### Table 2. Different computational logic issues in inductive logic programming.

"... In PAGE 6: ...1 An analysis The 31% of computational logic oriented work in inductive logic programming can be further classified according to the topic addressed. The most important results are shown in Table2 . The table clearly shows that the main issues are program synthesis and rules and frameworks for inductive inference.... ..."

### Table 1: Comparison of techniques for the induction of recursive logic programs

1999

"... In PAGE 11: ...iques are overviewed in Section 3.3. Finally, in Section 3.4, we point out cross-fertilisation opportunities and identify directions for future work. The comparison chart ( Table1 ) at the end of this section will be helpful towards this aim, and it may be a good idea for the reader to briefly study it right now. Techniques that are somehow related to some others, or representative thereof, and techniques that are somehow more sophisticated and powerful (in an absolute, application-independent sense) than others will obviously get more coverage here than those that are completely different from all others, or that feature highly specialised (sub-)machinery that is impossible to explain in the allotted space, or whose power is quite limited.... In PAGE 28: ...3.4 Summary We now summarise our overview by means of a chart (see Table1 ). The top five lines name classification criteria, whereas the bottom sixteen lines name actual comparison criteria and features, so that the techniques may be meas- ured up to each other.... In PAGE 35: ... The two approaches can thus be considered complementary, rather than rivals, and the ultimate decision of which one to use should lie with the specifier, not with the research community. So then, what is our statement on the future of the inductive synthesis of recursive programs applied towards pro- gramming assistance? We believe such techniques can be (made) viable, provided more research is done on over- coming the obstacles listed above, provided more realistic programming scenarios are aimed at, and provided the future work directions and cross-fertilisation opportunities of Table1 are pursued. We believe that some categories of programmers would use such techniques, provided it improves their productivity or increases the class of pro- grams they can write by themselves.... ..."

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### Table 2. Time and space performance of JITI on Inductive Logic Programming datasets (a) Time (in seconds)

2007

"... In PAGE 13: ...xtensional database. Several of these datasets are standard in the ILP literature. Time performance. We compare times for 10 runs of the saturation/refinement cycle of the ILP system; see Table2 (a). The Mesh and Pyrimidines applications are the only ones that do not benefit much from indexing in the database; they do benefit through from indexing in the dynamic representation of the search space, as their running times improve somewhat with demand-driven indexing.... In PAGE 13: ... Space performance. Table2 (b) shows memory usage when using demand-drivenindex- ing. The table presents data obtained at a point near the end of execution; memory us- age should be at the maximum.... ..."

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### Table 2. (Adapted from [313]) Complexity of Horn Logic Programs Notation: The complexity results in the above table refer to worst case analysis for skeptical reasoning, i.e. to determining if a given literal is true in every canonical model (with respect to a particular semantics) of the program. For logic programs with no function symbols, the data complexity over an EDB E is presented. The notation used is the following: jPj denotes the length of the program P; jAj denotes the number of propositional letters in P; jEj denotes the total number of symbols that occur in the EDB E.

"... In PAGE 12: ... Schlipf [313] has written a comprehensive survey article that summarizes the results. Some of these results, taken from [313], are listed in Table2 . A user may wish to determine which semantics to be used based upon the complexity expected to nd answers to queries.... In PAGE 17: ... Schlipf [313] and Eiter and Gottlob [106] have written comprehen- sive survey articles that summarize the complexity results that are known for alternative semantics. Some of these results, taken from [104, 106], are listed in Table2 . A user may wish to determine the semantics to be used based upon the complexity expected to nd answers to queries.... ..."

### Table 1: Results for nite element mesh design data To nd determinate literals, Foidl must perform an intensional equivalent of determining the number of extensional tuples produced by a literal. The tuples are equivalent to a set of bindings produced by the clause to which the literal is added. So Foidl considers a literal determinate if, given each set of bindings produced by the clause for every example, there is exactly one possible proof of the literal if the example is positive, and no more than one possible proof of the literal if the example is negative. When a clause is complete, Foidl greedily prunes literals that are not necessary to prove the positives covered or eliminate negatives.

1998

"... In PAGE 9: ... We ran ve trials; in each, the learning system was provided with the information about four of the objects, and the resulting program was tested on the edges from the remaining object. Table1 shows our results for Foidl along with those reported for several other systems. The numbers for Foil and FFoil are taken from (Quinlan, 1996).... ..."

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### Table 1: Results for nite element mesh design data To nd determinate literals, Foidl must perform an intensional equivalent of determining the number of extensional tuples produced by a literal. The tuples are equivalent to a set of bindings produced by the clause to which the literal is added. So Foidl considers a literal determinate if, given each set of bindings produced by the clause for every example, there is exactly one possible proof of the literal if the example is positive, and no more than one possible proof of the literal if the example is negative. When a clause is complete, Foidl greedily prunes literals that are not necessary to prove the positives covered or eliminate negatives.

1998

"... In PAGE 9: ... We ran ve trials; in each, the learning system was provided with the information about four of the objects, and the resulting program was tested on the edges from the remaining object. Table1 shows our results for Foidl along with those reported for several other systems. The numbers for Foil and FFoil are taken from (Quinlan, 1996).... ..."

Cited by 3

### Table 2. Program logic part of the ODL sequent calculus.

2006

"... In PAGE 10: ... For rst-order and propositional logic standard rule schemata are listed in Table 1, including an integer induction scheme. Within the rules for the program logic part ( Table2 ), state update rules R29{R30 constitute a peculiarity of ODL and will be discussed after de ning rule applications. Essentially, the ODL inference rules have the e ect of reducing more complex formulas to simpler ones.... ..."

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### Table 1 shows an example of Inductive Logic Programming based learn- ing from an attribute-value database (D zerovski 1996). The presented tables contain the database and the rules induced by the mining process.

1997

"... In PAGE 2: ...if Income(Person) 100 000 then Potential-Customer(Person) if Sex(Person) = F and Age(Person) 32 then Potential-Customer(Person) if Married(Person, Spouse) and Income(Person) 100 000 then Potential-Customer(Spouse) if Married(Person, Spouse) and Potential-Customer(Person) then Potential-Customer(Spouse) Table1... ..."

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### Table 2: Testability and literal count comparisons of multilevel implementations

1992

"... In PAGE 27: ... Multilevel logic circuits are of much greater utility. In Table2 , we compare literal counts and path-delay-fault testabilities obtained using algebraic factorization and unconstrained multilevel lo- gic optimization on the di erent examples. Unconstrained multilevel optimization implies that the initial two-level network was multiple-output minimized and Boolean factorization was used during logic optimization.... In PAGE 27: ... Unconstrained multilevel optimization implies that the initial two-level network was multiple-output minimized and Boolean factorization was used during logic optimization. Algebraic factorization was used on the circuits produced by the single-output minimization ( Table2 ). A modi ed version of the program mis [2] was used in both cases.... ..."

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### Table 5: Performance impact of the Generalized Induction technique. Analysis and Program Patterns In order to substitute induction variables, one must rst determine the value of the induction variable prior to the loop, nd all the induction sites, and determine the loop bounds of the loops enclosing the induction sites. From this information, we can then compute the value of the induction variable at each reference in the loop body. The rst step is the same for all types of loops. The second step is more di cult in triangular loops, although it can be considered an extension of the rectangular loop case. For example, assume a doubly-nested loop with indices I and J, loop bounds M and N, and an induction variable that increments in steps of 1. In each iteration of the outer loop, the induction variable 12

1998

"... In PAGE 12: ... In [BE94a], we have described a new analysis technique that can handle such nonlinear subscripts. The e ect of the Generalized Induction Variable transformation is shown in Table5 . The table shows the same type of information as Tables 3 and 4.... ..."

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