Disjunctive and conjunctive normal forms of pseudo-Boolean functions

The problem is the transformation of a conjunctive normal form (CNF) into a minimized (for the inclusion operator) disjunctive normal form (DNF) and vice versa. This operation is called the unionist product. For a CNF (resp. DNF), one pass of the unionist product provides the prime implicants (resp. implicates); two passes provide the prime implicates (resp. implicants). An algorithm built upon the classical Davis and Putnam procedure is presented for calculating, without the explicit minimization for the inclusion, this unionist product.

We consider the problem of efficiently indexing Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF) Boolean expressions over a high-dimensional multi-valued attribute space. The goal is to rapidly find the set of Boolean expressions that evaluate to true for a given assignment of values to attributes. A solution to this problem has applications in online advertising (where a Boolean expression represents an advertiser's user targeting requirements, and an assignment of values to attributes represents the characteristics of a user visiting an online page) and in general any publish/subscribe system (where a Boolean expression represents a subscription, and an assignment of values to attributes represents an event). All existing solutions that we are aware of can only index a specialized sub-set of conjunctive and/or disjunctive expressions, and cannot efficiently handle general DNF and CNF expressions (including NOTs) over multi-valued attributes. In this paper, we present a novel solution based on the inverted list data structure that enables us to index arbitrarily complex DNF and CNF Boolean expressions over multi-valued attributes. An interesting aspect of our solution is that, by virtue of leveraging inverted lists traditionally used for ranked information retrieval, we can efficiently return the top-N matching Boolean expressions. This capability enables emerging applications such as ranked publish/subscribe systems [16], where only the top subscriptions that match an event are desired. For example, in online advertising there is a limit on the number of advertisements that can be shown on a given page and only the "best" advertisements can be displayed. We have evaluated our proposed technique based on data from an online advertising application, and the results show a dramatic performance improvement over prior techniques.

Normal forms for fuzzy logic functions and their approximation ability

We describe the problem of mining set valued rules in large relational tables containing categorical attributes taking a finite number of values. Such rules allow for an interval of possible values to be selected for each attribute in condition instead of a single value for association rules, while conclusion contains a projection of the data restricted by the condition onto a target attribute. An example of such a rule might be “if HOUSEHOLDSIZE = {Two OR Tree} AND OCCUPATION={Professional OR Clerical} THEN PAYMENT_METHOD = {CashCheck (Max=249, Sum=4952) OR DebitCard (Max=175, Sum=3021)} WHERE Confidence=85%, Support=10%.}” We use an original conceptional and formal framework for representing multidimensional distribution induced from data by a number of so-called prime disjunctions upper bounding its surface. Each prime disjunction represents a wide multidimensional interval of impossible combinations of attribute values. This original formalism generalises the conventional boolean approach in two directions: (i) finite-valued attributes (instead of only 0 and 1), and (ii) continuous-valued semantics instead of (true and false). In addition, we describe an efficient algorithm, which carries out the generalised dual transformation from possibilistic disjunctive normal form (DNF) representing data into conjunctive normal form (CNF) representing knowledge and thus generates all the most interesting prime disjunctions. Once obtained they can be used to build different forms of rules or for other purposes (prediction, clustering etc.).

Fuzzy set theories allow us to represent our knowledge under various interpretations and axiomatic foundations from linguistic to computational representations. There are at least four levels of knowledge representation: (i) linguistic, (ii) meta-linguistic, (iii) propositional, and (iv) computational. There are three transformations, which depend on the particular interpretations put on a knowledge representation schema. There are various choices, corresponding to one's interpretation of: (a) type of set representation, (fuzzy or crisp), (b) type of propositional connectives and normal forms, and (c) type of computational connectives, i.e., weak or strong t-norms and co-norms. In the light of these selections, fuzzy disjunctive and conjunctive normal forms (FDNF, FCNF) are derived from fuzzy truth tables. It is shown that classical expressions such as excluded middle etc., when fuzzified, ought to be reinterpreted with a type II, second order, semantic uncertainty. The classical expressions should not be interpreted as to whether they are valid or not. One can only state that the well-known tautologies of classical logic are valid to many degrees specified by an interval defined by their FDNF and FCNF. FDNF and FCNF boundaries identify the nonspecificity measure associated with type II, second order, semantic uncertainty. Thus, those researchers who are not familiar or who are not concerned with type II semantic uncertainty work with a myopic understanding of fuzzy set and logic theories. >

Feature terms are a common denominator of basic data-structures in knowledge representation and computational linguistics. The adaptation of the usual unification algorithms for first order terms is not straightforward, because feature terms may contain logical disjunction. Expansion into disjunctive normal form would reduce the problem more or less to unification of first order terms. But for reasons of efficiency, rewriting into disjunctive normal form should not be compulsory. In this paper, a sequent calculus is defined which gives a clear formal basis for proof optimizations: inference steps which require more than one subproof, i.e. which lead towards a “disjunctive normal form”, are only performed when they are no longer unavoidable. It is shown that the calculus is sound and complete with respect to the so-called feature structure interpretations of feature terms.

An Algorithm is given which, as input, accepts a formula φ in prenex normal form, with prime formulas P=o, P≠o, for P ε Z [x 1 ,...,x 1 ], and some r ε N. It produces a quantifierfree formula ψ in disjunctive normal form (DNF) such that, in any algebraically closed field, φ ψ is true. The time required for a formula of length l with q quantifiers and r-q free variables is bounded by (c 1 ·l) c r 2 r q , for some constants c 1 , c 2 . If no ∀-quantifiers occur and the matrix of φ is in DNF, then the time is bounded by (c 1 ·l)rc q 2 .

A checking test for a monotone read-once function f depending essentially on all its n variables is a set of vectors M distinguishing f from all other monotone read-once functions of the same variables. We describe an inductive procedure for obtaining individual lower and upper bounds on the minimal number of vectors T(f) in a checking test for any function f. The task of deriving the exact value of T(f) is reduced to a combinatorial optimization problem related to graph connectivity. We show that for almost all functions f expressible by read-once conjunctive or disjunctive normal forms, T(f) ~n / ln n. For several classes of functions our results give the exact value of T(f).

We exhibit an unambiguous -DNF (disjunctive normal form) formula that requires conjunctive normal form width , which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon–Saks–Seymour problem in graph theory (posed in 1991), which asks, How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query/communication complexity, learning theory, and automata theory.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

The theory of Boolean functions, especially in respect to representing these functions in the disjunctive or conjunctive normal forms, is extended in this chapter onto the case of finite predicates. Finite predicates are decomposed by that into some binary units, which will correspond to components of Boolean vectors and matrices and are represented as combinations of these units. Further, the main concepts used for solving pattern recognition problems are defined, namely world model, data, and knowledge. The data presenting information about the existence of some objects with definite combinations of properties is considered, as well as the knowledge presenting information about the existence of regular relationships between attributes. These relationships prohibit some combinations of properties. In this way, the knowledge gives the information about the non-existence of objects with some definite (prohibited) combinations of attribute values. A special form of regularity representation, called implicative regularities, is introduced. Any implicative regularity generates an empty interval in the Boolean space of object descriptions, which do not contradict the data. The problem of plausibility evaluation of induced implicative regularities should be solved by that. The pattern recognition problem is solved by two steps. First, regularities are extracted from the database (inductive inference); second, the obtained knowledge is used for the object recognition (deductive inference).

One of the tasks of synthesis and analysis of the functioning of combinational devices is the representation of a Boolean function in various bases. The absolute completeness of the use and implementation of various transformations of Boolean functions is impossible without the implementation of transformations into universal bases of Schaeffer (AND-NOT) or Webb (OR-NOT). The Logic library of the Maple computer algebra system allows you to use only the operations &nand (Schaeffer basis) and &nor (Webb basis) to build a model of a digital device. Therefore, there is a need to expand the functionality of the Logic library of the Maple computer algebra system by representing Boolean functions in Schaeffer and/or Pierce (Webb) bases. In this regard, the authors of the article have developed procedures that allow the representation of disjunctive or conjunctive normal forms in the Schaeffer basis or in the Webb basis. The development was carried out taking into account the existing data processing functions and procedures. The article describes the technology of extending the Logic library of the Maple computer algebra system by developed procedures that implement the transformation of a given or received Boolean function in Schaeffer or Webb bases.

We study the learnability of read-k-satisfy-j (RkSj) DNF formulas. These are boolean formulas in disjunctive normal form (DNF), in which the maximum number of occurrences of a variable is bounded by k, and the number of terms satisfied by any assignment is at most j. After motivating the investigation of this class of DNF formulas, we present an algorithm that for any unknown RkSj DNF formula to be learned, with high probability finds a logically equivalent DNF formula using the well-studied protocol of equivalence and membership queries. The algorithm runs in polynomial time for $k\cdot j=O({\log n\over\log\log n})$, where n is the number of input variables.

CNF to DNF Conversion is considered as vast area of research by scientists for PLA's, circuit designs, FPGA's, artificial intelligence, etc. High dimension variable conversion has become a key demand in the current business standard. Various applications are in its requirement like gnome analysis, grid computing, bioinformatics, imaging system, rough sets requires higher variable processing algorithm. Problem statement is - Design and implementation of High dimension optimal conjunctive normal form to optimal (prime implicants) disjunctive normal form conversion which is an “NP hard problem conversion to an NP complete”. Thus CNF to DNF can only be considered to evaluate best performance for higher variable processing on high end systems. The best-known representations of Boolean functions f are those as disjunctions of terms (DNFs) and as conjunctions of clauses (CNFs). It is convenient to define the DNF size off as the minimal number of terms in a DNF representing f and the CNF size as the minimal number of clauses in a CNF representing f.

CNF to DNF Conversion is considered as vast area of research by scientists for PLA's, circuit designs, FPGA's, artificial intelligence, etc. High dimension variable conversion has become a key demand in the current business standard. Various applications are in its requirement like gnome analysis, grid computing, bioinformatics, imaging system, rough sets requires higher variable processing algorithm. Problem statement is - Design and implementation of High dimension optimal conjunctive normal form to optimal (prime implicants) disjunctive normal form conversion which is an “NP hard problem conversion to an NP complete”. Thus CNF to DNF can only be considered to evaluate best performance for higher variable processing on high end systems. The best-known representations of Boolean functions f are those as disjunctions of terms (DNFs) and as conjunctions of clauses (CNFs). It is convenient to define the DNF size of f as the minimal number of terms in a DNF representing f and the CNF size as the minimal number of clauses in a CNF representing f.