This is a fourth article in the series of works (see also first one, second one, and third one) describing Machine Learning system based on Lattice Theory named 'VKF-system'. The program uses Markov chain algorithms to generate causes of the target property through computing random subset of similarities between some subsets of training objects. This article describes bitset representations of objects to compute these similarities as bit-wise multiplications of corresponding encodings. Objects with discrete attributes require some technique from Formal Concept Analysis. The case of objects with continuous attributes asks for logistic regression, entropy-based separation of their ranges into subintervals, and a presentation corresponding to the convex envelope for subintervals those similarity is computed.  To encode object with only discrete attributes we need to compute an auxiliary bitset representations of values of each attribute. We assume that an expert is able to relate these values with respect to some 'general/special' partial order. The ordering must form a low semi-lattice after addition the special value 'null' (with shorthand '_' in some cases) to denote a trivial (absent) similarity between values of the given attribute of comparable objects. The representation of whole object is a concatenation of encodings of values of its attributes in some fixed order. Then bit-wise multiplication over long bitset strings reduces to multiplications over values of each attribute. Hence encoding must replace similarity between values by bit-wise multiplication.  Since any low semi-lattice easily converts into a lattice (with additional top element, if it is absent), Formal Concept Analysis (FCA) provides all essential tools.  Modern formulation of Fundamental Theorem of FCA asserts that
For every finite lattice  let  be a (super)set of all -irreducible elements and  be a (super)set of all -irreducible elements. For  the sample  generates the all candidates lattice  that is isomorphic to the original lattice . Element  of lattice  is called -irreducible, if  and for all   and  imply .
Element  of lattice  is called -irreducible, if  and for all   and  imply . The lattice below contains red vertices as -irreducible elements and blue vertices as -irreducible ones.  Fundamental theorem (initially proved by Prof. Rudolf Wille through the sample ) implies minimal sample of the form  to generate the lattice of all the candidates that is isomorphic to the original lattice. Note, that the Wille's sample uses 121 bites, and the new one needs only 24 bites!  The author proposed the following algorithm to encode values by bitsets: This algorithm is a part of both 'vkfencoder' (as vkfencoder.XMLImport class constructor) and 'vkf' (as vkf.FCA class constructor) CPython libraries. The difference is sources: vkf.FCA reads a MariaDB database table and vkfencoder.XMLImport reads an XML file. We discuss steps of encodings of continuous attributes case in the order of their inventions. At first, we apply an idea of C4.5 system of decision trees learning to separation of variable's domain into subintervals through entropy considerations.
After that we encode appearance of value in some subinterval by bitset in such a way that bit-wise multiplication corresponds to convex envelope of compared subintervals.
At last, we consider how to combine several attributes to obtain their disjunction and implications. The key is to compute logistic regression between them. When we have a continuous attribute its range must be separated into several subintervals of possible different influence on the target property. To choose correct thresholds we relate this attribute and the target property through entropy.  Let  be a disjoint union of training examples  and counter-examples . Interval  of values of continuous attribute  generates three subsets  , and
.  Entropy of interval  of values of continuous attribute  is    Mean information for partition  of interval  of values of continuous attribute  is   Threshold is a value  with minimal mean information.  For continuous attribute  denote  by  and let  be an arbitrary number greater than .
Thresholds  are computed sequentially by splitting the most entropy subinterval. We represent continuous attribute value by bitset string of length , where  is a number of thresholds. Bitset may be considered as a string of indicator (Boolean) variables   where .  Then string  is a bitset representation of continuous attribute  on element . The next Lemma asserts that the result of bit-wise multiplication of bitset representations is convex envelope of its arguments' intervals. Let  represent  and  represent . Then   corresponds to . Note that the trivial similarity  corresponds to the trivial condition . The FCA-based approach to Machine Learning naturally considers a conjunction of several binary attributes as a possible cause of the target property. In the case of discrete attribute an expert has opportunity to express disjunction of the values through additional values (see lattice structures in paragraph 1). The case of continuous attributes is different. So we need some technique to include this case too.  The key was the following Lemma The disjunction of propositional variables  is equivalent to satisfaction of inequality  for any . Since we restrict ourselves to two target classes, we look for a classifier Classifier is a map R, where  is a domain of objects to classify (described by  continuous attributes) and  are the target labels. As usual, we assume the existence of some probability distribution of  which can be decomposed as    where  is a marginal distribution of objects and  is a conditional distribution of labels on given object, i.e. for every  the following decomposition    holds. Error probability of classifier  is    Bayes classifier  with respect to 
corresponds to   We remind well-known Theorem on optimality of Bayes classifier The Bayes classifier  has the minimal error probability:    Bayes Theorem implies   where  and  is the well-known logistic function. Let approximate unknown  by linear combination  of basis functions  () with respect to unknown weights . For training sample  introduce signs . Then    Hence the logarithm of likelihood   is concave. Newton-Raphson method leads to iterative procedure    With help of  we obtain   where  is diagonal matrix with elements
 and  is vector with coordinates .   where  are iterative calculated weights.  As usual, the ridge regression helps to avoid ill-conditioned situation    In the computer program 'VKF system' we use standard basis: constant 1 and attributes themselves. At last, we need a criterion for significance of regression. For logistic regression two types of criteria were applied: Criterion of Cox-Snell declares attribute  significant, if    McFadden criterion declares attribute  significant, if    The 'VKF-system' was applied to Wine Quality dataset from Machine Learning repository (University California Irvine). The experiments demonstrated the prospects of the proposed approach. For high-quality red wines (with rating >7), all examples were classified correctly. The disjunction situation (from paragraph 2.3) arose with 'alcohol' and 'sulphates' relationship. Positive (although slightly different) weights correspond to different scales of measurement of different attributes, and the threshold was strictly between 0 and 1. The situation with 'citric acid' and 'alcohol' was similar. The situation with the pair ('pH', 'alcohol') was radically different. The weight of 'alcohol' was positive, whereas the weight for 'pH' was negative. But with an obvious logical transformation we get the implication ('pH'  'alcohol'). The author would like to thanks his colleagues and students for support and stimulus. Researcher