Indexing Boolean expressions

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.

A great amount of fault-based testing strategies have been proposed to generate test cases for detecting certain types of faults in Boolean specifications. However, most of the previous studies on these strategies were focused on the Boolean expressions in the disjunctive normal form (DNF), even the irredundant DNF (IDNF)-little work has been conducted to comprehensively investigate their performance on general Boolean specifications. In this study, we conducted a series of experiments to evaluate and compare 18 fault-based testing strategies using over 4000 randomly generated fault-seeded Boolean expressions. In the experiments, a testing strategy is regarded as effective and efficient if it can detect most of the seeded faults using a small number of test cases. Our experimental results show that if a testing strategy is highly effective and efficient when testing the Boolean expressions in the IDNF, it also shows high effectiveness and efficiency on general Boolean expressions. It is found that one family of fault-based testing strategies, namely MUMCUT, normally deliver the best performance among all the 18 strategies. Our study provides an in-depth understanding and insight of fault-based testing for general Boolean expressions.

Article Free Access Share on Refinement types for ML Authors: Tim Freeman School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania School of Computer Science, Carnegie Mellon University, Pittsburgh, PennsylvaniaView Profile , Frank Pfenning School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania School of Computer Science, Carnegie Mellon University, Pittsburgh, PennsylvaniaView Profile Authors Info & Claims PLDI '91: Proceedings of the ACM SIGPLAN 1991 conference on Programming language design and implementationMay 1991 Pages 268–277https://doi.org/10.1145/113445.113468Online:01 May 1991Publication History 293citation1,256DownloadsMetricsTotal Citations293Total Downloads1,256Last 12 Months93Last 6 weeks10 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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.).

Normal forms for fuzzy logic functions and their approximation ability

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. >

Fault based testing aims at detecting hypothesized faults based on specifications or program source. There are some fault based techniques for testing Boolean expressions which are commonly used to model conditions in specifications as well as logical decisions in program source. The MUMCUT strategy has been proposed to generate test cases from Boolean expressions. Moreover, it detects eight common types of hypothesized faults provided that the original expression is in irredundant disjunctive normal form, IDNF. Software practitioners are more likely to write the conditions and logical decisions in general form rather than IDNF. Hence, it is interesting to investigate the fault detecting capability of the MUMCUT strategy with respect to general form Boolean expressions. In this article, we perform empirical studies to investigate the fault detection capability of the MUMCUT strategy with respect to general form Boolean expressions as well as mutated expressions. A mutated expression can be obtained from the original given Boolean expression by making a syntactic change based on a particular type of fault.

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).

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.

Complexity analysis of the SAT engine: DNA algorithms as probabilistic algorithms

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 .

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.

We present some compositions of clones of Boolean functions that imply factorizations of Ω, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems, providing representations of Boolean functions analogous to the disjunctive normal form, the conjunctive normal form, and the Zhegalkin polynomial representations.