1 S = (X, A  {d[1],d[2],..,d[k]}, V), where: - X is a finite set of objects, - A is a finite set of classification attributes, - {d[1],d[2],..,d[k]} is a set of hierarchical decision attributes, and V =  {V a : a  A  {d[1],d[2],..,d[k]}} is a set of their values. We assume that: V a, V b are disjoint for any a, b  A  {d[1],d[2],..,d[k]}, such that a ≠ b, a : X →V a is a partial function for every a  A  {d[1],d[2],..,d[k]}. Decision queries (d-queries) for S - a least set T D such that: - 0, 1  T D, - if w   {V a : a  {d[1],d[2],..,d[k]}}, then w, ~w  T D, - if t 1, t 2  T D, then (t 1 + t 2 ), (t 1  t 2 )  T D. Multi-Hierarchical Decision System Incomplete Database Atomic Level

2 Example Xabcd x1a[1]b[2]c[1]d[3] x2a[1]b[1]c[1]d[3,1] x3a[1]b[2]c[2,2]d[1] x4a[2]b[2]c[2]d[1] C[1]C[2] C[2,1]C[2,2] d[1]d[2] d[3,1]d[3,2] 3 d[3]Level I Level II Classification AttributesDecision Attributes

3 Classification terms (c-terms) for S are defined as the least set T C : -0, 1  T C, -if w   {V a : a  A}, then w, ~w  T C, -if t 1, t 2  T C, then (t ), (t 1  t 2 )  T C. c-term t is called simple if t = t 1  t 2  …  t n and (  j  {1,2,…,n}) [(t j   {V a : a  A})]  (t j = ~w  w   {V a : a  A})]. d-query t is called simple if t = t 1  t 2  …  t n and (  j  {1,2,…,n}) [(t j   {V a : a  {d[1],d[2],..,d[k]}})  (t j = ~w  w   {V a : a  {d[1], d[2],.., d[k]})]. By a classification rule we mean an expression [t 1  t 2 ], - both t 1 and t 2 are simple. Simple Term to Simple Query Atomic Level

4 Semantics M S of c-terms in S is defined in a standard way as follows: - M S (0) = 0, M S (1) = X, - M S (w) = {x  X : w = a(x)} for any w  V a, a  A, - M S (~w) = {x  X : (  v  V a )[v = a(x)  v≠w]} for any w  V a, a  A, - if t1, t2 are terms, then M S (t1 + t2) = M S (t1)  M S (t2), M S (t1  t2) = M S (t1)  M S (t2). Classifier-based semantics M S of d-queries in S = (X, A  {d[1],d[2],..,d[k]}, V ), if t is a simple d-query in S and {r j = [t j  t]: j  J t } is a set of all rules defining t which are extracted from S by classifier, then M S (t) = {(x,p x ): (  j  J t )(x  M S (t j )[p x =  {conf(j)  sup(j): x  M S (t j ) & j  J t }/  {sup(j): x  M S (t j ) & j  J t }]}, where conf(j), sup(j) denote the confidence and the support of [t j  t], correspondingly. Classifier-based Semantics

5 Attribute value d[j 1, j 2,…j n ] in S is dependent on an attribute value which is either its ancestor or descendant in d[j 1 ]. d[j 1, j 2,…j n ] is independent from any other attribute value in S. Let S = (X, A  {d[1],d[2],..,d[k]}, V), w  V d[i], and IV d[i] be the set of all attribute values in V d[i] which are independent from w. Standard semantics N S of d-queries in S is defined as follows: -N S (0) = 0, N S (1) = X, -if w  V d[i], then N S (w) = {x  X : d[i](x)=w}, for any 1  i  k - if w  V d[i], then N S (~w) = {x  X : (  v  IV d[i] )[ d[i](x)=v]}, for any 1  i  k -if t 1, t 2 are terms, then -N S (t 1 + t 2 ) = N S (t 1 )  N S (t 2 ), N S (t 1  t 2 ) = N S (t 1 )  N S (t 2 ). Standard Semantics of D-queries

6 Let S = (X, A  {d[1],d[2],..,d[k]}, V), t is a d-query in S, N S (t) is its meaning under standard semantics, and M S (t) is its meaning under classifier-based semantics. Assume that N S (t) = X  Y, where X = {x i, i  I 1 }, Y = {y i, i  I 2 }. Assume also that M S (t) = {(x i, p i ): i  I 1 }  {(z i, q i ): i  I 3 } and {y i, i  I 2 }  {z i, i  I 3 }= . The Overlap of Semantics NSNS I2I2 I1I1 I3I3 MSMS

7 Precision & Recall Precision Prec(M S, t) of a classifier-based semantics M S on a d-query t: Prec(M S, t) = [  {p i : i  I 1 } +  {(1 – q i ) : i  I 3 }] [card(I 1 ) + card(I 3 )]. Recall Rec(M S, t) of a classifier-based semantics M S on a d-query t: Rec(M S, t) = [  {p i : i  I 1 }]____ [card(I 1 ) + card(I 2 )]. NSNS I2I2 I1I1 I3I3 MSMS