### Table 3. First-Order Sensitivity Derivatives

2001

"... In PAGE 6: ...ard-mode (Eq. (3) and (4)) and the reverse-mode (Eq. (6) and (7)) approaches. The calculated FO SDs from a hand-differentiated incremental-iterative (HDII) im- plementation of these two approaches are presented in Table3 , where the results are seen to agree, as ex- pected. ... In PAGE 8: ... American Institute of Aeronautics and Astronautics The FO SDs presented in Table3 have been thoroughly verified for accuracy through a meticulous implementa- tion of the method of central finite-differences, where agreement to six significant digits or greater is noted in all comparisons. The SO Method 3 is implemented by application (in the forward-mode) of ADIFOR to appropriate pieces of the FORTRAN code used earlier for hand-differentiated forward-mode calculation of the FO SDs.... ..."

Cited by 3

### Table 1: Correspondence Between MEBN and First-Order Logic Syntactic Elements

2003

"... In PAGE 8: ... The value of RV X when applied to instance V is written X(V); the expression X(V)=O denotes that RV X has outcome O when applied to instance V. Table1 shows the correspondence between the above MEBN syntactic elements and syntactic elements of first-order logic. Table 1 also shows MEBN constructs corresponding to logical connectives, nested function application, and quantification.... In PAGE 8: ... Table 1 shows the correspondence between the above MEBN syntactic elements and syntactic elements of first-order logic. Table1 also shows MEBN constructs corresponding to logical connectives, nested function application, and quantification. In first-order logic, logical connectives are used to compose terms into sentences.... ..."

Cited by 2

### 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 2.1. The Curry-Howard correspondence between natural deduction and typed calculus

in Proof Theory

### Table 1: Propositional Logic tests, Pelletier 1-17 3.2 First Order Tests

1995

"... In PAGE 9: ... 3.1 Propositional Tests Table1 compares the performance of `leanKE apos; with leanTAP [Beckert and Posegga, 1995] for the rst 17 Pelletier problems [Pelletier, 1986]. The times shown are calculated as an average of 25 runs on the platform described above, the space required (in terms of formulas derived and branches closed) is, of course, constant.... ..."

Cited by 10

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

1998

"... In PAGE 2: ... 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.... ..."

Cited by 52

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

1998

"... In PAGE 2: ... 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.... ..."

Cited by 52

### Table 1 First-order axioms for out-bu ered agents

"... In PAGE 5: ...4 Axioms for asynchrony The in- and out-bu ered agents have an interesting direct characterization in terms of properties of labeled transition systems. Up to weak bisimulation, the out-bu ered, respectively, in-bu ered agents are precisely those that satisfy the axioms in Table1 , respectively, Table 2. In stating these axioms, we use the convention that variables are implicitly existentially quanti ed if they occur only on the right-hand-side of an implication, and all other variables are implicitly universally quanti ed.... ..."