### Table 2. First-order model-checking results for the EMF EMF-1 EMF-2 EMF-3

"... In PAGE 9: ... It turned out that the EMF (as well as the ELV mentioned above) satis es the applicability conditions of rst-order model checking. Table2 lists some results for the EMF case study. It shows that due to the automatic transfor- mation, the veri cation e ort needed signi cantly less time and space compared with the results of Table 1.... In PAGE 17: ... The main point about all this is that once we have transformed system and speci cation, we can use a standard (symbolic) model checker to verify a system, as all rst-order operations are coded in the discrete boolean domain. The trans- formation has been implemented and used to produce the results of Table2 . In the following, we will present the algorithm in more detail, illustrate it via an example, and sketch the arguments proving its soundness.... ..."

### 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.... ..."

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### 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.... ..."

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### 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.... ..."

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### Table 5: First-order approximation. Economy with capital. pi and R reported in annualized percentage points.

2005

"... In PAGE 17: ....5.2 Optimal Inflation Persistence We complete the central focus of our study with numerical evidence that capital formation generates inertia in inflation. Table5 presents simulation-based moments using a first-order approximation to the policy function for the model with capital.6 6We also computed a second-order approximation for the model with capital but without habit and found very similar results.... ..."

### Table 1: Categorization of di erent RNN models according to the four aspects proposed above. First-order networks

"... In PAGE 5: ...re also others, e.g. (Watrous amp; Kuhn, 1992; Zeng, Goodman, amp; Smyth, 1993). On the other hand, some methods only use an approximation to the real gradient by truncating the computation of the backward recurrence. With these, some representative RNN models are categorized as shown in Table1 . As far as the models listed are concerned, it seems very consistent that rst-order networks follow the prediction paradigm using only positive examples for training whereas second-order networks follow the classi cation paradigm using both positive and negative examples for training.... In PAGE 6: ... Moreover, using the RTRL algorithm, each update of the forward-propagated gradient involves O(n2m4) terms. On the other hand, the rst-order RNN models listed in Table1 have serious problems mainly caused by the prediction paradigm, as will be discussed in detail in Section 2. Our objective in this study is to avoid the high computational requirements of second-order RNN models and the problems caused by the prediction paradigm in existing rst-order RNN models.... In PAGE 21: ...5{0.8 Initial weight range 0:2{ 1:7 (a) Parameter settings Training # strings 500{700 set % illegal strings 88{93% Test # strings 500 set % illegal strings 94% (b) Data sets Table1 1: The setting that led to 100% accuracy and required the smallest number of training epochs in learning the grammar with embedded structures. # hidden units 9 Learning rate 0.... In PAGE 21: ...2 Momentum 0.5 Initial weight range 1:7 Training # strings 600 set % illegal strings 90% # epochs 194 Table1 2: Results of learning the grammar with embedded structures. # trials 20 % converged trials 55% Recognition rate 96.... In PAGE 21: ...4{100% # epochs 194{3030 Average # epochs 787 As in the rst experiment, let us analyze more closely the hidden layer patterns learned by the network. As shown in Table1 3, the hidden layer patterns for substrings BP and BT are quite... In PAGE 22: ... In other words, the relevant path information has not been remembered correctly for the correct identi cation of the last symbol. Table1 3: Hidden layer patterns found in the ASCOC trained to learn the grammar with em- bedded structures. H Substring 0.... In PAGE 23: ...Table1 4: Hidden layer patterns found in the SRN trained to learn the grammar with embedded structures. H Substring 0.... In PAGE 24: ...Table1 5: Summary of results for di erent variants of SRN, including SRN and ASCOC as the two extreme cases. The symbol c refers to the existence of direct context-to-output connections, a refers to the use of auto-associative learning, and n refers to the use of negative examples for training in addition to positive examples.... ..."

### Table 3: Experiments With First Order Problems. Error Rate For The Train Check-out 3 Is An Average Of 3 Runs

1998

"... In PAGE 6: ....56% for EI, 8.47% for IE, and 4.62% for Neither. This comparison suggests that genetic search could be better suited to complex problems. Finally, Table3 reports the results of experiments aimed at con rming that G-NET (as its predecessor REGAL) is able to e ectively deal with more complex languages, such as predicate logic based ones. The rst row in Table 3 refers to the mutagenesis dataset, a challenging problem widely used in the ILP community for testing induction algorithms in First Order Logic (King et al.... In PAGE 6: ... Finally, Table 3 reports the results of experiments aimed at con rming that G-NET (as its predecessor REGAL) is able to e ectively deal with more complex languages, such as predicate logic based ones. The rst row in Table3 refers to the mutagenesis dataset, a challenging problem widely used in the ILP community for testing induction algorithms in First Order Logic (King et al., 1995).... In PAGE 7: ...oorly (Botta et al., 1997). G-NET has been run by discretizing every numeric attribute into a range of 30 intervals. As it appears from the last row in Table3 , it was able to nd two clauses which show an error rate around 11%. 7 DISCUSSION As it appears from the results reported above, G-NET is a very exible system, able to deal with many dif- ferent problems, producing good results.... ..."

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### Table 3: Experiments With First Order Problems. Error Rate For The Train Check-out 3 Is An Average Of 3

1998

"... In PAGE 6: ....56% for EI, 8.47% for IE, and 4.62% for Neither. This comparison suggests that genetic search could be better suited to complex problems. Finally, Table3 reports the results of experiments aimed at con rming that G-NET (as its predecessor REGAL) is able to e ectively deal with more complex languages, such as predicate logic based ones. The rst rowinTable 3 refers to the mutagenesis dataset, a challenging problem widely used in the ILP community for testing induction algorithms in First Order Logic (King et al.... In PAGE 6: ... Finally,Table 3 reports the results of experiments aimed at con rming that G-NET (as its predecessor REGAL) is able to e ectively deal with more complex languages, such as predicate logic based ones. The rst rowin Table3 refers to the mutagenesis dataset, a challenging problem widely used in the ILP community for testing induction algorithms in First Order Logic (King et al., 1995).... In PAGE 7: ...oorly (Botta et al., 1997). G-NET has been run by discretizing every numeric attribute into a range of 30 intervals. As it appears from the last rowin Table3 , it was able to nd two clauses whichshow an error rate around 11%. 7 DISCUSSION As it appears from the results reported above, G-NET isavery exible system, able to deal with many dif- ferent problems, producing good results.... ..."

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### Table 1. First-order logic part of the ODL calculus.

2006

"... In PAGE 10: ... Yet, rule applications for rst- order reasoning and program reasoning are not separated but intertwined. For rst-order and propositional logic standard rule schemata are listed in Table1 , including an integer induction scheme. Within the rules for the program logic part (Table 2), state update rules R29{R30 constitute a peculiarity of ODL and will be discussed after de ning rule applications.... ..."

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