Logic based Pattern Recognition extends the well known similarity models, where the distance 
measure is the base instrument for recognition. Initial part (1) of current publication in iTECH-06 reduces the logic 
based recognition models to the reduced disjunctive normal forms of partially defined Boolean functions. This 
step appears as a way to alternative pattern recognition instruments through combining metric and logic 
hypotheses and features, leading to studies of logic forms, hypotheses, hierarchies of hypotheses and effective 
algorithmic solutions. Current part (2) provides probabilistic conclusions on effective recognition by logic means in 
a model environment of binary attributes

Aslanyan, Levon

Ryazanov, Vladimir

Bulgarian Digital Mathematics Library at IMI-BAS

International Journal "Information Theories & Applications" Vol.15 / 2008   314Authors' Information Peretz Shoval (Department of Information Systems Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel; e-mail: shoval@bgu.ac.il) is a Professor at the Dept. of Information Systems Engineering of Ben-Gurion University. He earned his Ph.D. in Information Systems from the University of Pittsburgh, where he specialized in expert systems for information retrieval. In 1984 he joined Ben-Gurion University, where he founded the Information Systems Program and later on founded and headed the Dept. of Information Systems Engineering. Prior to moving to academia, Shoval held professional and managerial positions in computer and software companies. Shoval's research interests include information systems analysis and design methods, data modeling, and information retrieval and filtering. Veronica Maidel (Department of Information Systems Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel; e-mail: maidel@bgu.ac.il) received her B.Sc. from Tel-Aviv University in 2001 and is currently a graduate student at the Dept. of Information Systems Engineering of Ben-Gurion University. Her research is on content-based filtering. This paper is part of her research. Bracha Shapira (Department of Information Systems Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel; e-mail: bshapira@bgu.ac.il) is Senior Lecturer at the Department of Information Systems Engineering of Ben-Gurion University. She holds a M.Sc. in Computer Science from the Hebrew University in Jerusalem and a Ph.D. in Information Systems from Ben-Gurion University. Her research interests include Information Retrieval and Filtering, specializing in various aspects of user profiling and personalization. In addition, she has worked on privacy preservation while browsing and on formal models of Information Retrieval systems. She is leading research projects in these domains at the Deutche-Telekom research lab at Ben-Gurion University.    LOGIC BASED PATTERN RECOGNITION - ONTOLOGY CONTENT (2)1 Levon Aslanyan, Vladimir Ryazanov Abstract: Logic based Pattern Recognition extends the well known similarity models, where the distance measure is the base instrument for recognition. Initial part (1) of current publication in iTECH-06 reduces the logic based recognition models to the reduced disjunctive normal forms of partially defined Boolean functions. This step appears as a way to alternative pattern recognition instruments through combining metric and logic hypotheses and features, leading to studies of logic forms, hypotheses, hierarchies of hypotheses and effective algorithmic solutions. Current part (2) provides probabilistic conclusions on effective recognition by logic means in a model environment of binary attributes.  1. Introduction Pattern Recognition consists in reasonable formalization (ontology) of informal relations between object’s visible/measurable properties and of object classification by an automatic or a learnable procedure [1]. Similarity measure [1] is the basic instrument for many recognition formalisms but additional means are available such as logical terms discussed in part (1) of current research [2]. Huge number of recognition models follows the direct goal of increasing recognition speed and accuracy. Several models use control sets above ordinary learning sets, others use optimization and other direct forces. Besides, more alternative notions are available to describe algorithmic properties. In existing studies the role of these notions is underestimated and less attention is paid to these components. In part (1) the attention is paid to implementing the learning set through its pairs of elements rather than the elements separately. The following framework is considered: given a set of logical variables                                                           1 The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be  International Journal "Information Theories & Applications" Vol.15 / 2008   315(properties) nx,...,x,x 21  to code the studied objects, and let we have two types/classes for classification of objects: 1K  and 2K . Let 1K∈β , and 2K∈γ , and α  is an unknown object in sense of classification. We say, that γ  is separated by the information of β  for α  if αβγβ ⊕≤⊕ , where ⊕  is 2mod  summation. Formally, after this assumption, the reduced disjunctive normal forms of two complementary partially defined Boolean functions appear to describe the structure of information enlargement of the learning sets. The idea used is in knowledge comparison. α  is an object of interest. Relation αβγβ ⊕≤⊕  informs that the descriptive knowledge difference of β  and α  is larger than the same difference of β  and γ . This approach we call logic separation. While notion of similarity gives the measure of descriptive knowledge differences, the logic separation describes areas which are preferable for classes and learning set elements. In general the question is in better use of learning set. The learning set based knowledge, which is used by recognition procedure, at least is supposed to reconstruct the learning set itself. It is indeed negative when this information is not able to reconstruct the learning set. It is easy to check that the similarity knowledge can’t reconstruct an arbitrary learning set, and only special sets allow reconstructing of objects by their distances [3]. Restructuring power is high when comparison is used for the set of all attribute subsets. Theoretically such structures are studied in discrete tomography problems [4], but practically even the use of pairs draws to known hard computational area of disjunctive normal forms.  Consider pairs of elements of the learning set, where each pair contains elements of different classes (the case of 2 learning classes is supposed). It was shown [2] that the logical separators divide the object space into three areas, where only one of these areas needs to be treated afterward by AEA (algorithms of estimation analogies – voting algorithms) [1]. This set is large enough for almost all weakly defined Boolean functions, but for the functions with compactness property it is small. Let, for nkk ≤<≤ 100 , 10 k,k,nF  be the set of all Boolean functions defined as follows: each of them has zero (false) value on the vertices of 0k -sphere centered at 0~ , and has one (true) value on ( 1kn − )-sphere centered at 1~ . On the remainder vertices of n -cube the assignment/evaluation is arbitrary. These functions (for appropriate choice of 0k  and 1k ) satisfy the compactness assumptions [8], and their quantity is not less thann)n( 22ε  for an appropriate 0→)n(ε  with 0→n . For these functions we have also, that for recovering the full classification by means of logical separators procedure, it is enough to consider a learning set which consists of any n)n(nn ε−2  or more arbitrary points. This is an example of postulations which will be considered below. It is relating the metric and logic structures and suppositions, although separately studies of these structures are also important. The follow up articles will describe the mixed hierarchy of recognition metric-logic interpretable hypotheses, which helps to allocate classification algorithms to the application problems. 2. Structuring by Logic Separation Let f be a Boolean function (it might be partially or completely defined). Let fN  denotes the reduced disjunctive normal form of f  and sets fN 0 ,…,fN3  [2] define areas, in which fN  and fN  take values {0,1}, {1,0}, {0,0} and {1,1} correspondingly. Identical to fN0 ,…,fN3 , similar areas are defined by logic separation - fM 0 ,…,fM 3 . Let )(20 nPf ∈  (a completely defined Boolean function of n  variables) and 1)~(0 =αf . Denote by )~,( 0 αft  the number of k-subcubes included in 0fN  and covering the vertex α~ . Let km  is the average number of )~,( 0 αft  calculated for all )(20 nPf ∈  and 0~ fN∈α . It is easy to check that 12122222222−−−== knkn knnknnkCCm . Dispersion kd  of the same value )~,( 0 αft  is expressed as 2120122221⎟⎟⎠⎞⎜⎜⎝⎛−= −=++−−−∑ + kjk knkjjkknjkknkCCCCd . International Journal "Information Theories & Applications" Vol.15 / 2008   316Applying the Chebishev inequality to above measures )~,( 0 αft , km , kd  leads to the conclusion: Proposition 1(8). 120 2~)~,( −kknCft α  for almost all pairs )(20 nPf ∈  and 0~ fN∈α , when ∞→n  and ∞→kknC22.  Taking into account that for almost all Boolean functions the number of 1-vertices is equivalent to ∞→− nn ,2 1 , we obtain that for almost all functions )(20 nPf ∈ , almost all 1-vertices are covered by the number of k-intervals from 0fN , which is equivalent to 122 −kknC , when ∞→n and ∞→kknC22. Particularly, this fact might be used to adjust the postulation in Proposition 7, [2]. Indeed, the 122 −kknC  intervals, coming from a common fixed vertex, cover not less than 122 −kknC  vertices of an n-cube. Now consider arbitrary placement of any l  points into the vertices of an n-cube M . Estimate for almost all functions )(20 nPf ∈  (see Proposition 1(8)) the main value of the number of vertices 0~ fN∈α , which are not covered by any of the k-intervals included in 0fN  which is pricked by our l  vertices: llCnnnkknnCCnlkn22/))(1(2111222))(1(),,(−+−−+ εεμ ≺ , ∞→n , 0)(1 →nε , 0)(2 →nε , and ∞→kknC22. Proposition 2(9). If ∞→kknC22 and knnCnlk222)(ϕ≥ , where ∞→)(nϕ as ∞→n , then random l  vertices for almost all functions )(20 nPf ∈  prick such sets of k-subcubes included in 0fN , which cover almost all 0fN .  In case of ]log[log nk =  we conclude that the minimal number l  satisfying the above proposition, is not greater than ]log[log2 /2 nnn Cn .  Notice, that in conditions of Proposition 7 [2] and Proposition 2(9) only the usability of condition 0F  (logic separation) is mentioned, so that these are the conditions, when usage of 0F , as a rule, doesn’t imply to significant errors. Also, it is important, that we applied the condition 0F  to the whole class )(2 nP , although it was supposed for problems, satisfying compactness suppositions. So, it is interesting to know how completely the class )(2 nP  satisfies to these suppositions.  Let us bring now a particular justification of compactness conception [8]. Let )(20 nPf ∈ . We call the vertex M∈α~  boundary vertex for function 0f , if the sphere )1,~(αS  of radius 1 centered at α~ , contains a vertex for which 0f  has the opposite value to )~(0 αf . Denote by )( 0fΓ  the set of all boundary vertices of 0f . We will say that the function hipping (completion) procedure obeys the compactness conditions, if .),2()( 0 ∞→=Γ nof n  It is easy to calculate that the average number of boundary vertices of functions )(20 nPf ∈  is almost n2 . This shows that )(2 nP  contradicts the compactness conditions. The same time we proved that the use of the 0F  rule in a very wide area )(2 nP  doesn’t move to a sensitive error. Below we consider an example problem, which obeys the compactness assumptions, and will follow the action of the rule 0F  on that class. Before that we justify some estimates for the set fM 3 . International Journal "Information Theories & Applications" Vol.15 / 2008   317Consider the class ))(),(,(2 nlnknΦ  of all of partial Boolean functions, for which )(0 nlM =  and )(1 nkM = . We'll deal with the case )2()( nonl =  and )2()( nonk = . Estimate now the quantitative characteristics of sets fM 0 ,fM 1  and fM 3 . First estimate the average number of vertices of the cube, which are achievable from set 0M : ",1,0,12 )(2)(2)()(2)(2)(2030 =⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∑−≥−−−− jCCCCC nlnlCnknlnknlnnjjnnnjn  Proposition 3(10). If )(nk  and )(nl  are )2( no , ∞→n  and there exists a 0j , that 02)( 0 →−njnk  and ∑ ∞→− 00)(2jinn nlC , then for almost all functions of class ))(),(,(2 nlnknΦ , ∞→≈ noM nf ),2(1 .  To except the trivial cases in the pattern recognition problems we have to suppose, that ∞→≅ nnlnk ),()( . Then it is clear that choosing appropriate values for 0j  we get )2(1nf oM =  and )2(0 nf oM =  for almost all functions of class ∞→Φ nnlnkn )),(),(,(2 . Let us give an other estimation of 03c :  )(12)(22003 nlnlnjjnnjnCCCc−−=∑≥ . If )(nλ  is the minimal value for which ∞→∑+−=nC nnnnniin ,2~)(2/)(2/λλ, then nn ≈)(λ . From here we conclude: Proposition 4(11). If 0)( ≥nl  and 02)(2 )(2 →− nnCn nk  as ∞→n  for )(nc∀  - restricted, then almost ever )2(~1nf oM . So, for the small values of )(nk  and )(nl  from the each vertex of set 10 MM ∪ , almost all vertices of the n-unite-cube almost ever are achievable. Comparing this, for example with [5] we find that for these classes 0F  works ineffectively. 3. Logic Separation on Compact Classes Consider problems, satisfying the compactness assumptions. First of all it is evident, that for )( 010 fMM Γ⊇∪  the continuation of function f  made on base of 0F , exactly correspond to the final result 0f . Taking into account that by the given description of the compactness assumptions ∞→=Γ nof n ),2()( 0 , we receive that in problems, satisfying the compactness assumptions we can point out learning sets of size ∞→no n ),2( , which allow to complete and exact continuation of function 0f  on base of condition 0F only. Let M∈α~  and nkk ≤≤≤ 210 . Consider functions )(20 nPf ∈ , for which ),~()( 200 knSfM −⊇ α , ),~()( 101 knSfM −⊇ α  and which receive arbitrary values on vertices of sets ),~()1,~( 12 kSkS αα ⊇− . Denote the class of these functions by )(nK . It is evident, that for )2(),~(\)1,~( 12nokSkS =− αα  all the constructed functions satisfy the given formalisms for the compactness assumptions, and that the quantity of these functions is not less than nn 2)(12ε , where )(1 nε  is an arbitrary function of 0)(, 1 →nn ε  with the ∞→n . International Journal "Information Theories & Applications" Vol.15 / 2008   318Take a point 1,)~,~(,~ kkkM <=∈ βαρβ . It is evident that no more than [ ] ∞→≅ nnC nnn ,222/ π  subsets of any fixed size are coming out from any point of n -cube. From the other hand it is evident, that it is enough to take )(1 nokk =−  as ∞→n  to get the )2(),~(\),~( 1 nokSkS =αα . Suppose, that lfMfM =∪ )()( 0100 , and that l  points appear as the result of their appropriate placement on the vertices of the n -cube M , when all of these placements are equally probable. Estimate the probability of reaching of this point β~  from zeros of function 0f . 0,22 )(2222 )(2 →≤ − nlnlnnnnCCnεπτε  with ∞→n . From here we conclude the Proposition 5(12). Let ),,( 210 kknKf ∈  and 1f  -- is the continuation of function f  on base of condition 0F . If )2(2 )(2 nnnn onl =≥ −ε , ∞→n  and the set )()( 10 fMfM ∪  is formed as a random collection of points of size l  from the set M , then almost ever the function 1f  is the continuation for f , which converges to the 0f  by the accuracy, tending to 1 with the ∞→n . Conclusion Logic Separation is an alternative approach to pattern recognition hypotheses and formalisms, while the base concept uses the similarity approach. Structures appearing in this relation are based on terms of Reduced Disjunctive Normal Forms of Boolean Functions. Propositions 1-5(8-12) provide additional knowledge on quantitative properties of areas appearing in extending classification by means of compactness and logic separation principles.  Bibliography 1. Zhuravlev Yu. I. On an algorithmic approach to the problems of recognition and classification. Problemi Kibernetiki, 33, (1978) 5--68. 2. Aslanyan L. and Castellanos J., Logic based pattern recognition – ontology content (1), iTECH-06, 20-25 June 2006, Varna, Bulgaria, Proceedings, pp. 61-66. 3. Gavrilov G. and Sapojenko A., Collection of exercises of discrete mathematics, NAUKA, Moscow, 1973, 368 p. 4. Herman G.T. and Kuba A., editors. Discrete Tomography: Foundations, Algorithms and Applications. Birkhauser, 1999. 5. Zhuravlev Yu. I. Selected Scientific Works, Publishing House Magister, Moscow, (1998) 417p. 6. Aslanyan L. H. On a pattern recognition method based on the separation by the disjunctive normal forms. Kibernetika, 5, (1975), 103--110. 7. Vapnik V. and Chervonenkis A. Theory of Pattern Recognition. “Nauka”, 1974. 8. Aslanyan L. H. The Discrete Isoperimetric Problem and Related Extremal Problems for Discrete Spaces. Problemy Kibernetiki, 36, (1979), 85--128. 9. Nechiporuk E. I. On topological principles of self-correcting. Problemy Kibernetiki, 21, (1969), 5--102. 10. Graham N., Harary F., Livingston M. and Stout Q. Subcube Fault-Tolerance in Hypercubes. Information and Computation 102 (1993), pp. 280{314. 11. Glagolev V. V. Some Estimations of D.N.F. for functions of Algebra of Logic. Problemy Kibernetiki, 19, (1967), 75--94. Authors' Information Levon Aslanyan – Institute for Informatics and Automation Problems, NAS Armenia, P.Sevak St. 1,  Yerevan-14, Armenia; e-mail: lasl@sci.am  Vladimir Ryazanov - Computing Centre of the Russian Academy of Sciences, 40 Vavilova St., Moscow, GSP-1, 119991, Russian Federation, rvvccas@mail.ru  

Logic Based Pattern Recognition - Ontology Content (2)

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Logic Based Pattern Recognition - Ontology Content (2)

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