### Tableau Caching for Description Logics

2006

Cited by 4

### Table 1: Constructors in First-Order Description Logics

1999

"... In PAGE 3: ... The for- mer are interpreted as subsets of a given domain, and the latter as binary relations on the domain. Table1 lists constructors that allow one to build #28complex#29 concepts and roles from #28atomic#29 concept names and role names. For instance, the concept Man u9Child:#3Eu8Child:Human denotes the set of... In PAGE 3: ...Table 1: Constructors in First-Order Description Logics Description logics di#0Ber in the constructions they admit. By combining constructors taken from Table1 , two well-known hierarchies of description logics may be obtained. The logics we consider here are extensions of FL , ; this is the logic with #3E, ?, universal quanti#0Ccation, conjunction and un- quali#0Ced existential quanti#0Ccation 9R:#3E.... In PAGE 4: ... For instance, FLEU , is FL , with #28full#29 existential quanti#0Ccation and disjunction. Description logics are interpreted on interpretations I =#28#01 I ; #01 I #29, where #01 I is a non-empty domain, and #01 I is an interpretation function assigning subsets of #01 I to concept names and binary relations over #01 I to role names; complex concepts and roles are interpreted using the recipes speci#0Ced in Table1 . The semantic value of an expression E in an interpretation I is simply the set E I .... In PAGE 4: ...ome page at http:#2F#2Fdl.kr.org#2Fdl#2F. 3 De#0Cning Expressive Power In this section we de#0Cne our notion of expressive power, and explain our method for determining the expressivepower of a given description logic. Our aim in this paper is to determine the expressive power of concept expressions of every extension of FL , and AL that can be de#0Cned using the constructors in Table1 . Wesay that a logic L 1 is at least as expressive as a logic L 2 if for every concept expression in L 2 there is an equivalent concept expression in L 1 ; notation: L 2 #14 L 1 .... In PAGE 4: ... First, item 1 is next to trivial. The semantics given in Table1 induces translations #28#01#29 #1C and #28#01#29 #1B taking concepts and roles, respectively, to formulas in a #0Crst-order language whose signature consists of unary predicate symbols corresponding... In PAGE 7: ... Hence, ALC #3C ALCR, ALCN, ALCRN. a Now, what do we need to do to adapt the above result for other exten- sions of FL , de#0Cned by Table1 ? For logics less expressive than ALC we... In PAGE 8: ... We #0Crst consider the `minimal apos; logic FL , ,char- acterize its concepts semantically, and use the characterization to separate FL , from richer logics. After that, we treat each of the constructors in Table1 that are not in FL , , and examine which changes are needed to characterize the concepts de#0Cnable in the resulting logics. This is followed by a brief section in which we consider combinations of constructors.... In PAGE 18: ... FL , FLE , FLU , AL FLN , FLR , FLEU , ALE FLEN , FLER , ALU FLUN , FLUR , ALN ALR FLNR , ALC FLEUN , FLEUR , ALEN ALER FLENR , ALUN ALUR FLUNR , ALNR ALCN ALCR FLEUNR , ALENR ALUNR ALCNR Figure 2: Classifying Description Logics Several comments are in order. First, the diagram does not mention all possible combinations of the constructors listed in Table1 . The reason for... In PAGE 21: ... A second important di#0Berence between Baader apos;s work and ours lies in the type of results that have been obtained. Baader only establishes a small number of separation results, whereas we provide a complete classi#0Ccation of all languages de#0Cnable using the constructors in Table1 . More importantly, our separation results are based on semantic characterizations; this gives a deeper insightinto the properties of logics than mere separation results.... In PAGE 35: ... B.6 Classifying an Arbitrary Description Logic To obtain a characterization of an arbitrary description logic #28de#0Cned from Table1 #29, simply combine the observations listed in Sections B.... ..."

Cited by 3

### Tableau Algorithms for Description Logics 23

2000

Cited by 119

### Tableau Algorithms for Description Logics 25

2000

Cited by 119

### Table 1. Description logic ALC

1997

"... In PAGE 2: ... We use the notation of DLs, focusing on the well-known DL ALC, corresponding to the standard PDL with atomic programs only. Table1 summarizes the syntax and the semantics of ALC and the corresponding PDL. In addition, weusethetwo nonmonotonic modal operators: a minimal knowledge operator K and a default assumption operator A.... ..."

Cited by 30

### Table 2: Final results on test set. The first set of results show our HPSG baseline and HPSG with soft dependency constraints using three different sources of dependency constraints. The second set of results show the accuracy of the same parsers when gold part-of-speech tags are used. The third set of results is from existing published models on the same data.

2007

Cited by 2

### Table 2: Final results on test set. The first set of results show our HPSG baseline and HPSG with soft dependency constraints using three different sources of dependency constraints. The second set of results show the accuracy of the same parsers when gold part-of-speech tags are used. The third set of results is from existing published models on the same data.

2007

Cited by 2

### Table 2 Descriptive Statistics of asset based factors

2006

"... In PAGE 10: ... The summary statistics for these premia are given in Table 2. lt; Insert Table2 here gt; B. Option based factors The logic underlying the focus on option-based strategies to explain hedge fund returns is twofold.... ..."

### Table 1: Selectional restrictions in the parser

"... In PAGE 3: ...Table 1: Selectional restrictions in the parser The parser stores a subcategorization frame for each verb and adjective in the lexicon, indicating its obligatory arguments. It also speci#0Ces selectional re- strictions, classi#0Ccations of entities which are appro- priate for the argument slots #28 Table1 #29. For example, the verb todokeru #28#5Cdeliver quot;#29 is assigned the subcate- gorization frame below.... In PAGE 4: ... Parse U i and construct a #0Crst-order propositional semantic representation #1E i suchasgo to#28robot, Loc#29. Each argument in #1E i is associated with a set of selectional restrictions #28 Table1 #29. Ar- guments which are Prolog-style logical variables such as Loc correspond to pronouns #28zero pro- nouns in the case of Japanese#29.... ..."