded by an integer k. We identify a general property which, when satisfied by a chase variant, leads to the decidability of k-boundedness. We then show that the main chase variants satisfy this property, namely the oblivious, semi-oblivious (aka Skolem), and restricted chase, as well as their breadth-first versions.

Complexity and Expressive Power of Disjunction and Negation in Limit Datalog

Limit Datalog is a fragment of Datalogℤ—the extension of Datalog with arithmetic functions over the integers—which has been proposed as a declarative language suitable for capturing data analysis tasks. In limit Datalog programs, all intensional predicates with a numeric argument are limit predicates that keep maximal (or minimal) bounds on numeric values. Furthermore, to ensure decidability of reasoning, limit Datalog imposes a linearity condition restricting the use of multiplication in rules. In this paper, we study the complexity and expressive power of limit Datalog programs extended with disjunction in the heads of rules and non-monotonic negation under the stable model semantics. We show that allowing for unrestricted use of negation leads to undecidability of reasoning. Decidability can be restored by stratifying the use of negation over predicates carrying numeric values. We show that the resulting language is Π2EXP -complete in combined complexity and that it captures Π2P over ordered structures in the sense of descriptive complexity.We also provide a study of several fragments of this language: we show that the complexity and expressive power of the full language are already reached for disjunction-free programs; furthermore, we show that semi-positive disjunctive programs are coNEXPcomplete and that they capture coNP.

Determining inference semantics for disjunctive logic programs

In a seminal paper, Gelfond and Lifschitz [34] introduced simple disjunctive logic programs, where in rule heads the disjunction operator “|” is used to express incomplete information, and defined the answer set semantics (called GL-semantics for short) based on a program transformation (called GL-reduct) and the minimal model requirement. Our observations reveal that the requirement of the GL-semantics, i.e., an answer set should be a minimal model of rules of the GL-reduct, may sometimes be too strong a condition and exclude some answer sets that would be reasonably acceptable. To address this, we present an alternative, more permissive answer set semantics, called the determining inference (DI) semantics. Specifically, we introduce a head selection function to formalize the operator | and define answer sets as follows: (i) Given an interpretation I and a selection function sel, we transform a disjunctive program Π into a normal program ΠselI, called a disjunctive program reduct; (ii) given a base answer set semantics X for normal programs, we define I to be a candidate answer set of Π w.r.t. X if I is an answer set of ΠselI under X; and (iii) we define I to be an answer set of Π w.r.t. X if I is a minimal candidate answer set. The DI-semantics is general and applicable to extend any answer set semantics X for normal programs to disjunctive programs. By replacing X with the GLnlp-semantics defined by Gelfond and Lifschitz [33], we induce a DI-semantics for simple disjunctive programs, and by replacing X with the well-justified semantics defined by Shen et al. [65], we further induce a DI-semantics for general disjunctive programs. We also establish a novel characterization of the GL-semantics in terms of a disjunctive program reduct, which reveals the essential difference of the DI-semantics from the GL-semantics and leads us to giving a satisfactory solution to the open problem presented by Hitzler and Seda [36] about characterizing split normal derivatives of a simple disjunctive program Π such that answer sets of the normal derivatives are answer sets of Π under the GL-semantics. Finally we give computational complexity results; in particular we show that in the propositional case deciding whether a simple disjunctive program Π has some DI-answer set is NP-complete. This is in contrast to the GL-semantics and equivalent formulations such as the FLP-semantics [24], where deciding whether Π has some answer set is Σ2p-complete, while brave and cautious reasoning are Σ2p- and Π2p-complete, respectively, for both GL- and DI-answer sets. For general disjunctive programs with compound formulas as building blocks, the complexity of brave and cautious reasoning increases under DI-semantics by one level of the polynomial hierarchy, which thus offers higher problem solving capacity.

Finite model reasoning over existential rules

AbstractOntology-based query answering asks whether a Boolean conjunctive query is satisfied by all models of a logical theory consisting of a relational database paired with an ontology. The introduction of existential rules (i.e., Datalog rules extended with existential quantifiers in rule heads) as a means to specify the ontology gave birth to Datalog+/-, a framework that has received increasing attention in the last decade, with focus also on decidability and finite controllability to support effective reasoning. Five basic decidable fragments have been singled out: linear, weakly acyclic, guarded, sticky, and shy. Moreover, for all these fragments, except shy, the important property of finite controllability has been proved, ensuring that a query is satisfied by all models of the theory iff it is satisfied by all its finite models. In this paper, we complete the picture by demonstrating that finite controllability of ontology-based query answering holds also for shy ontologies, and it therefore applies to all basic decidable Datalog+/- classes. To make the demonstration, we devise a general technique to facilitate the process of (dis)proving finite controllability of an arbitrary ontological fragment.

Using linear constraints for logic program termination analysis

AbstractIt is widely acknowledged that function symbols are an important feature in answer set programming, as they make modelling easier, increase the expressive power, and allow us to deal with infinite domains. The main issue with their introduction is that the evaluation of a program might not terminate and checking whether it terminates or not is undecidable. To cope with this problem, several classes of logic programs have been proposed where the use of function symbols is restricted but the program evaluation termination is guaranteed. Despite the significant body of work in this area, current approaches do not include many simple practical programs whose evaluation terminates. In this paper, we present the novel classes ofrule-boundedandcycle-bounded programs, which overcome different limitations of current approaches by performing a more global analysis of how terms are propagated from the body to the head of rules. Results on the correctness, the complexity, and the expressivity of the proposed approach are provided.

Rewriting recursive aggregates in answer set programming: back to monotonicity

AbstractAggregation functions are widely used in answer set programming for representing and reasoning on knowledge involving sets of objects collectively. Current implementations simplify the structure of programs in order to optimize the overall performance. In particular, aggregates are rewritten into simpler forms known as monotone aggregates. Since the evaluation of normal programs with monotone aggregates is in general on a lower complexity level than the evaluation of normal programs with arbitrary aggregates, any faithful translation function must introduce disjunction in rule heads in some cases. However, no function of this kind is known. The paper closes this gap by introducing a polynomial, faithful, and modular translation for rewriting common aggregation functions into the simpler form accepted by current solvers. A prototype system allows for experimenting with arbitrary recursive aggregates, which are also supported in the recent version 4.5 of the groundergringo, using the methods presented in this paper.

Fuzzy answer set computation via satisfiability modulo theories

AbstractFuzzy answer set programming (FASP) combines two declarative frameworks, answer set programming and fuzzy logic, in order to model reasoning by default over imprecise information. Several connectives are available to combine different expressions; in particular the Gödel and Łukasiewicz fuzzy connectives are usually considered, due to their properties. Although the Gödel conjunction can be easily eliminated from rule heads, we show through complexity arguments that such a simplification is infeasible in general for all other connectives. The paper analyzes a translation of FASP programs into satisfiability modulo theories (SMT), which in general produces quantified formulas because of the minimality of the semantics. Structural properties of many FASP programs allow to eliminate the quantification, or to sensibly reduce the number of quantified variables. Indeed, integrality constraints can replace recursive rules commonly used to force Boolean interpretations, and completion subformulas can guarantee minimality for acyclic programs with atomic heads. Moreover, head cycle free rules can be replaced by shifted subprograms, whose structure depends on the eliminated head connective, so that ordered completion may replace the minimality check if also Łukasiewicz disjunction in rule bodies is acyclic. The paper also presents and evaluates a prototype system implementing these translations.

Disjunctive logic programs with existential quantification in rule heads

AbstractWe consider disjunctive logic programs without function symbols but with existential quantification in rule heads, under the semantics of general stable models. There are at least two interesting prospects in these programs. The first is that a program can be made more succinct by using existential variables, and the second is on the potential in representing defeasible ontological knowledge by these logic programs. This paper studies some of the properties of these programs. First, we show a simple yet intuitive definition of stable models for these programs that does not resort to second-order logic. Second, the stable models of these programs can be characterized by an extension of progression for disjunctive programs, which provides a native characterization of justification for stable models. We then study the decidability issue. While the stable model existence problem for safe disjunctive programs is decidable, with existential quantification allowed in rule heads the problem becomes undecidable. We identify an interesting decidable fragment by exploring a new notion of stratification over existential quantification.

A practical analysis of non-termination in large logic programs

AbstractA large body of work has been dedicated to termination analysis of logic programs but relatively little has been done to analyze non-termination. In our opinion, explaining non-termination is a much more important task because it can dramatically improve a user's ability to effectively debug large, complex logic programs without having to abide by punishing syntactic restrictions. Non-termination analysis examines program execution history when the program is suspected to not terminate and informs the programmer about the exact reasons for this behavior. In Liang and Kifer (2013), we studied the problem of non-termination in tabled logic engines with subgoal abstraction, such as XSB, and proposed a suite of algorithms for non-termination analysis, called Terminyzer. These algorithms analyze forest logging traces and output sequences of tabled subgoal calls that are the likely causes of non-terminating cycles. However, this feedback was hard to use in practice: the same subgoal could occur in multiple rule heads and in even more places in rule bodies, so Terminyzer left too much tedious, sometimes combinatorially large amount of work for the user to do manually.Here we propose a new suite of algorithms, Terminyzer+, which closes this usability gap. Terminyzer+ can detect not only sequences of subgoals that cause non-termination, but, importantly, the exact rules where they occur and the rule sequences that get fired in a cyclic manner, thus causing non-termination. This makes Terminyzer+ suitable as a back-end for user-friendly graphical interfaces on top of Terminyzer+, which can greatly simplify the debugging process. Terminyzer+ back-ends exist for the SILK system as well as for the open-source ${\cal F}$lora-2 system. A graphical interface has been developed for SILK and is currently underway for ${\cal F}$lora-2. We also report experimental studies, which confirm the effectiveness of Terminyzer+ on a host of large real-world knowledge bases. All tests used in this paper are available online.1In addition, we make a step towards automatic remediation of non-terminating programs by proposing an algorithm that heuristically fixes some causes of misbehavior. Furthermore, unlike Terminyzer, Terminyzer+ does not require the underlying logic engine to support subgoal abstraction, although it can make use of it.

The expressive power of CHR with priorities

Constraint Handling Rules (CHR) is a committed-choice declarative language which has been originally designed for writing constraint solvers and which is nowadays a general purpose language.Recently the language has been extended by introducing user-definable (static or dynamic) rule priorities. The resulting language allows a better control over execution while retaining a declarative and flexible style of programming.In this paper we study the expressive power of this language. We first show that, in the presence of priorities, differently from the case of standard CHR, considering more than two atoms in the heads of rules does not augment the expressive power of the language. Next we show that also dynamic priorities do not augment the expressive power w.r.t. static priorities. These results are proved by providing explicitly a translation of one language into another one, which preserves a reference semantics.Finally we show that CHR with priorities is strictly more expressive than standard CHR (under the theoretical operational semantics). This result is obtained by adapting to the CHR case a notion of language encoding which allows to compare Turing powerful languages.

Disjunctive datalog with existential quantifiers: Semantics, decidability, and complexity issues

AbstractDatalog is one of the best-known rule-based languages, and extensions of it are used in a wide context of applications. An important Datalog extension is Disjunctive Datalog, which significantly increases the expressivity of the basic language. Disjunctive Datalog is useful in a wide range of applications, ranging from Databases (e.g., Data Integration) to Artificial Intelligence (e.g., diagnosis and planning under incomplete knowledge). However, in recent years an important shortcoming of Datalog-based languages became evident, e.g. in the context of data-integration (consistent query-answering, ontology-based data access) and Semantic Web applications: The language does not permit any generation of and reasoning with unnamed individuals in an obvious way. In general, it is weak in supporting many cases of existential quantification. To overcome this problem, Datalog∃ has recently been proposed, which extends traditional Datalog by existential quantification in rule heads. In this work, we propose a natural extension of Disjunctive Datalog and Datalog∃, called Datalog∃,˅, which allows both disjunctions and existential quantification in rule heads and is therefore an attractive language for knowledge representation and reasoning, especially in domains where ontology-based reasoning is needed. We formally define syntax and semantics of the language Datalog∃,˅, and provide a notion of instantiation, which we prove to be adequate for Datalog∃,˅. A main issue of Datalog∃ and hence also of Datalog∃,˅ is that decidability is no longer guaranteed for typical reasoning tasks. In order to address this issue, we identify many decidable fragments of the language, which extend, in a natural way, analog classes defined in the non-disjunctive case. Moreover, we carry out an in-depth complexity analysis, deriving interesting results which range from Logarithmic Space to Exponential Time.

A general Datalog-based framework for tractable query answering over ontologies

Ontologies and rules play a central role in the development of the Semantic Web. Recent research in this context focuses especially on highly scalable formalisms for the Web of Data, which may highly benefit from exploiting database technologies. In this paper, as a first step towards closing the gap between the Semantic Web and databases, we introduce a family of expressive extensions of Datalog, called Datalog±, as a new paradigm for query answering over ontologies. The Datalog± family admits existentially quantified variables in rule heads, and has suitable restrictions to ensure highly efficient ontology querying. We show in particular that Datalog± encompasses and generalizes the tractable description logic EL and the DL-Lite family of tractable description logics, which are the most common tractable ontology languages in the context of the Semantic Web and databases. We also show how stratified negation can be added to Datalog± while keeping ontology querying tractable. Furthermore, the Datalog± family is of interest in its own right, and can, moreover, be used in various contexts such as data integration and data exchange. It paves the way for applying results from databases to the context of the Semantic Web.

A General Datalog-Based Framework for Tractable Query Answering Over Ontologies

Ontologies and rules play a central role in the development of the Semantic Web. Recent research in this context focuses especially on highly scalable formalisms for the Web of Data, which may highly benefit from exploiting database technologies. In this paper, as a first step towards closing the gap between the Semantic Web and databases, we introduce a family of expressive extensions of Datalog, called Datalog±, as a new paradigm for query answering over ontologies. The Datalog± family admits existentially quantified variables in rule heads, and has suitable restrictions to ensure highly efficient ontology querying. We show in particular that Datalog± encompasses and generalizes the tractable description logic ℇ 𝓛 and the DL-Lite family of tractable description logics, which are the most common tractable ontology languages in the context of the Semantic Web and databases. We also show how stratified negation can be added to Datalog± while keeping ontology querying tractable. Furthermore, the Datalog± family is of interest in its own right, and can, moreover, be used in various contexts such as data integration and data exchange. It paves the way for applying results from databases to the context of the Semantic Web.