In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While the embeddings all correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and provide useful guidance in the choice of a particular embedding for knowledge combination.
University of Science and Technology of China（中国科学技术大学）；Leipzig University（莱比锡大学）
The logic of knowledge and justified assumptions, also known as logic of grounded knowledge (GK), was proposed by Lin and Shoham as a general logic for nonmonotonic reasoning. To date, it has been used to embed in it default logic (propositional case), autoepistemic logic, Turner's logic of universal causation, and general logic programming under stable model semantics. Besides showing the generality of GK as a logic for nonmonotonic reasoning, these embeddings shed light on the relationships among these other logics. In this paper, for the first time, we show how the logic of GK can be embedd…
The paper presents a constructive fixpoint semantics for autoepistemic logic (AEL). This fixpoint characterizes a unique but possibly three-valued belief set of an autoepistemic theory. It may be three-valued in the sense that for a subclass of formulas F, the fixpoint may not specify whether F is believed or not. The paper presents a constructive 3-valued semantics for autoepistemic logic (AEL). We introduce a derivation operator and define the semantics as its least fixpoint. The semantics is 3-valued in the sense that, for some formulas, the least fixpoint does not specify whether they are …
Defined by Gelfond in 1991 (G91), epistemic specifications (or programs) are an extension of logic programming under stable models semantics that introducessubjective literals. A subjective literal al-lows checking whether some regular literal is true in all (or in some of) the stable models of the program, being those models collected in a setcalledworld view. One epistemic program may yield several world views but, under the original G91 semantics, some of them resulted from self-supported derivations. During the last eight years, several alternative approaches have been proposed to get rid …
We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the abov…
This paper analyses the declarative readings of logic programming. Logic programming - and negation as failure - has no unique declarative reading. One common view is that logic programming is a logic for default reasoning, a sub-formalism of default logic or autoepistemic logic. In this view, negation as failure is a modal operator. In an alternative view, a logic program is interpreted as a definition. In this view, negation as failure is classical objective negation. From a commonsense point of view, there is definitely a difference between these views. Surprisingly though, both types of de…
The Autoepistemic Logic of Knowledge and Belief (AELB) is a powerful nonmonotic formalism introduced by Teodor Przymusinski in 1994. In this paper, we specialize it to a class of theories called `super logic programs'. We argue that these programs form a natural generalization of standard logic programs. In particular, they allow disjunctions and default negation of arbibrary positive objective formulas. Our main results are two new and powerful characterizations of the static semant ics of these programs, one syntactic, and one model-theoretic. The syntactic fixed point characterization is mu…
Open Universiteit（开放大学）；Tel-Aviv Academic College（特拉维夫学术学院）；Vrije Universiteit Brussel（布鲁塞尔自由大学）
Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to dealing with non-deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT to non-deterministic operators, whose ranges are sets of…
Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given …
IRIT University of Toulouse（图卢兹大学IRIT研究所）；CNRS（法国国家科学研究中心）；Universität Potsdam（波茨坦大学）
In a recent line of research, two familiar concepts from logic programming semantics (unfounded sets and splitting) were extrapolated to the case of epistemic logic programs. The property of epistemic splitting provides a natural and modular way to understand programs without epistemic cycles but, surprisingly, was only fulfilled by Gelfond's original semantics (G91), among the many proposals in the literature. On the other hand, G91 may suffer from a kind of self-supported, unfounded derivations when epistemic cycles come into play. Recently, the absence of these derivations was also formalis…
University of Naples Federico II（那不勒斯费德里科二世大学）；University of Bremen（不来梅大学）；University of Liverpool（利物浦大学）
As fragments of first-order logic, Description logics (DLs) do not provide nonmonotonic features such as defeasible inheritance and default rules. Since many applications would benefit from the availability of such features, several families of nonmonotonic DLs have been developed that are mostly based on default logic and autoepistemic logic. In this paper, we consider circumscription as an interesting alternative approach to nonmonotonic DLs that, in particular, supports defeasible inheritance in a natural way. We study DLs extended with circumscription under different language restrictions …
Distributed Autoepistemic Logic with Inductive Definitions (dAEL(ID)) is a recently proposed non-monotonic logic for says-based access control. We define a query-driven decision procedure for dAEL(ID) that is implemented in the knowledge-base system IDP. The decision procedure is designed in such a way that it allows one to determine access rights while avoiding redundant information flow between principals in order to enhance security and reduce privacy concerns. Given that the decision procedure has in the worst case an exponential runtime, it is to be regarded as a proof of concept that inc…
KU Leuven（鲁汶大学）；University of Nebraska at Omaha（内布拉斯加大学奥马哈分校）；University of Kentucky（肯塔基大学）
In Knowledge Representation, it is crucial that knowledge engineers have a good understanding of the formal expressions that they write. What formal expressions state intuitively about the domain of discourse is studied in the theory of the informal semantics of a logic. In this paper we study the informal semantics of Answer Set Programming. The roots of answer set programming lie in the language of Extended Logic Programming, which was introduced initially as an epistemic logic for default and autoepistemic reasoning. In 1999, the seminal papers on answer set programming proposed to use this…
Answer set programming (ASP) is an efficient problem-solving approach, which has been strongly supported both scientifically and technologically by several solvers, ongoing active research, and implementations in many different fields. However, although researchers acknowledged long ago the necessity of epistemic operators in the language of ASP for better introspective reasoning, this research venue did not attract much attention until recently. Moreover, the existing epistemic extensions of ASP in the literature are not widely approved either, due to the fact that some propose unintended res…
To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of ICLP 2015 Recent advances in knowledge compilation introduced techniques to compile \emph{positive} logic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited to \emph{n…
Institute of Philosophy II, Ruhr Universität Bochum（波鸿鲁尔大学第二哲学研究所）；Ruhr Universität Bochum（波鸿鲁尔大学）
In this paper we make a contribution to the unification of formal models of defeasible reasoning. We present several translations between formal argumentation frameworks and nonmonotonic logics for reasoning with plausible assumptions. More specifically, we translate adaptive logics into assumption-based argumentation and ASPIC+, ASPIC+ into assumption-based argumentation and a fragment of assumption-based argumentation into adaptive logics. Adaptive logics are closely related to Makinson's default assumptions and to a significant class of systems within the tradition of preferential semantics…
Recently there have been some unexpected results concerning Fuzzy Description Logics (FDLs) with General Concept Inclusions (GCIs). They show that, unlike the classical case, the DL ALC with GCIs does not have the finite model property under Lukasiewicz Logic or Product Logic and, specifically, knowledge base satisfiability is an undecidable problem for Product Logic. We complete here the analysis by showing that knowledge base satisfiability is also an undecidable problem for Lukasiewicz Logic.
Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order…
Universitat Jaume I de Castelló（卡斯特利翁海梅一世大学）；European Bioinformatics Institute（欧洲生物信息研究所）；University of Oxford（牛津大学）
In this paper we present a preliminary logic-based evaluation of the integration of post-composed phenotypic descriptions with domain ontologies. The evaluation has been performed using a description logic reasoner together with scalable techniques: ontology modularization and approximations of the logical difference between ontologies.
Universität Tübingen（图宾根大学）；Wilhelm-Schickard-Institut（威廉·席卡德研究所）
Every definite logic program has as its meaning a least Herbrand model with respect to the program-independent ordering "set-inclusion". In the case of normal logic programs there do not exist least models in general. However, according to a recent approach by Rondogiannis and Wadge, who consider infinite-valued models, every normal logic program does have a least model with respect to a program-independent ordering. We show that this approach can be extended to formula-based logic programs (i.e., finite sets of rules of the form A\leftarrowF where A is an atom and F an arbitrary first-order f…
Institute of Mathematics and Statistics, Federal University of Bahia - UFBA（巴伊亚联邦大学数学与统计学院）
AGM's belief revision is one of the main paradigms in the study of belief change operations. In this context, belief bases (prioritised bases) have been primarily used to specify the agent's belief state. While the connection of iterated AGM-like operations and their encoding in dynamic epistemic logics have been studied before, few works considered how well-known postulates from iterated belief revision theory can be characterised by means of belief bases and their counterpart in dynamic epistemic logic. Particularly, it has been shown that some postulates can be characterised through transfo…
German University in Cairo（开罗德国大学）；Cairo University（开罗大学）
Tremendous research effort has been dedicated over the years to thoroughly investigate non-monotonic reasoning. With the abundance of non-monotonic logical formalisms, a unified theory that enables comparing the different approaches is much called for. In this paper, we present an algebraic graded logic we refer to as LogAG capable of encompassing a wide variety of non-monotonic formalisms. We build on Lin and Shoham's argument systems first developed to formalize non-monotonic commonsense reasoning. We show how to encode argument systems as LogAG theories, and prove that LogAG captures the no…
An important problem to be addressed within Event-Driven Architecture (EDA) is how to correctly and efficiently capture and process the event/action-based logic. This paper endeavors to bridge the gap between the Knowledge Representation (KR) approaches based on durable events/actions and such formalisms as event calculus, on one hand, and event-condition-action (ECA) reaction rules extending the approach of active databases that view events as instantaneous occurrences and/or sequences of events, on the other. We propose formalism based on reaction rules (ECA rules) and a novel interval-based…
This paper presents complexity analysis and variational methods for inference in probabilistic description logics featuring Boolean operators, quantification, qualified number restrictions, nominals, inverse roles and role hierarchies. Inference is shown to be PEXP-complete, and variational methods are designed so as to exploit logical inference whenever possible.
We study the problem of learning probabilistic first-order logical rules for knowledge base reasoning. This learning problem is difficult because it requires learning the parameters in a continuous space as well as the structure in a discrete space. We propose a framework, Neural Logic Programming, that combines the parameter and structure learning of first-order logical rules in an end-to-end differentiable model. This approach is inspired by a recently-developed differentiable logic called TensorLog, where inference tasks can be compiled into sequences of differentiable operations. We design…
We introduce a lazy approach to the explanation-based approximation of probabilistic logic programs. It uses only the most significant part of the program when searching for explanations. The result is a fast and anytime approximate inference algorithm which returns hard lower and upper bounds on the exact probability. We experimentally show that this method outperforms state-of-the-art approximate inference.
Recent work on neuro-symbolic inductive logic programming has led to promising approaches that can learn explanatory rules from noisy, real-world data. While some proposals approximate logical operators with differentiable operators from fuzzy or real-valued logic that are parameter-free thus diminishing their capacity to fit the data, other approaches are only loosely based on logic making it difficult to interpret the learned "rules". In this paper, we propose learning rules with the recently proposed logical neural networks (LNN). Compared to others, LNNs offer strong connection to classica…
Centrale Supelec（中央理工-高等电力学院）；Université Paris-Saclay（巴黎萨克雷大学）；Université Paris-Dauphine（巴黎多芬大学）；CNRS（法国国家科学研究中心）；Télécom ParisTech（巴黎电信学院）
Belief revision of knowledge bases represented by a set of sentences in a given logic has been extensively studied but for specific logics, mainly propositional, and also recently Horn and description logics. Here, we propose to generalize this operation from a model-theoretic point of view, by defining revision in an abstract model theory known under the name of satisfaction systems. In this framework, we generalize to any satisfaction systems the characterization of the well known AGM postulates given by Katsuno and Mendelzon for propositional logic in terms of minimal change among interpret…
Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been considered, e.g., extension with default rules, extension with modal belief operators, or modification of the semantics. In this survey we consider a logical formalism from each of the above possibilities, namely Reiter's default logic, Moore's autoepistemic logic and McCarthy's circumscription. Additionally, we consider abduction, where one is not interested in …
{\em Computability logic} (CoL) is a powerful computational model which views computational problems as games played by a machine and its environment. In this paper, we show that CoL naturally supports multiagent programming models with distributed control. To be specific, we discuss a distributed logic programming model based on CoL (CL1 to be exact), which we call CL1^Ω. The key feature of this model is that it supports $dynamic/evolving$ knowledgebase of an agent. This model turns out to be a promising approach to reaching both general AI and future computing model.
University of Bonn（波恩大学）；Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）
Knowledge graph embedding models have gained significant attention in AI research. Recent works have shown that the inclusion of background knowledge, such as logical rules, can improve the performance of embeddings in downstream machine learning tasks. However, so far, most existing models do not allow the inclusion of rules. We address the challenge of including rules and present a new neural based embedding model (LogicENN). We prove that LogicENN can learn every ground truth of encoded rules in a knowledge graph. To the best of our knowledge, this has not been proved so far for the neural …
Logic-based paradigms are nowadays widely used in many different fields, also thank to the availability of robust tools and systems that allow the development of real-world and industrial applications. In this work we present LoIDE, an advanced and modular web-editor for logic-based languages that also integrates with state-of-the-art solvers.
We introduce a new semantics for a logic of explicit and implicit beliefs based on the concept of multi-agent belief base. Differently from existing Kripke-style semantics for epistemic logic in which the notions of possible world and doxastic/epistemic alternative are primitive, in our semantics they are non-primitive but are defined from the concept of belief base. We provide a complete axiomatization and prove decidability for our logic via a finite model argument. We also provide a polynomial embedding of our logic into Fagin & Halpern's logic of general awareness and establish a complexit…
The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata.…
Knowledge bases theory provide an important example of the field where applications of universal algebra and algebraic logic look very natural, and their interaction with practical problems arising in computer science might be very productive. In this paper we study the equivalence problem for knowledge bases. Our interest is to find out how the informational equivalence is related to the logical description of knowledge. Studying various equivalences of knowledge bases allows us to compare different knowledge bases. The main objective of this paper is logically automorphically equivalent know…
We give the first ExpTime (complexity-optimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHIQ when numbers are coded in unary. Our procedure is based on global state caching and integer linear feasibility checking.
This volume constitutes the pre-proceedings of the 28th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2018), held on 4-6th September 2018 in Frankfurt am Main, Germany and co-located with the 20th International Symposium on Principles and Practice of Declarative Programming (PPDP 2018) and the 26th International Workshop on Functional and Logic Programming (WFLP 2018).
This paper focuses on resolution in linguistic first order logic with truth value taken from linear symmetrical hedge algebra. We build the basic components of linguistic first order logic, including syntax and semantics. We present a resolution principle for our logic to resolve on two clauses having contradictory linguistic truth values. Since linguistic information is uncertain, inference in our linguistic logic is approximate. Therefore, we introduce the concept of reliability in order to capture the natural approximation of the resolution inference rule.
This work presents three increasingly expressive Dynamic Logics in which the programs are CCS processes (sCCS-PDL, CCS-PDL and XCCS-PDL). Their goal is to reason about properties of concurrent programs and systems described using CCS. In order to accomplish that, CCS's operators and constructions are added to a basic modal logic in order to create dynamic logics that are suitable for the description and verification of properties of communicating, concurrent and non-deterministic programs and systems, in a similar way as PDL is used for the sequential case. We provide complete axiomatizations …
We give the first cut-free ExpTime (optimal) tableau decision procedure for checking satisfiability of a knowledge base in the description logic SHI, which extends the description logic ALC with transitive roles, inverse roles and role hierarchies.
We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applicati…
A detailed exposition of foundations of a logic-algebraic model for reasoning with knowledge bases specified by propositional (Boolean) logic is presented. The model is conceived from the logical translation of usual derivatives on polynomials (on residue rings) which is used to design a new inference rule of algebro-geometric inspiration. Soundness and (refutational) completeness of the rule are proved. Some applications of the tools introduced in the paper are shown.
In this paper we present a dependency graph-based method for computing the various semantics of normal logic programs. Our method employs \textit{conjunction nodes} to unambiguously represent the dependency graph of normal logic programs. The dependency graph can be transformed suitably in a semantics preserving manner and re-translated into an equivalent normal logic program. This transformed normal logic program can be augmented with a few rules written in answer set programming (ASP), and the CLINGO system used to compute its answer sets. Depending on how these additional rules are coded in…
Concolic testing mixes symbolic and concrete execution to generate test cases covering paths effectively. Its benefits have been demonstrated for more than 15 years to test imperative programs. Other programming paradigms, like logic programming, have received less attention. In this paper, we present a concolic-based test generation method for logic programs. Our approach exploits SMT-solving for constraint resolution. We then describe the implementation of a concolic testing tool for Prolog and validate it on some selected benchmarks.
Let $Θ$ be a variety of algebras, $(H, Ψ, f)$ be a model, where $H$ is an algebra from $Θ$, $Ψ$ is a set of relation symbols $φ$, $f$ is an interpretation of all $φ$ in $H$. Let $X^0$ be an infinite set of variables, $Γ$ be a collection of all finite subsets in $X^0$ (collection of sorts), $\widetildeΦ$ be the multi-sorted algebra of formulas. These data define a knowledge base $KB(H,Ψ, f)$. In the paper the notion of isomorphism of knowledge bases is considered. We give sufficient conditions which provide isomorphism of knowledge bases. We also study the problem of necessary and sufficient co…
This paper presents a distributed agent-based automated theorem proving framework based on order-sorted first-order logic. Each agent in our framework has its own knowledge base, communicating to its neighboring agent(s) using message-passing algorithms. The communication language between agents is restricted in such a manner that each agent can only communicate to its neighboring agent(s) by means of their common language. In this paper we provide a refutation-complete report procedure for automated theorem proving in order-sorted first-order logic in a subclass of distributed agent-based net…
This volume contains the papers presented at the 19th Workshop on Logic- based methods in Programming Environments (WLPE'09), which was held in Pasadena, USA, on July 14th, 2009. WLPE aims at providing an informal meeting for researchers working on logic-based methods and tools which support program development and analy- sis. This year, we have continued and consolidated the shift in focus from en- vironmental tools for logic programming to logic-based environmental tools for programming in general, so that this workshop can be possibly interesting for a wider scientific community. All the pa…
Continuing to pursue a research direction that we already explored in connection with Gödel-Dummett logic and Ruspini partitions, we show here that Łukasiewicz logic is able to express the notion of pseudo-triangular basis of fuzzy sets, a mild weakening of the standard notion of triangular basis. En route to our main result we obtain an elementary, logic-independent characterisation of triangular bases of fuzzy sets.
