A First-Order Logic of Limited Belief Based on Possible Worlds

2020 ◽  
Author(s):  
Gerhard Lakemeyer ◽  
Hector J. Levesque
Keyword(s):  
Knowledge Base ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Order Logic ◽  
First Order Logic ◽  
Possible World ◽  
Possible World Semantics ◽  
First Order ◽  
Epistemic States

In a recent paper Lakemeyer and Levesque proposed a first-order logic of limited belief to characterize the beliefs of a knowledge base (\KB). Among other things, they show that their model of belief is expressive, eventually complete, and tractable. This means, roughly, that a \KB\ may consist of arbitrary first-order sentences, that any sentence which is logically entailed by the \KB\ is eventually believed, given enough reasoning effort, and that reasoning is tractable under reasonable assumptions. One downside of the proposal is that epistemic states are defined in terms of sets of clauses, possibly containing variables, giving the logic a distinct syntactic flavour compared to the more traditional possible-world semantics found in the literature on epistemic logic. In this paper we show that the same properties as above can be obtained by defining epistemic states as sets of three-valued possible worlds. This way we are able to shed new light on those properties by recasting them using the more familiar notion of truth over possible worlds.

scholarly journals Conservative Intensional Extension of Tarski's Semantics

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zoran Majkić
Keyword(s):  
Free Algebra ◽  
Possible Worlds ◽  
Order Logic ◽  
Point Of View ◽  
Relational Algebra ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
Possible World ◽  
Logic Formula ◽  
First Order

We considered an extension of the first-order logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term “intension” derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the “extension” of an idea consists of the subjects to which the idea applies, and the “intension” consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

Logical Tools

2021 ◽  
pp. 14-52
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
System P

This chapter presents the system of classical higher-order modal logic which will be employed throughout this book. Nothing more than a passing familiarity with classical first-order logic and standard systems of modal logic is presupposed. We offer some general remarks about the kind of commitment involved in endorsing this logic, and motivate some of its more non-standard features. We also discuss how talk about possible worlds can be represented within the system.

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 Encoding hybridized institutions into first-order logic

2014 ◽  
Vol 26 (5) ◽  
pp. 745-788 ◽  
Author(s):  
RĂZVAN DIACONESCU ◽  
ALEXANDRE MADEIRA
Keyword(s):  
Possible Worlds ◽  
Sufficient Conditions ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Institution Theory ◽  
Wide Range ◽  
Characteristic Features ◽  
Verification Process ◽  
Base Logic

A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridized institutions into (many-sorted) first-order logic (abbreviated $\mathcal{FOL}$) as a ‘hybridization’ process of abstract encodings of institutions into $\mathcal{FOL}$, which may be seen as an abstraction of the well-known standard translation of modal logic into $\mathcal{FOL}$. The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover, we consider the so-called theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accommodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridization process, which provides the possibility to shift a formal verification process from the hybridized institution to $\mathcal{FOL}$.

scholarly journals On the Expressiveness of Levesque's Normal Form

2008 ◽  
Vol 31 ◽  
pp. 259-272
Author(s):  
Y. Liu ◽  
G. Lakemeyer
Keyword(s):  
Normal Form ◽  
Knowledge Base ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Method ◽  
Closed World Assumption ◽  
First Order ◽  
Closed World ◽  
Polynomial Size

Levesque proposed a generalization of a database called a proper knowledge base (KB), which is equivalent to a possibly infinite consistent set of ground literals. In contrast to databases, proper KBs do not make the closed-world assumption and hence the entailment problem becomes undecidable. Levesque then proposed a limited but efficient inference method V for proper KBs, which is sound and, when the query is in a certain normal form, also logically complete. He conjectured that for every first-order query there is an equivalent one in normal form. In this note, we show that this conjecture is false. In fact, we show that any class of formulas for which V is complete must be strictly less expressive than full first-order logic. Moreover, in the propositional case it is very unlikely that a formula always has a polynomial-size normal form.

scholarly journals Creation of a consulting tool and implementation of an ontology for a Master’s Degree Program in Computer Sciences

2018 ◽  
Vol 19 (1) ◽  
pp. 29-38
Author(s):  
Cecilia Reyes Peña ◽  
Mireya Tovar Vidal ◽  
Concepción Stephanie Vázquez González
Keyword(s):  
Knowledge Base ◽  
Master's Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Class Diagram ◽  
Design Phase ◽  
Master Program ◽  
First Order ◽  
Python Language ◽  
Master’S Degree

In this paper, a manual ontology for a Computer Sciences Master program constructed, that uses some elements from the METHONTOLOGY, Grüninger and Fox, and Bravo’s methodologies, is presented. A series of steps to identify and represent the Master’s Degree program’s knowledge base has been followed. Afterwards, first order logic axioms and competency questions to evaluate the ontology are used. The development of a module written in Python language is used for evaluating the ontology through competency questions defined during design phase. This module is flexible enough to present predefined or defined questions by the user in running time and to obtain results to the queries representing the competency questions. Elements as a hierarchy class diagram and a description of the relations and attributes are used in this ontology’s construction. Keywords: Ontology; Python tool; SPARQL language.

scholarly journals Counting Query Answers over a DL-Lite Knowledge Base

2020 ◽  
Author(s):  
Diego Calvanese ◽  
Julien Corman ◽  
Davide Lanti ◽  
Simon Razniewski
Keyword(s):  
Knowledge Base ◽  
Data Access ◽  
Upper Bounds ◽  
Order Logic ◽  
Query Answering ◽  
First Order Logic ◽  
Management Systems ◽  
Query Rewriting ◽  
Data Complexity ◽  
First Order

Counting answers to a query is an operation supported by virtually all database management systems. In this paper we focus on counting answers over a Knowledge Base (KB), which may be viewed as a database enriched with background knowledge about the domain under consideration. In particular, we place our work in the context of Ontology-Mediated Query Answering/Ontology-based Data Access (OMQA/OBDA), where the language used for the ontology is a member of the DL-Lite family and the data is a (usually virtual) set of assertions. We study the data complexity of query answering, for different members of the DL-Lite family that include number restrictions, and for variants of conjunctive queries with counting that differ with respect to their shape (connected, branching, rooted). We improve upon existing results by providing PTIME and coNP lower bounds, and upper bounds in PTIME and LOGSPACE. For the LOGSPACE case, we have devised a novel query rewriting technique into first-order logic with counting.

scholarly journals Possibility Semantics for Intuitionistic Logic

2004 ◽  
Vol 2 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Intuitionistic Logic ◽  
Possible Worlds ◽  
Predicate Logic ◽  
Order Logic ◽  
Semantic Interpretation ◽  
First Order Logic ◽  
First Order ◽  
Partial Indices ◽  
Closure Conditions ◽  
First Order Predicate Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

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