### Table 1. S-T CONNECTIVITY problems

2007

"... In PAGE 3: ... An entry of RL means that such a solution would imply RL = L. It is interesting to note the very different behavior of the oblivious setting and explicit setting, as summa- rized in Table1 . Our result shows that in the explicit setting, the gap between L and RL centers around whether the stationary distribution is known.... ..."

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### Table 1. Cars and Dealers combined DL and Datalog KB

2004

"... In PAGE 3: ... Our choice is to have the sellsDiscount, customer and pays con- cepts in the Datalog component, while all the rest of the concepts would be represented in the DL component. The KB can be seen in Table1 . Following the GAV approach, and using the mappings between source and domain concepts, each concept in the domain ontology can be expressed as a view of the source ontologies.... ..."

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### Table 5. From DL-safe rules (Datalog safeness condition)

"... In PAGE 15: ...3. Examples DL-safe Rules In Table5 , the hybrid rule of \BadChild quot; has a variable ?Y as the pure-DL variable, and the binary DL queries of parent(?X, ?Y) and parent(?Z, ?Y) are chained by this ?Y, con icting with our deflnition of independent properties. That means, our system fails to infer \BadChild(Cain) quot; under the Datalog safeness condition, lacking of expressions for nominals such as 9parent.... ..."

### Tabled Datalog

1994

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### Table 2. An attribute grammar for translating EPL expressions to Datalog1S rules. EvtType E PPS EVar(E) IVar

"... In PAGE 17: ... This is because in a sequence expression of the form (F,!G), the subexpression G is evaluated with respect to the basic event history starting right after the satisfaction of F. Thus, using Table2 , we get, for instance:PPS(2) = PPS(4) = PPS(5) = f1g Finally, observe that while variables de ned in a negated event are not exported outside it, variables de ned in preceding events can be used inside the negated event. For instance, in Example 14, variable X can be used in the conditions qb and qc (see also Example 6).... In PAGE 21: ... The same reasoning applies to relaxed sequences, which can be treated similarlyto immediate sequences. In particular, the rules of Table2 for an immediate sequence remain intact in the case of a relaxed sequence. The only di erence is that in [F; G], an instance of G can begin at any later stage, rather than immediately after an occurrence of F.... In PAGE 22: ... Consider now the more complicated case of Example 9, where star expressions are nested. Referring to Figure 3 and using Table2 , we get: PPS(7) = ; PPS(1) = PPS(7) = ; PPS(6) = f1g PPS(4) = f4g [ PPS(6) = f1;4g PPS(5) = f4g PPS(2) = f2g [ PPS(4) = f1;2;4g PPS(3) = f2g Using the PPS sets of the basic events, as well as the scope rules of EPL, the... In PAGE 24: ...15 various subevents, and (ii) the transmission of variable bindings according to the scope rules of the language. Table2 describes how this information is derived for each basic EPL construct. Formally, it de nes a simple attribute grammar [20, 11].... In PAGE 24: ...The rst column of Table2 lists the various EPL event types. For each subevent Q of a EPL event E, the second column in the Table de nes the Possible Predecessors Set of Q, denoted as PPS(Q).... In PAGE 24: ...n which events may precede E|i.e., F inherits E apos;s possible predecessors. The remaining two columns of Table2 describe the scope rules for variables in EPL. For each event expression E, the third column speci es the set of exported variables from E.... ..."

### Table 3. Datalog1S rule templates for the basic constructs of EPL. Event Type E Datalog1S Rule Templates Any Basic Event any(J) hist Mod( ; ;J):

"... In PAGE 18: ... Since a negated event instance has single stage duration, negated events are treated similarly to basic events, as far as temporal ordering is concerned. Specif- ically, one rule of the form shown in the 8-th row of Table3 is created for each of the possible predecessors of a negated event. In this example, the only possible predecessor of the negated event E5 is E1.... In PAGE 22: ... In the case of an immediate sequence or a star sequence expression, an extra copy- rule is added after the rules for the components of this expression are generated. The copy-rule (shown in the fourth and fth rows of Table3 ) expresses the fact that the sequence is satis ed when the last component event is satis ed|see, for instance, the third rule in Example 8. The rules for disjunction and conjunction are also straightforward.... ..."

### Table 3: An attribute grammar for syntax-directed translation from TREPL to Datalog1S. which returns the last value (tuple) in the event sequence represented by a star sequence goal. Using this attribute grammar, the generation of the ac- tual rules is simple, as shown in Table 4. Observe that except for basic events, X and Y denote sets of exported variables de ned in various subevents, and IV denotes the set of imported variables into a particular event type E. The anonymous variable

"... In PAGE 4: ... As demonstrated by Example 14, the Datalog1S rules de- rived from the translation of a composite TREPL expression must model (i) the transmission of variable bindings accord- ing to the scopes of the various TREPL constructs, so that variables can be matched and conditions can be checked, and (ii) the temporal precedences among the various subevents. Table3 describes how this information is derived for each basic TREPL construct. Formally, it de nes a simple at- tribute grammar [20, 9].... In PAGE 4: ... Formally, it de nes a simple at- tribute grammar [20, 9]. For each subevent Q of an TREPL event E, the second column in Table3 de nes the Possible Predecessors Set of Q, denoted as PPS(Q). A subevent P is a possible predecessor of Q within E, if in an instance of E, the satisfaction of P can immediately precede the rst basic event of an instance of Q (i.... In PAGE 4: ...ay precede E|i.e., F inherits E apos;s possible predecessors. The remaining two columns of Table3 describe the scope rules for variables in TREPL. The third column shows the set of exported variables of an TREPL expression.... ..."

### Table 1: The Complexity of Brave Reasoning in various Extensions of Datalog with Con- straints (Propositional Case under Stable Model Semantics)

"... In PAGE 23: ...Table 1: The Complexity of Brave Reasoning in various Extensions of Datalog with Con- straints (Propositional Case under Stable Model Semantics) Table1 sumarizes the complexity results of the previous section, complemented with other results (on the complexity of programs without constraints) already known in the literature. Therein, each column refers to a speci c form of constraints, namely: fg = no constraints, s = strong constraints, w lt; = weak constraints with priorities, w = weak constraints without priorities (i.... In PAGE 23: ...riorities (i.e., W has only one component). The lines of Table1 specify the allowance of disjunction and negation; in particular, :s stands for strati ed negation [45] and _h stands for HCF disjunction [3] (see AppendixB). Each entry of the table provides the complexity class of the corresponding fragment of the language.... In PAGE 24: ...i.e., relevance can be reduced to brave reasoning on DATALOG_;:;c). 8 Considering that DATALOG_;:;c is a linguistic extension of DATALOG_;: by constraints, it turns out that constraints do add expressive power to DATALOG_;:. However, it is not the case of strong constraints, as it can be seen from Table1 . Indeed, if we look at the various fragments of the language that di er only for the presence of strong constraints, we can note that complexity is constant (compare column 1 to 2, or 3 to 4, or 5 to 6).... In PAGE 26: ... Comparing the above DATALOG_;: program with the DATALOG_;:;c version of Example 9,10 it is quite apparent the advantage that weak constraints provide in terms of simplicity and naturalness of programming. Concluding, we would like to bring reader apos;s attention to the fragment of DATALOG_;:;c with HCF disjunction and strati ed negation ((5,6) in Table1 ): it has a very clear and easy- to-understand semantics and, at the same time, allows us to express several hard problems (up to P 2 -complete problems) in a natural and compact fashion. (In our opinion, recursion through disjunction or negation makes programs more di cult to understand).... ..."

### Table 1: Decidability and relationships Some of these results are immediate consequences of those in [6]. For the others, rst note that every computable function over the natural numbers Z can be expressed as a Datalog query. The undecidability of domain independence and niteness for Datalog queries can then be proved by reduction from the undecidable \empty domain quot; and \ nite range quot; problems for computable functions over Z. The conjunctive query n 0 is domain independent, but not nite. The rst-order query 8x p(x; y) is nite, but not domain independent. The Datalog query (p; P), where P is the Datalog program

1991

"... In PAGE 10: ... 5 Domain independence and niteness We now turn to the decidability of domain independence and niteness, and to the rela- tionships between these two properties, in the di erent query languages we have described. Brie y, these results are summarized in Table1 . For simplicity, we only consider the... ..."

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