Inductive Logic Programming (ILP) is a promising technol-ogy for knowledge extraction applications. ILP has produced intelligiblesolutions for a wide variety of domains where it has been applied. TheILP lack of eciency is, however, a major impediment for its scalabilityto applications requiring large amounts of data. In this paper we pro-pose a set of techniques that improve ILP systems eciency and makethen more likely to scale up to applications of knowledge extraction fromlarge datasets. We propose and evaluate the lazy evaluation of examples,to improve the eciency of ILP systems. Lazy evaluation is essentiallya way to avoid or postpone the evaluation of the generated hypotheses(coverage tests).The techniques were evaluated using the IndLog system on ILP datasetsreferenced in the literature. The proposals lead to substantial eficiencyimprovements and are generally applicable to any ILP system

A. Srinivasan

C. Nédellec

H. Blockeel

M. Botta

P. Laag van der

S. Muggleton

V. Santos Costa

Inductive Logic Programming (ILP) is a promising technol-
ogy for knowledge extraction applications. ILP has produced intelligible
solutions for a wide variety of domains where it has been applied. The
ILP lack of eciency is, however, a major impediment for its scalability
to applications requiring large amounts of data. In this paper we pro-
pose a set of techniques that improve ILP systems eciency and make
then more likely to scale up to applications of knowledge extraction from
large datasets. We propose and evaluate the lazy evaluation of examples,
to improve the eciency of ILP systems. Lazy evaluation is essentially
a way to avoid or postpone the evaluation of the generated hypotheses
(coverage tests).
The techniques were evaluated using the IndLog system on ILP datasets
referenced in the literature. The proposals lead to substantial eficiency
improvements and are generally applicable to any ILP system

Rui Camacho

Repositório Aberto da Universidade do Porto

Improving the efficiency of ILP systems ?Rui CamachoLIACC, Rua do Campo Alegre, 823, 4150 Porto, PortugalFEUP, Rua Dr Roberto Frias, 4200-465 Porto, Portugalrcamacho@fe.up.pthttp://www.fe.up.pt/∼rcamachoAbstract. Inductive Logic Programming (ILP) is a promising technol-ogy for knowledge extraction applications. ILP has produced intelligiblesolutions for a wide variety of domains where it has been applied. TheILP lack of efficiency is, however, a major impediment for its scalabilityto applications requiring large amounts of data. In this paper we pro-pose a set of techniques that improve ILP systems efficiency and makethen more likely to scale up to applications of knowledge extraction fromlarge datasets. We propose and evaluate the lazy evaluation of examples,to improve the efficiency of ILP systems. Lazy evaluation is essentiallya way to avoid or postpone the evaluation of the generated hypotheses(coverage tests).The techniques were evaluated using the IndLog system on ILP datasetsreferenced in the literature. The proposals lead to substantial efficiencyimprovements and are generally applicable to any ILP system.key words: Inductive Logic Programming, Efficiency1 IntroductionInductive Logic Programming (ILP) has achieved considerable success in a widerange of domains. It is recognised however that efficiency is a major obstacle tothe use of ILP systems in applications requiring large amounts of data. RelationalData Mining applications are an example where efficiency is an important issue.In this paper we address the problem of efficiency in ILP systems by proposinga technique to speedup the evaluation of hypotheses.A typical ILP system carries out a search through an ordered hypothesisspace. During the search hypothesis are generated and their quality estimatedagainst the given examples. Improving the efficiency of such search proceduremay be done by avoiding to generate useless hypothesis or/and improving theevaluation procedures.? The work presented in this paper has been partially supported by Universidade doPorto, project APRIL (Project POSI/SRI/40749/2001), funds granted to LIACCthrough the Programa de Financiamento Plurianual, Fundac¸a˜o para a Cieˆncia eTecnologia and Programa POSI.Avoiding to generate useless hypotheses may be achieved with the specifi-cation of language bias limiting therefore the size of the search space ([?]). Analternative approach considers the study of refinement operators that allow toefficiently navigate through a hypothesis space ([?]).The problem of efficient testing of candidate hypotheses has been tackled bythe following techniques. Work of a stochastic nature (see [?,?]). These reducethe evaluation effort at the cost of being correct only with high probability. Astudy on exact transformations of queries when evaluating hypotheses may befound in [?] and [?]. In [?], the authors illustrated that query execution was avery high percentage of total running time.In this paper we address the problem of ILP system efficiency by avoiding orreducing the computational cost in the evaluation of the hypotheses using theexamples. The proposed techniques are globally called lazy evaluation of exam-ples and reduce considerably the amount of examples necessary to evaluate eachhypothesis. The proposed techniques may be adopted in any ILP system.The structure of the rest of the paper is as follows. In Section ?? we presentthe lazy evaluation of examples techniques. The experiments that empiricallyevaluate our proposals are described in Section ??. The last section draws theconclusions.2 Lazy evaluation of examplesLanguage bias may be used to avoid the generation, and therefore, the evalua-tion of a significant number of hypotheses. However, once an hypothesis has beengenerated the problem then is how to evaluate it efficiently using the availabledata (examples and background knowledge). This problem is specially criticalif either the number of examples is large, like in Data Mining applications, orthe evaluation of individual hypothesis (theorem proving effort) is hard [?]. Thesecond problem has been addressed recently by means of techniques like querypacks [?] or query transformations [?] and [?]. To handle the first situation, aprobabilistic evaluation has been proposed [?] with the advantage of avoidingthe use of all of the examples. This approach however prevents the system fromhaving a correct measure of the hypothesis value.We propose lazy evaluation of examples as a way to avoid unnecessary use ofexamples and therefore speed up the evaluation of each hypothesis. As the prob-abilistic approach we do not use all the examples to evaluate each hypothesis.Contrary to the probabilistic approach we get an exact count but only when itis absolutely necessary to do so. We can still profit from improvements due toquery transformations and therefore combine the two technique for increasingspeedup. We distinguish between lazy evaluation of positive examples, lazy eval-uation of negative examples and positive evaluation avoidance. The techniquesare now described.Some systems like Progol [?], Aleph1 or IndLog [?] rely on heuristics toguide the search during refinement. If the search is complete then the role of theheuristic is to improve speed. The final hypothesis should be the same. For someheurstics it is important to determine the exact number of examples covered.However, an hypothesis will only be accepted if it is consistent with the negativeexamples. A partial clause2 must be specialised otherwise the search for furtherrefinements of the partial clause terminates there. In this circumstances “lazy”evaluation of the negative examples is useful. When using the lazy evaluationof negatives we are only interesting in knowing if the hypothesis covers morethan the allowed number of negative examples. We are not really interesting inknowing how much above the noise level the hypothesis is. Testing stops as soonas there are no more negative examples to be tested or the number of negativeexamples covered exceeds the allowed number for consistency (noise level). Thenoise level is quite often very close to zero and therefore the number of negativeexamples used in the test of each clause is very small. If the heuristic does notuse the negative counting then this produces exactly the same results (clausesand accuracy) of the non-lazy approach but with a very significant speedup. Itis also very common that the negative examples outnumber the positives. Wemay still use heuristics based on length and on positive cover.IndLog also allows the positive cover to be evaluated lazily. A partial clausemust be either specialised (if it covers more positives that the best consistentclause found so far) or is justifiably pruned away otherwise. When using lazyevaluation of positives it is relevant only to determine if an hypothesis coversmore positives or not than the current best consistent hypothesis. We mightthen evaluate the positive examples just until the best cover so far is exceeded.If the best cover is exceeded we retain the hypothesis (either accept it as final isconsistent or refine it otherwise) if not we may justifiably discard it. Only whenaccepting a consistent hypothesis we need to evaluate its exact positive cover.We may go a bit further and simply do not evaluate the positive cover at allof partial clauses. The advantage of the first version of lazy evaluation of posi-tives is that we may discard hypotheses that are worse than the best consistentso far, while in the second version we may keep around some partial clauses withvery poor positive cover. The advantage of the second version however is thatfor each hypothesis we only test it on the negatives until it covers at least thenoise level. Generating hypotheses is very efficient and although we may generatemore hypotheses we may still gain by the increase in speed of their evaluationprocess. This technique may be very useful in domains where the evaluation ofeach hypothesis is very time-consuming.When performing lazy evaluation of positive and negative examples we mayuse a breadth-first search strategy for example. This is not a too bad choice if,like in most applications, one is looking for short clauses that are very close to1 Available from http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/2 A clause that covers more than the allowed number of negative examples.the top of the subsumption lattice.Lazy evaluation of either negative or positive examples cannot be used whenthe technique called lazy evaluation of literals [?] is used. To compute the con-stant values all of the positive and negative coverage has to be computed exactly.It is also not applicable in data sets with positives only examples and the use ofcompression measure ([?]) or a user defined cost function like Aleph and IndLogmight use ([?]).3 The experimentsTo empirically evaluate our proposals we run IndLog [?] encoded in Yap Pro-log [?] (version 4.21) and run on a PC. For each dataset the induced clause(s)are the same with and without lazy evaluation. Therefore the accuracy of theinduced theories does not change by adopting any of the proposed techniques.The datasets used in the experiments were downloaded from the Oxford 3 andYork 4 Universities Machine Learning repositories.In the experiments the IndLog settings were such that all lazy evaluationsfor the same dataset produce the same final theory and construct the same setof hypotheses. For the lazy evaluation of negatives we measured the numberof theorem proving calls used to evaluate the negative examples and also theCPU time spent. For the lazy evaluation of positives we measured the number oftheorem proving calls used to evaluate the positive examples and also the CPUtime needed. For the no positive evaluation form of lazy evaluation we measuredthe number of theorem proving calls used to evaluate both the negative andpositive examples, the spent CPU time and the number of nodes constructedduring the search. The results are shown in Table ??.lazy negs lazy pos no posDataset negs calls cpu time pos calls cpu time nodes negs calls pos calls cpu time(%) (%) (%) (%) (%) (%) (%) (%)amine uptake 25 75 69 99 146 56 14 97carcinogenesis 27 79 80 115 105 76 23 68choline 27 74 54 92 100 44 13 46mesh 86 99 93 99 105 125 77 81multiplication 31 99 91 104 152 95 44 127mutagenesis 40 99 66 101 100 48 10 98pyrimidines 38 76 63 99 120 62 13 58triazines 8 56 39 77 148 55 26 48Table 1. Percentage of theorem proving calls and CPU time savings when using lazyvaluation of examples (lazy / no lazy × 100%).3 URL: http://www.comlab.ox.ac.uk/oucl/groups/machlearn/4 URL:http://www.cs.york.ac.uk/mlg/index.htmlTable ?? shows the percentage of the measured values when compared withthe run when not using any kind of lazy evaluation. Except for the small (num-ber of examples) datasets there is significant gain in using the lazy evaluationtechnique.4 ConclusionsWe proposed and evaluated a set of techniques globaly called lazy evaluation ofexamples that improve the efficiency of ILP systems. As shown by the empiricalresults the lazy evaluation may produce substantial improvements to an ILPsystem.References1. Ne´dellec, C. and Rouveirol, C. and Ade´, H. and Bergadano, F. and Tausend, B.Declarative Bias in ILP ed. De Raedt, L., in Advances in Inductive Logic Pro-gramming, 82-103, 1996, IOS2. Patrick van der Laag and Shan-Hwei Nienhuys-Cheng Completeness and Proper-ness of Refinement Operators Journal of Logic Programming, 34, 3, 201–226, 19983. A. Srinivasan A study of two sampling methods for analysing large datasets withILP Data Mining and Knowledge Discovery,vol. 3, N. 1, 95–123, 19994. M. Sebag and C. Rouveirol Tractable induction and classification in first-orderlogic via stochastic matching (IJCAI97) Proceedings of the 15th InternationalJoint Conference on Artificial Intelligence, 888–893, 19975. Santos Costa, Vı´tor and Srinivasan, Ashwin and Camacho, Rui, A note on twosimple transformations for improving the efficiency of an ILP system Proceedingsof the 10th International Conference on Inductive Logic Programming, LNAI, vol1866, ed. J. Cussens and A. Frisch, 225–242, 20006. Vı´tor Costa and Ashwin Srinivasan and Rui Camacho and Hendrik Blockeel andBart Demoen and and Gerda Janssens and Jan Struyf and Henk Vandecasteeleand Wim Van Laer Query Transformations for Improving the Efficiency of ILPSystems Journal of Machine Learning Research, 20027. M. Botta and A. Giordana and L. Saitta and M. Sebag Relational learning: hardproblems and phase transitions Proc. of the 6th Congress AI*IA, LNAI 1792,178–189, 19998. Blockeel, Hendrik and Dehaspe, Luc and Demoen, Bart and Janssens, Gerda andRamon, Jan and Vandecasteele, Henk Executing Query Packs in ILP Proceedingsof the 10th International Conference on Inductive Logic Programming,LNAI 1866,ed. J. Cussens and A. Frisch, 60–77, 20009. Muggleton, S. Inverse Entailment and Progol New Generation Computing, Specialissue on Inductive Logic Programming, vol. 13, N. 3-4, 245–286, 199510. R. Camacho, Inducing Models of Human Control Skills using Machine LearningAlgorithms PhD thesis,Department of Electrical Engineering and Computation,Universidade do Porto, 200011. A. Srinivasan and R.C. Camacho Numerical reasoning with an ILP program capableof lazy evaluation and customised search Journal of Logic Programming, vol. 40,N. 2-3, 185–214, 199912. Costa, V. and Damas, L. and Reis, R. and Azevedo, R., YAP Prolog User’s ManualUniversidade do Porto, 1989

Improving the efficiency of ILP systems

Crossref

Improving the Efficiency of ILP Systems

Open Repository of the University of Porto

Improving the efficiency of ILP systems ?
Rui Camacho
LIACC, Rua do Campo Alegre, 823, 4150 Porto, Portugal
FEUP, Rua Dr Roberto Frias, 4200-465 Porto, Portugal
rcamacho@fe.up.pt
http://www.fe.up.pt/∼rcamacho
Abstract. Inductive Logic Programming (ILP) is a promising technol-
ogy for knowledge extraction applications. ILP has produced intelligible
solutions for a wide variety of domains where it has been applied. The
ILP lack of efficiency is, however, a major impediment for its scalability
to applications requiring large amounts of data. In this paper we pro-
pose a set of techniques that improve ILP systems efficiency and make
then more likely to scale up to applications of knowledge extraction from
large datasets. We propose and evaluate the lazy evaluation of examples,
to improve the efficiency of ILP systems. Lazy evaluation is essentially
a way to avoid or postpone the evaluation of the generated hypotheses
(coverage tests).
The techniques were evaluated using the IndLog system on ILP datasets
referenced in the literature. The proposals lead to substantial efficiency
improvements and are generally applicable to any ILP system.
key words: Inductive Logic Programming, Efficiency
1 Introduction
Inductive Logic Programming (ILP) has achieved considerable success in a wide
range of domains. It is recognised however that efficiency is a major obstacle to
the use of ILP systems in applications requiring large amounts of data. Relational
Data Mining applications are an example where efficiency is an important issue.
In this paper we address the problem of efficiency in ILP systems by proposing
a technique to speedup the evaluation of hypotheses.
A typical ILP system carries out a search through an ordered hypothesis
space. During the search hypothesis are generated and their quality estimated
against the given examples. Improving the efficiency of such search procedure
may be done by avoiding to generate useless hypothesis or/and improving the
evaluation procedures.
? The work presented in this paper has been partially supported by Universidade do
Porto, project APRIL (Project POSI/SRI/40749/2001), funds granted to LIACC
through the Programa de Financiamento Plurianual, Fundac¸a˜o para a Cieˆncia e
Tecnologia and Programa POSI.
Avoiding to generate useless hypotheses may be achieved with the specifi-
cation of language bias limiting therefore the size of the search space ([?]). An
alternative approach considers the study of refinement operators that allow to
efficiently navigate through a hypothesis space ([?]).
The problem of efficient testing of candidate hypotheses has been tackled by
the following techniques. Work of a stochastic nature (see [?,?]). These reduce
the evaluation effort at the cost of being correct only with high probability. A
study on exact transformations of queries when evaluating hypotheses may be
found in [?] and [?]. In [?], the authors illustrated that query execution was a
very high percentage of total running time.
In this paper we address the problem of ILP system efficiency by avoiding or
reducing the computational cost in the evaluation of the hypotheses using the
examples. The proposed techniques are globally called lazy evaluation of exam-
ples and reduce considerably the amount of examples necessary to evaluate each
hypothesis. The proposed techniques may be adopted in any ILP system.
The structure of the rest of the paper is as follows. In Section ?? we present
the lazy evaluation of examples techniques. The experiments that empirically
evaluate our proposals are described in Section ??. The last section draws the
conclusions.
2 Lazy evaluation of examples
Language bias may be used to avoid the generation, and therefore, the evalua-
tion of a significant number of hypotheses. However, once an hypothesis has been
generated the problem then is how to evaluate it efficiently using the available
data (examples and background knowledge). This problem is specially critical
if either the number of examples is large, like in Data Mining applications, or
the evaluation of individual hypothesis (theorem proving effort) is hard [?]. The
second problem has been addressed recently by means of techniques like query
packs [?] or query transformations [?] and [?]. To handle the first situation, a
probabilistic evaluation has been proposed [?] with the advantage of avoiding
the use of all of the examples. This approach however prevents the system from
having a correct measure of the hypothesis value.
We propose lazy evaluation of examples as a way to avoid unnecessary use of
examples and therefore speed up the evaluation of each hypothesis. As the prob-
abilistic approach we do not use all the examples to evaluate each hypothesis.
Contrary to the probabilistic approach we get an exact count but only when it
is absolutely necessary to do so. We can still profit from improvements due to
query transformations and therefore combine the two technique for increasing
speedup. We distinguish between lazy evaluation of positive examples, lazy eval-
uation of negative examples and positive evaluation avoidance. The techniques
are now described.
Some systems like Progol [?], Aleph1 or IndLog [?] rely on heuristics to
guide the search during refinement. If the search is complete then the role of the
heuristic is to improve speed. The final hypothesis should be the same. For some
heurstics it is important to determine the exact number of examples covered.
However, an hypothesis will only be accepted if it is consistent with the negative
examples. A partial clause2 must be specialised otherwise the search for further
refinements of the partial clause terminates there. In this circumstances “lazy”
evaluation of the negative examples is useful. When using the lazy evaluation
of negatives we are only interesting in knowing if the hypothesis covers more
than the allowed number of negative examples. We are not really interesting in
knowing how much above the noise level the hypothesis is. Testing stops as soon
as there are no more negative examples to be tested or the number of negative
examples covered exceeds the allowed number for consistency (noise level). The
noise level is quite often very close to zero and therefore the number of negative
examples used in the test of each clause is very small. If the heuristic does not
use the negative counting then this produces exactly the same results (clauses
and accuracy) of the non-lazy approach but with a very significant speedup. It
is also very common that the negative examples outnumber the positives. We
may still use heuristics based on length and on positive cover.
IndLog also allows the positive cover to be evaluated lazily. A partial clause
must be either specialised (if it covers more positives that the best consistent
clause found so far) or is justifiably pruned away otherwise. When using lazy
evaluation of positives it is relevant only to determine if an hypothesis covers
more positives or not than the current best consistent hypothesis. We might
then evaluate the positive examples just until the best cover so far is exceeded.
If the best cover is exceeded we retain the hypothesis (either accept it as final is
consistent or refine it otherwise) if not we may justifiably discard it. Only when
accepting a consistent hypothesis we need to evaluate its exact positive cover.
We may go a bit further and simply do not evaluate the positive cover at all
of partial clauses. The advantage of the first version of lazy evaluation of posi-
tives is that we may discard hypotheses that are worse than the best consistent
so far, while in the second version we may keep around some partial clauses with
very poor positive cover. The advantage of the second version however is that
for each hypothesis we only test it on the negatives until it covers at least the
noise level. Generating hypotheses is very efficient and although we may generate
more hypotheses we may still gain by the increase in speed of their evaluation
process. This technique may be very useful in domains where the evaluation of
each hypothesis is very time-consuming.
When performing lazy evaluation of positive and negative examples we may
use a breadth-first search strategy for example. This is not a too bad choice if,
like in most applications, one is looking for short clauses that are very close to
1 Available from http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/
2 A clause that covers more than the allowed number of negative examples.
the top of the subsumption lattice.
Lazy evaluation of either negative or positive examples cannot be used when
the technique called lazy evaluation of literals [?] is used. To compute the con-
stant values all of the positive and negative coverage has to be computed exactly.
It is also not applicable in data sets with positives only examples and the use of
compression measure ([?]) or a user defined cost function like Aleph and IndLog
might use ([?]).
3 The experiments
To empirically evaluate our proposals we run IndLog [?] encoded in Yap Pro-
log [?] (version 4.21) and run on a PC. For each dataset the induced clause(s)
are the same with and without lazy evaluation. Therefore the accuracy of the
induced theories does not change by adopting any of the proposed techniques.
The datasets used in the experiments were downloaded from the Oxford 3 and
York 4 Universities Machine Learning repositories.
In the experiments the IndLog settings were such that all lazy evaluations
for the same dataset produce the same final theory and construct the same set
of hypotheses. For the lazy evaluation of negatives we measured the number
of theorem proving calls used to evaluate the negative examples and also the
CPU time spent. For the lazy evaluation of positives we measured the number of
theorem proving calls used to evaluate the positive examples and also the CPU
time needed. For the no positive evaluation form of lazy evaluation we measured
the number of theorem proving calls used to evaluate both the negative and
positive examples, the spent CPU time and the number of nodes constructed
during the search. The results are shown in Table ??.
lazy negs lazy pos no pos
Dataset negs calls cpu time pos calls cpu time nodes negs calls pos calls cpu time
(%) (%) (%) (%) (%) (%) (%) (%)
amine uptake 25 75 69 99 146 56 14 97
carcinogenesis 27 79 80 115 105 76 23 68
choline 27 74 54 92 100 44 13 46
mesh 86 99 93 99 105 125 77 81
multiplication 31 99 91 104 152 95 44 127
mutagenesis 40 99 66 101 100 48 10 98
pyrimidines 38 76 63 99 120 62 13 58
triazines 8 56 39 77 148 55 26 48
Table 1. Percentage of theorem proving calls and CPU time savings when using lazy
valuation of examples (lazy / no lazy × 100%).
3 URL: http://www.comlab.ox.ac.uk/oucl/groups/machlearn/
4 URL:http://www.cs.york.ac.uk/mlg/index.html
Table ?? shows the percentage of the measured values when compared with
the run when not using any kind of lazy evaluation. Except for the small (num-
ber of examples) datasets there is significant gain in using the lazy evaluation
technique.
4 Conclusions
We proposed and evaluated a set of techniques globaly called lazy evaluation of
examples that improve the efficiency of ILP systems. As shown by the empirical
results the lazy evaluation may produce substantial improvements to an ILP
system.
References
1. Ne´dellec, C. and Rouveirol, C. and Ade´, H. and Bergadano, F. and Tausend, B.
Declarative Bias in ILP ed. De Raedt, L., in Advances in Inductive Logic Pro-
gramming, 82-103, 1996, IOS
2. Patrick van der Laag and Shan-Hwei Nienhuys-Cheng Completeness and Proper-
ness of Refinement Operators Journal of Logic Programming, 34, 3, 201–226, 1998
3. A. Srinivasan A study of two sampling methods for analysing large datasets with
ILP Data Mining and Knowledge Discovery,vol. 3, N. 1, 95–123, 1999
4. M. Sebag and C. Rouveirol Tractable induction and classification in first-order
logic via stochastic matching (IJCAI97) Proceedings of the 15th International
Joint Conference on Artificial Intelligence, 888–893, 1997
5. Santos Costa, Vı´tor and Srinivasan, Ashwin and Camacho, Rui, A note on two
simple transformations for improving the efficiency of an ILP system Proceedings
of the 10th International Conference on Inductive Logic Programming, LNAI, vol
1866, ed. J. Cussens and A. Frisch, 225–242, 2000
6. Vı´tor Costa and Ashwin Srinivasan and Rui Camacho and Hendrik Blockeel and
Bart Demoen and and Gerda Janssens and Jan Struyf and Henk Vandecasteele
and Wim Van Laer Query Transformations for Improving the Efficiency of ILP
Systems Journal of Machine Learning Research, 2002
7. M. Botta and A. Giordana and L. Saitta and M. Sebag Relational learning: hard
problems and phase transitions Proc. of the 6th Congress AI*IA, LNAI 1792,
178–189, 1999
8. Blockeel, Hendrik and Dehaspe, Luc and Demoen, Bart and Janssens, Gerda and
Ramon, Jan and Vandecasteele, Henk Executing Query Packs in ILP Proceedings
of the 10th International Conference on Inductive Logic Programming,LNAI 1866,
ed. J. Cussens and A. Frisch, 60–77, 2000
9. Muggleton, S. Inverse Entailment and Progol New Generation Computing, Special
issue on Inductive Logic Programming, vol. 13, N. 3-4, 245–286, 1995
10. R. Camacho, Inducing Models of Human Control Skills using Machine Learning
Algorithms PhD thesis,Department of Electrical Engineering and Computation,
Universidade do Porto, 2000
11. A. Srinivasan and R.C. Camacho Numerical reasoning with an ILP program capable
of lazy evaluation and customised search Journal of Logic Programming, vol. 40,
N. 2-3, 185–214, 1999
12. Costa, V. and Damas, L. and Reis, R. and Azevedo, R., YAP Prolog User’s Manual
Universidade do Porto, 1989


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Improving the efficiency of ILP systems

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