scholarly journals A Knowledge Level Account of Forgetting

2017 ◽  
Vol 60 ◽  
pp. 1165-1213 ◽  
Author(s):  
James P. Delgrande
Keyword(s):  
Belief Change ◽  
Description Logics ◽  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
Consequence Relation ◽  
First Order ◽  
Formal Systems ◽  
High Level ◽  
Main Definition

Forgetting is an operation on knowledge bases that has been addressed in different areas of Knowledge Representation and with respect to different formalisms, including classical propositional and first-order logic, modal logics, logic programming, and description logics. Definitions of forgetting have been expressed in terms of manipulation of formulas, sets of postulates, isomorphisms between models, bisimulations, second-order quantification, elementary equivalence, and others. In this paper, forgetting is regarded as an abstract belief change operator, independent of the underlying logic. The central thesis is that forgetting amounts to a reduction in the language, specifically the signature, of a logic. The main definition is simple: the result of forgetting a portion of a signature in a theory is given by the set of logical consequences of this theory over the reduced language. This definition offers several advantages. Foremost, it provides a uniform approach to forgetting, with a definition that is applicable to any logic with a well-defined consequence relation. Hence it generalises a disparate set of logic-specific definitions with a general, high-level definition. Results obtained in this approach are thus applicable to all subsumed formal systems, and many results are obtained much more straightforwardly. This view also leads to insights with respect to specific logics: for example, forgetting in first-order logic is somewhat different from the accepted approach. Moreover, the approach clarifies the relation between forgetting and related operations, including belief contraction.

Quine on Ontological Commitment in Light of Predicate-Functor Logic

2020 ◽  
pp. 56-81
Author(s):  
John Collins
Keyword(s):  
Ontological Commitment ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Formal Systems

Quine is renowned for offering a criterion of ontological commitment in terms of the values of bound variables. Simultaneously, however, Quine had a long-standing interest in formal systems that dispense with variables and quantifiers that permit variables to be explained away. The chapter seeks to resolve this often-neglected tension in a way that is sensitive to Quine’s allegiance to first-order logic, but is somewhat relaxed about the connection between semantics (/truth) and ontology.

scholarly journals Characterizing Boundedness in Chase Variants

2020 ◽  
Vol 21 (1) ◽  
pp. 51-79
Author(s):  
STATHIS DELIVORIAS ◽  
MICHEL LECLÈRE ◽  
MARIE-LAURE MUGNIER ◽  
FEDERICO ULLIANA
Keyword(s):  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Universal Models ◽  
Knowing That ◽  
Significant Interest ◽  
Horn Rules ◽  
Boundedness Problem ◽  
Undecidable Problem

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.

Logics for the Semantic Web

2011 ◽  
pp. 24-43
Author(s):  
J. Bruijn
Keyword(s):  
Semantic Web ◽  
Logic Programming ◽  
Description Logic ◽  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Horn Logic ◽  
First Order ◽  
Logical Language ◽  
Rule Languages

This chapter introduces a number of formal logical languages which form the backbone of the Semantic Web. They are used for the representation of both ontologies and rules. The basis for all languages presented in this chapter is the classical first-order logic. Description logics is a family of languages which represent subsets of first-order logic. Expressive description logic languages form the basis for popular ontology languages on the Semantic Web. Logic programming is based on a subset of first-order logic, namely Horn logic, but uses a slightly different semantics and can be extended with non-monotonic negation. Many Semantic Web reasoners are based on logic programming principles and rule languages for the Semantic Web based on logic programming are an ongoing discussion. Frame Logic allows object-oriented style (frame-based) modeling in a logical language. RuleML is an XML-based syntax consisting of different sublanguages for the exchange of specifications in different logical languages over the Web.

scholarly journals Towards closed world reasoning in dynamic open worlds

2010 ◽  
Vol 10 (4-6) ◽  
pp. 547-563 ◽  
Author(s):  
MARTIN SLOTA ◽  
JOÃO LEITE
Keyword(s):  
Semantic Web ◽  
State Of The Art ◽  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
Static Case ◽  
First Order ◽  
Closed World ◽  
Hybrid Knowledge ◽  
First Time

AbstractThe need for integration of ontologies with nonmonotonic rules has been gaining importance in a number of areas, such as the Semantic Web. A number of researchers addressed this problem by proposing a unified semantics forhybrid knowledge basescomposed of both an ontology (expressed in a fragment of first-order logic) and nonmonotonic rules. These semantics have matured over the years, but only provide solutions for the static case when knowledge does not need to evolve.In this paper we take a first step towards addressing the dynamics of hybrid knowledge bases. We focus on knowledge updates and, considering the state of the art of belief update, ontology update and rule update, we show that current solutions are only partial and difficult to combine. Then we extend the existing work on ABox updates with rules, provide a semantics for such evolving hybrid knowledge bases and study its basic properties.To the best of our knowledge, this is the first time that an update operator is proposed for hybrid knowledge bases.

scholarly journals Summing up Smart Transitions

2021 ◽  
pp. 317-340
Author(s):  
Neta Elad ◽  
Sophie Rain ◽  
Neil Immerman ◽  
Laura Kovács ◽  
Mooly Sagiv
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Smart Contracts ◽  
Automate Reasoning ◽  
Arbitrary Length ◽  
First Order ◽  
Smt Solvers ◽  
Transfer Operation ◽  
Global Invariants ◽  
High Level

AbstractSome of the most significant high-level properties of currencies are the sums of certain account balances. Properties of such sums can ensure the integrity of currencies and transactions. For example, the sum of balances should not be changed by a transfer operation. Currencies manipulated by code present a verification challenge to mathematically prove their integrity by reasoning about computer programs that operate over them, e.g., in Solidity. The ability to reason about sums is essential: even the simplest ERC-20 token standard of the Ethereum community provides a way to access the total supply of balances.Unfortunately, reasoning about code written against this interface is non-trivial: the number of addresses is unbounded, and establishing global invariants like the preservation of the sum of the balances by operations like transfer requires higher-order reasoning. In particular, automated reasoners do not provide ways to specify summations of arbitrary length.In this paper, we present a generalization of first-order logic which can express the unbounded sum of balances. We prove the decidablity of one of our extensions and the undecidability of a slightly richer one. We introduce first-order encodings to automate reasoning over software transitions with summations. We demonstrate the applicability of our results by using SMT solvers and first-order provers for validating the correctness of common transitions in smart contracts.

Principles and practice in verifying rule-based systems

1992 ◽  
Vol 7 (2) ◽  
pp. 115-141 ◽  
Author(s):  
Alun D. Preece ◽  
Rajjan Shinghal ◽  
Aïda Batarekh
Keyword(s):  
Expert System ◽  
Knowledge Base ◽  
State Of The Art ◽  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
Rule Based ◽  
First Order ◽  
Rule Bases ◽  
System Knowledge

AbstractThis paper surveys the verification of expert system knowledge bases by detecting anomalies. Such anomalies are highly indicative of errors in the knowledge base. The paper is in two parts. The first part describes four types of anomaly: redundancy, ambivalence, circularity, and deficiency. We consider rule bases which are based on first-order logic, and explain the anomalies in terms of the syntax and semantics of logic. The second part presents a review of five programs which have been built to detect various subsets of the anomalies. The four anomalies provide a framework for comparing the capabilities of the five tools, and we highlight the strengths and weaknesses of each approach. This paper therefore provides not only a set of underlying principles for performing knowledge base verification through anomaly detection, but also a survey of the state-of-the-art in building practical tools for carrying out such verification. The reader of this paper is expected to be familiar with first-order logic.

scholarly journals Probabilistic Description Logics for Subjective Uncertainty

2017 ◽  
Vol 58 ◽  
pp. 1-66 ◽  
Author(s):  
Victor Gutierrez-Basulto ◽  
Jean Christoph Jung ◽  
Carsten Lutz ◽  
Lutz Schröder
Keyword(s):  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Two Dimensional ◽  
Subjective Probabilities ◽  
Subjective Uncertainty ◽  
First Order ◽  
New Family ◽  
Probabilistic Description

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable.

Robot Task Planning Using First-Order Logic

2000 ◽  
Author(s):  
Lei Yan ◽  
K. Krishnamurthy
Keyword(s):  
Inference Engine ◽  
Knowledge Bases ◽  
Order Logic ◽  
First Order Logic ◽  
Task Planning ◽  
Inference Method ◽  
First Order ◽  
Simulation Results ◽  
Robot Task ◽  
Domain Information

Abstract Task planning for a robot operating in an unknown environment using first-order logic is considered in this study. The approach is to use one agent to simulate the robot and a second agent to simulate the environment. Both agents employ knowledge bases and an inference engine. The rules for the knowledge bases are developed using first-order logic and the inference method is based on hyper-resolution. A weighting scheme is used by the robot to decide on the action to be taken. After enough domain information is obtained, a task planner, which is also based on rules, is employed. Simulation results validating the methodology are presented for a robot moving inside a warehouse with four rooms. In this example, the environment is initially unknown to the robot. But after mapping the environment, the robot can efficiently plan tasks such as moving an object from one room to another.

Mediating RDF/S Queries to Relational and XML Sources

2009 ◽  
pp. 596-614 ◽  
Author(s):  
I. Koffina ◽  
G. Serfiotis ◽  
V. Christophides ◽  
V. Tannen
Keyword(s):  
Semantic Web ◽  
Relational Databases ◽  
Large Scale ◽  
Order Logic ◽  
Data Sources ◽  
First Order Logic ◽  
Main Design ◽  
First Order ◽  
High Level ◽  
The Web

Semantic Web (SW) technology aims to facilitate the integration of legacy data sources spread worldwide. Despite the plethora of SW languages (e.g., RDF/S, OWL) recently proposed for supporting large-scale information interoperation, the vast majority of legacy sources still rely on relational databases (RDB) published on the Web or corporate intranets as virtual XML. In this article, we advocate a first-order logic framework for mediating high-level queries to relational and/or XML sources using community ontologies expressed in a SW language such as RDF/S. We describe the architecture and reasoning services of our SW integration middleware, termed SWIM, and we present the main design choices and techniques for supporting powerful mappings between different data models, as well as reformulation and optimization of queries expressed against mediator ontologies and views.

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