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software testing aims at understanding the risks of software implementation and providing proper quality of the system under test .
one of fundamental problems is that testing under all possible combinations of inputs is not feasible .
this holds for functional testing , while non - functional aspects ( performance , compatibility , etc . )
are left apart . for business critical applications
black - box testing is a widely - used approach .
it examines the functionality of a system under test without dealing with its implementation .
it is important to notice that static testing involves verification , while dynamic testing involves validation .
one of the biggest challenges in black - box dynamic testing is choosing the proper approach to enumerate all possible cases .
domain testing helps quality assurance engineer to define classes of input values , that are crucial to test .
the naive way is just to check for possible combinations of such parameters , and it immediately leads to exponential complexity . even in the case of 6 boolean parameters an expert has to check 64 combinations .
a possible solution is to generate test - cases in manual way .
a software test engineer considers all possible cases ordered in a certain way .
the main risk is to lose some information behind the cases . actually , it could be rather exhausting to cover all possibilities for a large number of parameters and not to skip some scenarios .
a popular alternative way is pairwise testing  @xcite , or its generalization , n - wise testing .
we define parameters and domains , and pass them like a model into a black - box algorithm  @xcite , which gives us a set of test - cases satisfying a certain condition ( for each pair of input parameters all possible discrete combinations of those parameters are tested ) . in general case it can produce different cases in different runs , while it can be fixed by passing a random seed to it .
the main advantage of the approach is its insensitivity to parameter values .
however , it can be a rather computationally demanding task .    for quality assurance engineers
it is sufficient to have knowledge about possible behavior of the program .
usually , there are dependencies between input parameters .
a natural form to express such dependencies in mathematical terms is implication , a statement of the form : ` if ... , then ... ' .
the ` if'-part is called premise , and the ` then ' is called conclusion .
consideration of parameter s interdependence can decrease the complexity of result test - cases in terms of their quantity , by excluding some of possible parameter combinations .
proposed approach to test - case generation is focused on implications .
we use an approach based on formal concept analysis ( fca)@xcite .
fca provides software engineers with a tool for exploring the domain of interest in semi - automatic way .
the algorithm outputs sound and complete description of the problem , but it is still highly dependent on the accuracy of expert s answers
. surveys on fca techniques and applications can be found in  @xcite .
fca - based approaches have already been applied in software engineering , e.g. , for inference of class hierarchies  @xcite , class design  @xcite , refactoring  @xcite .
another application of fca technique is analysis of object - oriented approach for a given system  @xcite .
the questions of mapping lines of source code to the functionality from requirements is of crucial importance for big systems  @xcite .
the rest of the paper is organized as follows : in section 2 we introduce basic notions of formal concept analysis . in section 3
we focus on the procedure of attribute exploration .
section 4 provides example of attribute exploration in the field of positive integers .
we make conclusions in section 5 .
formal concept analysis  @xcite ( fca ) provides mathematical technique for deriving applied ontologies from data .
fca relies on lattice and order theories  @xcite numerous applications are found in the field of machine learning , data mining , text mining and biology , see  @xcite .
a formal context @xmath0 is a triple @xmath1 , where g denotes a set of objects , m is a set of attributes , and @xmath2 is a binary relation between @xmath3 and @xmath4 .    it can be interpreted in the following way : for objects in @xmath3 there exists a description in terms of attributes in @xmath4 , and relation @xmath5 reflects that an object has an attribute : @xmath6 object @xmath7 possesses  @xmath8 .
an example of a formal context is provided below :        ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(1,2)5 ( 10,0)(-1,2)5    & @xmath9 & & & @xmath9 +    ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(0,1)10 ( 0,10)(1,-1)10    & @xmath9 & & @xmath9 & +    ( 20,10 ) ( 0,0)(1,0)20 ( 0,0)(0,1)10 ( 0,10)(1,0)20 ( 20,0)(0,1)10    & & @xmath9 & @xmath9 & +    ( 10,10 ) ( 0,0)(1,0)10 ( 0,0)(0,1)10 ( 0,10)(1,0)10 ( 10,0)(0,1)10    & & @xmath9 & @xmath9 & @xmath9 +    objects : attributes :    p4cmp4 cm    *  equilateral triangle *  right triange , *  rectangle , *  square ,    &    *  3 vertices , *  4 vertices , *  has a right angle , * 
all sides are equal    for a given context two following mappings are considered :    * @xmath10 * @xmath11    for all @xmath12 , @xmath13    1 .
@xmath14 2 .
@xmath15 3 .
@xmath16    mappings @xmath17 and @xmath18 , satisfying properties 1 - 3 above , define a galois connection between @xmath19 and @xmath20 , which means : @xmath21    traditionally , notation @xmath22 is used instead of @xmath17 and @xmath18 .
@xmath23 stands both for @xmath24 and @xmath25 ( depending on the argument ) . for arbitrary @xmath26 ,
@xmath27 @xmath28 @xmath29    _ ( formal ) concept _ is a pair @xmath30 : @xmath31 , @xmath32 , @xmath33 , @xmath34    in the example with geometric figures a pair @xmath35 is a formal concept . for a formal context @xmath36 ,
@xmath37 - set of objects , @xmath27 - set of attributes , the following statements hold for operation ( @xmath38) :    1 .
@xmath39 2 .
@xmath40 3 .
@xmath41 4 .
@xmath42 and @xmath43 5 .
@xmath44 6 .
@xmath45    _ closure operator _ on set @xmath3 is a mapping @xmath46 , which maps every @xmath47 to _ closure _ @xmath48 , under the following conditions :    1 .
@xmath49 ( _ idempotence _ ) 2 .
@xmath50 ( _ extensivity _ ) 3 .
@xmath51 ( _ monotonicity _ )    _ implication _ @xmath52 , where @xmath53 , takes place if @xmath54 , in other words if each object having @xmath55 also has all attributes from @xmath56 .
implications comply with armstrong axioms : @xmath57 @xmath58 @xmath59
one of advantages of building lattices of formal concepts is to get effective navigation from more general concepts to more specific .
for example , the line diagram for context with figures from previous section is shown in fig . 1 .    [ figures_lat ]     that property could be beneficial in two ways : regression testing , system description .
first of all , for the context of regression testing it implements the algorithm to determine the classes of tests that fail .
it could be considered in the following way .
let us define set of attributes for all tests that are being run .
[ cols="^,^,^,^,^",options="header " , ]     implication list :    1 .   * if * iqv * then * iqp 2 .   * if * isv * then * isp 3 .   * if * iiv * then * iip 4 .   *
if * = * then * ibs , not < , not > 5 .   *
if * < * then * ibs , not = , not > 6 .   *
if * ibs * then * not > 7 .   *
if * not iqp * then * not iqv 8 .
* if * not isp * then * not isv 9 .   *
if * not iip * then * not iiv 10 . *
if * not iiv , not isv , not iqv * then * not = , not < , not > , not ibs 11 . *
if * not < , not = * then * not ibs 12 . * if * not > , iqv , iqp * then * ibs 13 . * if * not > , isv , isp * then * ibs 14 . *
if * not > , iiv , iip * then * ibs 15 . * if * not > , not isv , not iqv , ibs * then * iiv 16 . *
if * not > , not isv , not iiv , ibs * then * iqv 17 .
* if * not > , not iiv , not iqv , ibs * then * isv 18 . *
if * not > , not = , ibs * then * < 19 . * if * not > , not < , ibs * then * = 20 .
* if * not ibs * then * not = , not < 21 . * if * not ibs , not < , not = , iqv , iqp * then * > 22 . * if * not ibs , not < , not = , isv , isp * then * > 23 . *
if * not ibs , not < , not = , iiv , iip**then * * > 24 . *
if * not ibs , not < , not = , > , not iqv , not isv**then * * iiv , iip 25 . *
if * not ibs , not < , not = , > , not iiv , not iqv**then * * isv , isp 26 . *
if * not ibs , not < , not = , > not iiv , not isv**then * * iqv , iqp 27 . *
if * not ibs , not > , not = , not < * * then * * not iqv , not isv , not iiv
the most important advantage of the proposed approach is the plug - in design .
it can be easily incorporated in existing process of test - case generation . for manual test - case design
one can develop test - cases in proposed system .
moreover , the step of implication extraction could be postponed up to the review of obtained cases .
it is important to notice that attribute exploration is a good technique to extract dependencies that could be formulated as requirements for the system under test . for the case of automatic test case generation ,
extraction of dependencies could eliminate the step of test debug and replace it with review of obtained implications and the model of the system under test .
also the proposed technique is applicable to verify the correctness of defining type of model , precisely wiseness of it . since big values of @xmath60 in n - wise modeling impose more test cases and it results in the growing time of test run execution .
formal concept analysis provides us with useful framework for software testing tasks .
it is especially beneficial in regression testing meta report construction , feature navigation , test case analysis and derivation .
it unites best practices of manual development and automatic generation of test scenarios .
it provides sound and complete description of the investigated domain , based on expert knowledge .
the output of the system consists of two main parts : the description of typical objects in the domain , and interdependence between parameters in terms of implications .
an important advantage of proposed technique is extensibility .
if we add a new attribute , we can just copy the previous examples into new formal context , assuming that new attribute is absent for all objects and proceed with the procedure of attribute exploration .
it holds even for the very start of procedure .
we can start with non - empty set of objects and implications simultaneously .
the described algorithm could be used as a standalone solution for the test case design , as well as , tool to get exiting dependencies in the domain .
the obtained implications could be valuable in pairwise testing to adjust the model description .
however , we should admit that the current approach is limited in terms of attribute description .
for now , it is highly dependent on the boolean nature of attributes .
one of the main directions of future work is to work with descriptions of general form by means of patterns structures  @xcite , an extension of fca .
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software testing uses wide range of different tools to enhance the complicated process of defining quality of the system under test . 
 formal concept analysis ( fca ) provides us with algorithms of deriving formal ontology from a set of objects and their attributes . with the use of fca 
 we can considerably improve the efficiency of test case derivation . 
 moreover , an fca - based machine learning system supports the analysis of regression testing results .