### Table 3. Six new derivation schemas used to test generalization of the natural deduction network.

1994

"... In PAGE 20: ... The network was not required to generalize to derivations in which the inferential steps would be put together in different orders. In a further test of the generalization capacities of this last network, I tested it on a new set of derivations constructed following the patterns shown in Table3 . These derivation patterns are modeled on those used in the training, but introduce variations.... In PAGE 21: ... -C. --------------------------------------------- Insert Table3 about here --------------------------------------------- There were a total of 252 inferential steps in these derivations, of which the network was correct on 198 (78.6% correct).... ..."

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### Table 3. Six new derivation schemas used to test generalization of the natural deduction network.

1994

"... In PAGE 23: ... The network was not required to generalize to derivations in which the inferential steps would be put together in different orders. In a further test of the generalization capacities of this last network, I tested it on a new set of derivations constructed following the patterns shown in Table3 . These derivation patterns are modeled on those used in the training, but introduce variations.... In PAGE 23: ... 12C. --------------------------------------------- Insert Table3 about here --------------------------------------------- There were a total of 252 inferential steps in these derivations, of which the network was correct on 198 (78.6% correct).... ..."

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### Table II. Six new derivation schemas used to test generalization of the natural deduction network.

### Table 1. The k sat algorithm schema.

"... In PAGE 4: ... The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [7] is based, for deciding the satis ability of formulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 1 The k sat algorithm schema.

"... In PAGE 5: ....0.2 The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [6] is based, for deciding the satis ability of formulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 1. The k sat algorithm schema.

"... In PAGE 5: ... The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [8] is based, for deciding the satis ability of for- mulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 1. Study of the horizontal position accuracy Ga7G20 Shape preservation can be studied at the operator level, where the average shape change values for different algorithms are compared or at the algorithm level, where the average shape change values for groups of lines of different complexity are compared. The creation of graphs and tables with the items shown in Table 2 is proposed. In addition, the construction of a table showing the clustering results (hierarchical and non-hierarchical) of different generalization schemas (different algorithms and a range of tolerance values) and the comparison with the initial segments clustering is considered useful.

"... In PAGE 7: ...7 Ga7G20 Horizontal position accuracy can be studied at the operator level, where the average horizontal position error values for different algorithms are compared or at the algorithm level, where the average horizontal position error values for groups of lines of different complexity are compared. The creation of graphs and tables with the items shown in Table1 is proposed (Table 1). Study Level Item 1 Item 2 1.... ..."

### Table 2. Statistical data about the structure of schemas

2002

"... In PAGE 7: ... As suggested in [18] and [25], richer, cross-domain (top-level) ontologies (schemas) are needed to provide more elaborate forms of semantic interoperability between various application domains. Table2 illustrates the statistics extracted by our testbed. The columns of this table correspond to various structural characteristics of a schema.... In PAGE 7: ... The gathered statistics are reported for all schema hierarchies. One general observation we can make from the data of Table2 is that most of the schemas define few classes and properties, with the exception of Real Estate Data Consortium [16], Basic Semantic Registry [7], UNSPSC [53] and Gene Ontology [24]. We can consider these schemas as rich domain models of the application to ... In PAGE 10: ... One last corollary refers to the correlation between the richness of modeling techniques used and the semantic depth. As we can observe from Table2 , the majority of schemas classified as Reference Models in Table 1 exhibit a rather complete use of RDF/S modeling constructs (e.g.... ..."

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### Table 1. However, where the argument and result-types are the same (as we are assuming for now), it may be that some analogue of relative complement (set di erence) will form a useful de nition for an excess function. One possible general schema for this is: x y , ufz j z v y ^ z u x = ?g (the greatest lower bound of the substructures of y that exclude information that is in x). (Instantiating this for sets gives x y = ynx, as desired.) Once excess functions are de ned, we can de ne at least four similarity func- tions that correspond to the four developed in the previous section. They are given in Table 2, in which f1 and f2 denote functions for combining orders, such as product or prioritisation. (Obviously, corresponding di erence functions can be de ned by inverting the lattices.)

1998

"... In PAGE 2: ... A good understanding of this paper is possible for those who know what a partial order 1 The requirement for complete lattices can be weakened but only by outlawing certain operators, such as prioritisation (see later), which require complete lattices. (In fact, to be more accurate, an axiom that excess functions must satisfy ( Table1 ) will only... In PAGE 4: ... Giving full axioma- tisations is not an aim of this paper, but some discussion of axioms is needed because any ways we de ne for building new functions from existing ones (such as using products and prioritisations, above) should ensure that the new func- tions will satisfy the axioms. (For example, an operator similar to product, but based on disjunction rather than conjunction, is unlikely to violate any axioms we might place on similarity functions but will not necessarily respect the axiom that we give in Table1 for excess functions.) Many authors claim that similarity functions must be re exive [11], which, in... In PAGE 5: ...degrees of similarity for di erent `identical apos; objects. Consequently, in [9] and here (see Table1 ), we use a weaker axiom: x x w x y, i.e.... In PAGE 6: ...xiom to be satis ed. The axiom is inspired by triangle inequality. It is given, along with other axioms, in Table 1. Type Axioms Similarity ! !(S; v) x x w x y Di erence ! !(S; v) x !x w x! y Excess ! !(S; v) ((x y v y x) ^ (y z v z y)) ) (x z v z x) Table1 . Types and axioms We leave open the question of whether excess functions should (like prefer- ence relations) be expected to satisfy (generalised versions of) re exivity and antisymmetry.... ..."

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