scholarly journals Making Cross Products and Guarded Ontology Languages Compatible

2017 ◽  
Author(s):  
Pierre Bourhis ◽  
Michael Morak ◽  
Andreas Pieris
Keyword(s):  
Description Logics ◽  
Order Logic ◽  
Query Answering ◽  
First Order Logic ◽  
First Order ◽  
Ontology Languages ◽  
Guarded Fragment

Cross products form a useful modelling tool that allows us to express natural statements such as "elephants are bigger than mice", or, more generally, to define relations that connect every instance in a relation with every instance in another relation. Despite their usefulness, cross products cannot be expressed using existing guarded ontology languages, such as description logics (DLs) and guarded existential rules. The question that comes up is whether cross products are compatible with guarded ontology languages, and, if not, whether there is a way of making them compatible. This has been already studied for DLs, while for guarded existential rules remains unanswered. Our goal is to give an answer to the above question. To this end, we focus on the guarded fragment of first-order logic (which serves as a unifying framework that subsumes many of the aforementioned ontology languages) extended with cross products, and we investigate the standard tasks of satisfiability and query answering. Interestingly, we isolate relevant fragments that are compatible with cross products.

scholarly journals The Triguarded Fragment of First-Order Logic

10.29007/m8ts ◽  
2018 ◽  
Author(s):  
Sebastian Rudolph ◽  
Mantas Simkus
Keyword(s):  
Description Logics ◽  
Past Research ◽  
Order Logic ◽  
First Order Logic ◽  
Crucial Importance ◽  
Natural Assumption ◽  
First Order ◽  
Natural Extensions ◽  
Guarded Fragment ◽  
Definition Of

Past research into decidable fragments of first-order logic (FO) has produced two very prominent fragments: the guarded fragment GF, and the two-variable fragment FO2. These fragments are of crucial importance because they provide significant insights into decidabil- ity and expressiveness of other (computational) logics like Modal Logics (MLs) and various Description Logics (DLs), which play a central role in Verification, Knowledge Represen- tation, and other areas. In this paper, we take a closer look at GF and FO2, and present a new fragment that subsumes them both. This fragment, called the triguarded fragment (denoted TGF), is obtained by relaxing the standard definition of GF: quantification is required to be guarded only for subformulae with three or more free variables. We show that, in the absence of equality, satisfiability in TGF is N2ExpTime-complete, but becomes NExpTime-complete if we bound the arity of predicates by a constant (a natural assumption in the context of MLs and DLs). Finally, we observe that many natural extensions of TGF, including the addition of equality, lead to undecidability.

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 Undecidable relativizations of algebras of relations

1999 ◽  
Vol 64 (2) ◽  
pp. 747-760 ◽  
Author(s):  
Szabolcs Mikulás ◽  
Maarten Marx
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Similarity Type ◽  
First Order ◽  
Equational Theories ◽  
Algebras Of Relations ◽  
Guarded Fragment ◽  
Cylindric Set Algebras

AbstractIn this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

scholarly journals The Triguarded Fragment with Transitivity

10.29007/z359 ◽  
2020 ◽  
Author(s):  
Emanuel Kieronski ◽  
Adam Malinowski
Keyword(s):  
Decision Procedure ◽  
Order Logic ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
First Order ◽  
Free Variables ◽  
Guarded Fragment

The triguarded fragment of first-order logic is an extension of the guarded fragment in which quantification for subformulas with at most two free variables need not be guarded. Thus, it unifies two prominent decidable logics: the guarded fragment and the two-variable fragment. Its satisfiability problem is known to be undecidable in the presence of equality, but becomes decidable when equality is forbidden. We consider an extension of the tri- guarded fragment without equality by transitive relations, allowing them to be used only as guards. We show that the satisfiability problem for the obtained formalism is decidable and 2-ExpTime-complete, that is, it is of the same complexity as for the analogous exten- sion of the classical guarded fragment. In fact, in our satisfiability test we use a decision procedure for the latter as a subroutine. We also show how our approach, consisting in exploiting some existing results on guarded logics, can be used to reprove some known facts, as well as to derive some other new results on triguarded logics.

Interpolation for extended modal languages

2005 ◽  
Vol 70 (1) ◽  
pp. 223-234 ◽  
Author(s):  
Balder ten Cate
Keyword(s):  
Difference Operator ◽  
Relation Algebra ◽  
Order Logic ◽  
Hybrid Logic ◽  
First Order Logic ◽  
Modal Language ◽  
First Order ◽  
Language L ◽  
Guarded Fragment

AbstractSeveral extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language L′ is the least expressive extension of L with interpolation. For instance, let ℳ(D) be the extension of the basic modal language with a difference operator [7], First-order logic is the least expressive extension of ℳ(D) with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.

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.

scholarly journals SOME MODEL THEORY OF GUARDED NEGATION

2018 ◽  
Vol 83 (04) ◽  
pp. 1307-1344
Author(s):  
VINCE BÁRÁNY ◽  
MICHAEL BENEDIKT ◽  
BALDER TEN CATE
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Description Logics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Beth Definability ◽  
Order Translations ◽  
Certain Answers

AbstractThe Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this article we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics.

scholarly journals A Generalisation of AGM Contraction and Revision to Fragments of First-Order Logic

2019 ◽  
Vol 64 ◽  
pp. 147-179
Author(s):  
Zhiqiang Zhuang ◽  
Zhe Wang ◽  
Kewen Wang ◽  
James Delgrande
Keyword(s):  
Propositional Logic ◽  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Epistemic Entrenchment ◽  
First Order ◽  
General Logic ◽  
Horn Fragment ◽  
Point Of Departure ◽  
Recovery Postulate

AGM contraction and revision assume an underlying logic that contains propositional logic. Consequently, this assumption excludes many useful logics such as the Horn fragment of propositional logic and most description logics. Our goal in this paper is to generalise AGM contraction and revision to (near-)arbitrary fragments of classical first-order logic. To this end, we first define a very general logic that captures these fragments. In so doing, we make the modest assumptions that a logic contains conjunction and that information is expressed by closed formulas or sentences. The resulting logic is called first-order conjunctive logic or FC logic for short. We then take as the point of departure the AGM approach of constructing contraction functions through epistemic entrenchment, that is the entrenchment-based contraction. We redefine entrenchment-based contraction in ways that apply to any FC logic, which we call FC contraction. We prove a representation theorem showing its compliance with all the AGM contraction postulates except for the controversial recovery postulate. We also give methods for constructing revision functions through epistemic entrenchment which we call FC revision; which also apply to any FC logic. We show that if the underlying FC logic contains tautologies then FC revision complies with all the AGM revision postulates. Finally, in the context of FC logic, we provide three methods for generating revision functions via a variant of the Levi Identity, which we call contraction, withdrawal and cut generated revision, and explore the notion of revision equivalence. We show that withdrawal and cut generated revision coincide with FC revision and so does contraction generated revision under a finiteness condition.

scholarly journals The Complexity of Circumscription in DLs

2009 ◽  
Vol 35 ◽  
pp. 717-773 ◽  
Author(s):  
P. A. Bonatti ◽  
C. Lutz ◽  
F. Wolter
Keyword(s):  
Description Logics ◽  
Default Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Alternative Approach ◽  
Default Rules ◽  
Logic Description ◽  
Interesting Alternative ◽  
Natural Way

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 and under different constraints on the sets of minimized, fixed, and varying predicates, and pinpoint the exact computational complexity of reasoning for DLs ranging from ALC to ALCIO and ALCQO. When the minimized and fixed predicates include only concept names but no role names, then reasoning is complete for NExpTime^NP. It becomes complete for NP^NExpTime when the number of minimized and fixed predicates is bounded by a constant. If roles can be minimized or fixed, then complexity ranges from NExpTime^NP to undecidability.

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