### Table 1: Syntax of First-Order System

"... In PAGE 21: ...A A A A0 A0 A00 -trans A A00 A1 B1 : : : An Bn m n -record fl1:A1; : : : ; ln:Amg fl1:B1; : : : ; ln:Bng A0 A B B0 -! A!B A0!B0 A B -sig Sig (X)A Sig (X)B -class (Class I with meth; init) Sig I Table1 . Subtyping empty-ok ` ok ? ` ok ? ` A ok weaken-ok ?; x : A ` ok ?ok ? ` init : Rep ? ` m : Rep!I(Rep) class-ok ? ` (Class I with s; m) ok Table 2.... In PAGE 89: ... 2.1 Syntax The language, whose syntax appears in Table1 , derives largely from the object calculi of Abadi and Cardelli [1], Fisher, Honsell, and Mitchell [10], and Liquori [18]. The types of the language in- clude base types, function types, and object types.... ..."

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

"... In PAGE 2: ... 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 (complex) concepts and roles from (atomic) concept names and role names.... In PAGE 3: ...Table 1: Constructors in First-Order Description Logics Description logics di er 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 gt;, ?, universal quanti cation, conjunction and un- quali ed existential quanti cation 9R: gt;.... In PAGE 3: ... For instance, FLEU? is FL? with (full) existential quanti cation and disjunction. Description logics are interpreted on interpretations I = ( I; I), where I is a non-empty domain, and I is an interpretation function assigning subsets of I to concept names and binary relations over I to role names; complex concepts and roles are interpreted using the recipes speci ed in Table1 . The semantic value of an expression E in an interpretation I is simply the set EI.... In PAGE 4: ... First, item 1 is next to trivial. The semantics given in Table1 induces translations ( ) and ( ) taking concepts and roles, respectively, to formulas in a rst-order language whose signature consists of unary predicate symbols corresponding to atomic concepts names, and binary predicate symbols corresponding to... In PAGE 7: ... Hence, ALC lt; ALCR, ALCN, ALCRN. a Now, what do we need to do to adapt the above result for other exten- sions of FL? de ned by Table1 ? For logics less expressive than ALC we can not just use bisimulations, as such logics lack negation or disjunction, and these are automatically preserved under bisimulations; moreover, the proof of Theorem 3.3 uses the presence of the booleans in an essential way.... In PAGE 8: ...Table1 that are not in FL?, and examine which changes are needed to characterize the resulting logics. This is followed by a section in which we consider combina- tions of constructors.... In PAGE 20: ...7.6 Classifying an Arbitrary Description Logic To obtain a characterization of an arbitrary description logic (de ned from Table1 ), somply combine the observations listed in Sections 4.... In PAGE 20: ... Several comments are in order. First, the diagram does not mention all possible combinations of the constructors listed in Table1 . The reason for... ..."

### 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

### Table 2: Local Reduction Steps of First-Order System

"... In PAGE 20: ... The subtyping and typing rules of OO are given in ta- bles 1 and 3. The rules for well-formedness of contexts and types are given in Table2 . These are needed because there are types with terms as subexpressions, namely the class types.... In PAGE 21: ...A A A A0 A0 A00 -trans A A00 A1 B1 : : : An Bn m n -record fl1:A1; : : : ; ln:Amg fl1:B1; : : : ; ln:Bng A0 A B B0 -! A!B A0!B0 A B -sig Sig (X)A Sig (X)B -class (Class I with meth; init) Sig I Table 1. Subtyping empty-ok ` ok ? ` ok ? ` A ok weaken-ok ?; x : A ` ok ?ok ? ` init : Rep ? ` m : Rep!I(Rep) class-ok ? ` (Class I with s; m) ok Table2 . Well-formedness ?; x : A; ?0 ` ok var ?; x : A; ?0 ` x : A ? ` a : A A B sub ? ` a : B ?; x : A ` b : B !-I ? ` x:A: b : A!B ? ` f : A!B ? ` a : A !-E ? ` fa : B ? ` a1 : A1 : : : ? ` an : An record-I ? ` fl1 = a1; : : : ; ln = ang : fl1:A1; : : : ; ln:Ang ? ` a : fl1:A1; : : : ; ln:Ang li 2 fl1; : : : ; lng record-E ? ` a:li : Ai ? ` s : Rep ? ` m : Rep!I(Rep) object-I1 ? ` objectI hs; mi : Sig I ? ` s : Rep ? ` m : Rep!I(Rep) object-I2 ? ` objectI hs;mi : (Class I with s; m) For SC Sig I or SC (Class I with meth;init) with I(X) = f: : : ; l : A!B; : : :g: ? ` o : SC ? ` a : A l 2 Lobs object-E-obs ? ` o l(a) : B ? ` o : SC ? ` a : A l 2 Lmut object-E-mut ? ` o l(a) : SC Table 3.... In PAGE 89: ... Code for methods in method override and object extension is then coerced to expect such stripped objects. The relation in Table2 is extended to a one-step evaluation relation on programs by e ; e0 () 9e1;e2: e = E[e1] ^ e1 ; e2 ^ E[e2] = e0: We can prove Proposition 1 (Determinacy) The relation ; is a partial function. We use ; to denote the reflexive, transitive closure of ;.... ..."

### Table 1: The four classes of first-order reactions considered in the stochastic model

2005

"... In PAGE 7: ... Our aim is to separate the effects of various types of reactions (catalytic, conversion) on the distribution of the chemical species, and to this end we divide the set of all reactions, represented by the directed edges a29 a30 a25 a24a51 a22 a28 a25 a1a0 a25 a32a30a25a30a25a30 a25 a21a4 into four subsets corresponding to the following reactions: production from a constant source, degradation, conversion to another species, and production catalyzed by another species. These four types are summarized in Table1 . The first type represents an explicit input to the system,... In PAGE 7: ... Since all reactant and product complexes are species, the stoichiometric matrix is a5 a22 a8a5a6 a10 0 a12 if at least one reaction of type I, II, or IV is present, and a5 a22 a8a5a6 a12 if the system is closed. The corresponding incidence matrices for the different types are equally simple, and if we order the types as in Table1 , then a34 can be written as follows. a34 a22 a3 a6 a12 a5a4 a10 a6 a6 a6 a6 a12 a6 a4 a10 a7a6 a6 a6 a6 a34 a18 a26 a6 a6 a6 a6 a6 a6 a12 a5a4 a10 a9a8 (21) where a4 a10 a22 a3 a8a28 a25 a28 a25 a15 a15a17a15 a25 a28 a8 , a6 is the identity matrix of the appropriate dimension, and a34 a18 a26 a6 is the incidence matrix for the conversion network.... In PAGE 15: ...1 Open and closed conversion systems In the context of first-order reaction dynamics, catalytic systems are necessarily open because they involve production from a source catalyzed by a time-dependent species (cf. Table1 ). Thus the comparison of open and closed systems can only be made for those in which there are no catalytic reactions.... ..."

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### TABLE 7. First-order transition probability matrix for the database of the M.Sc.class in 1994.*

1997

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### Table 2. First-Order Correlation Detection for 10-Mbyte Samples ( room temperature)

"... In PAGE 6: ... Again, this calculation on a sample collected from a hardware RNG will vary from 0 slightly, but the use of calculated bounds allows statistical confidence that the ideal can be approached over the infinite set. Table2... ..."

### Table 2: First order trees, their expression and coe cient for the Magnus expansion.

in We

"... In PAGE 12: ...order greater than three, the order conditions of the Magnus expansion are generally represented by a collection of order trees, not all of them independent. Table2 illustrates the trees and relative coe cients to order up to ve.1 For higher order conditions, we refer the reader to [17].... ..."

### Table 2. Apparent psuedo-first order rate constants for of benzo-

2005

"... In PAGE 6: ...900 h in all sediment types and redox conditions (Fig. 3). standards, (ii) interpretation of molecular ions and logical It is clear from Fig. 3 and Table2 that BT was transformed fragment losses, and (iii) reference to mass spectra from pub- in stirred marine sediments, with rates significantly higher lished libraries and databases. under oxidized conditions except for the course-grained Sediment III, which showed similar transformation rates under both oxidized and reduced conditions.... ..."