Efficient Algorithms for Description Problems over Finite Totally Ordered Domains

Generalized operations of fuzzy logic are considered. New methods of generation of involutive, contracting and expanding negations are proposed. The methods of generation of simple parametric classes of non-associative conjunctions are discussed. The ways of embedding of strict monotonic conjunction and disjunction operations on ordinal scales based on the concept of lexicographic valuation (uncertainty with memory) and aimed to application in expert systems are described.

There are three main families of inference algorithms in first-order logic: direct inference and its application to deductive databases and production systems; backward inference procedures and logic programming systems; theorem proving systems based on the resolution method. When solving specific problems, the most effective algorithms are those that allow you to cover all the facts and axioms and must be taken into account in the process of inference. An example is considered in which it is necessary to prove the guilt of a person in murder. On the basis of statements, a knowledge base is formed from expressions, with the help of which an expression of first-order logic is compiled and proved using direct logical inference. The proof of the reasoning obtained in direct inference using the proof tree is given. However, direct inference provides for the implementation of all admissible stages of logical inference based on all known facts. The article also considers a method based on the resolution when implementing the reverse inference, taking into account the expression obtained in the direct inference. This expression is converted into a conjunctive normal formula using the laws of Boolean algebra and is proved by the elimination of events using the conjunction operation.

We deal with the followingon-line 2-satisfiability problemP(m, n): starting fromC(0)=true, consider a sequence ofm Boolean formulasC(k) (inn variables and in conjunctive normal form), each of them being the intersection of the previous one with a single clause which is the union of two literals. Solve the sequence of 2-satisfiability problemsC(k)=true,k=1,...,m. It is well known that a 2-satisfiability problem involvingm clauses can be solved inO(m) time. Thus, by a naive approach one can solveP(m, n) in overallO(m 2) time. We present an algorithm with overallO(nm) time complexity, which for every formula not only checks its satisfiability, but also actually computes a solution (if any), and moreover, detects all forced and all identical variables. Our algorithm makes use of an efficient on-line transitive closure procedure by Italiano. We discuss two applications to the design of integrated electronic circuits and to edge classification in automated perception.

One aspect that has been poorly studied in multiple-valued logics, and in particular in Lukasiewicz logics, is the generation of instances of varying difficulty for evaluating, comparing and improving satisfiability solvers. In this paper we present a new class of clausal forms, called Lukasiewicz (L-)clausal forms, motivate their usefulness, study their complexity, and report on an empirical investigation that shows an easy-hard-easy pattern and a phase transition phenomenon when testing the satisfiability of L-clausal forms.

Resolution-based Theorem Proving for Many-valued Logics

Considers polynomial time testability of combinational circuits generated by input decomposition, especially those generated by the logic synthesis tool FACTOR. First, the complexity of the fault detection problem in this class of circuits is explored using a stuck-at fault model. An O(2/sup k/m) algorithm for detecting a single stuck-at fault is given that is faster than the O(16/sup k/m), previously reported best algorithm proposed by Fujiwara(1990), where k is the number of inputs in a subcircuit and m the number of signal lines in the circuit. Efficient, polynomial time algorithms are described for generating a test set for all single stuck-at faults in the circuit. The basic strategy is to eliminate backtracks during line justification by constructing tables or vector sets in each subcircuit, which makes the fault propagation procedure very simple and eventually results in an efficient test generation procedure. This presentation of efficient polynomial time test generation algorithms for FACTOR-generated circuits is important, since it shows that it is possible to synthesize circuits that are optimized for area and are polynomial time testable at the same time. >

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).

Signed logic is a way of expressing the semantics of many-valued connectives and quantifiers in a formalism that is well-suited for automated reasoning. In this paper we consider propositional, finitely-valued formulas in clausal normal form. We show that checking the satisfiability of formulas with three or more literals per clause is either NP-complete or trivial, depending on whether the intersection of all signs is empty or not. The satisfiability of bijunctive formulas, i.e., formulas with at most two literals per clause, is decidable in linear time if the signs form a Helly family, and is NP-complete otherwise. We present a polynomial-time algorithm for deciding whether a given set of signs satisfies the Helly property. Our results unify and extend previous results obtained for particular sets of signs.

Itpsilas well known that parametric fuzzy conjunction and disjunction operations have not effective hardware implementation because they use product and computing powers of operations in their definitions. New approach of generation of non-associative conjunctions fuzzy operations has been developed. Based on this new method parametric classes of fuzzy operations, suitable for effective hardware implementation, are obtained. The methods of Altera FPGA implementation of some of these operations are given.

We present a fuzzy similarity approach to address a text classification problem. The effectiveness of several fuzzy conjunction and disjunction operators used in a fuzzy similarity formula and on several document representations were evaluated using test sets from three text document collections. Based on empirical results obtained from using these collections, a special case of the fuzzy similarity formula performs very well.

The SAT-Problem fot boolean formulas in conjunctive normal form often arises in the area of artificial intelligence, its solution is known as mechanical theorem proving. Most mechanical theorem provers perform this by using a variant of the resolution principle. The time and space complexity of resolution strongly depends on the class of formulas. Horn formulas, where each clause may contain at most one positive literal, represent the most important class where resolution proofs with polynomial length exist.

Multi-valued Model Checking is an extension of classical, two-valued model checking with multi-valued logic. Multi-valuedness has been proved useful in expressing additional information such as incompleteness, uncertainty, and many others, but with the cost of time and space complexity. This paper addresses this problem, and proposes a new algorithm for Multi-valued Model Checking. While Chechik et al. have extended BDD-based Symbolic Model Checking algorithm to the multi-valued case, our algorithm extends Bounded Model Checking (BMC), which can generate a counterexample of minimum length efficiently (if any). A notable feature of our algorithm is that it directly generates conjunctive normal forms, and never reduces multi-valued formulas into many slices of two-valued formulas. To achieve this feature, we extend the BMC algorithm to the multi-valued case and also devise a new translation of multi-valued propositional formulas. Finally, we show experimental results and compare the performance of our algorithm with that of a reduction-based algorithm.

The ever-increase in the quality and quantity of data generated from day-to-day businesses operations in conjunction with the continuously imported related social data have made the traditional statistical approaches inadequate to tackle such data floods. This has dictated researchers to design and develop advance and sophisticated analytics that can be incorporated to gain valuable insights that benefit the business domain. This chapter sheds the light on core aspects that lay the foundations for social big data analytics. In particular, the significance of predictive analytics in the context of SBD is discussed fortified with presenting a framework for SBD predictive analytics. Then, various predictive analytical algorithms are introduced with their usage in several important application and top-tier tools and APIs. A case study on using predictive analytics to social data is provided supported with experiments to substantiate the significance and utility of predictive analytics.

The two-element set plays a vital role in computer science, both as the bits 0 and 1 of binary arithmetic (corresponding to the two stable states of the basic subcomponents of computer circuitry), and as the two truth values, T (true) and F (false), of the logical analysis of circuitry, and the tests of programs. Section 4.1 introduces the basic operations of negation, conjunction, and disjunction on truth values, and shows how these operations can be used to express Boolean functions in normal form. We briefly discuss the use of components corresponding to these basic operations in building up computer circuits for addition. Section 4.2 looks at proof techniques from two perspectives. The first makes explicit the various “everyday” techniques we use in proving theorems in this book. The second gives a formal discussion of the notion of proof in propositional logic, and closes with a brief introduction to the quantifiers of predicate logic.

The computational complexity study of algorithms is highly germane to the design and development of high-speed computing devices. The whole essence of computation is principally influenced by efficiency of algorithms; this is more so the case with the algorithms whose solution space explodes exponentially. Cryptographic algorithms are good examples of such algorithms. The goal of this study is to compare the computational speeds of RSA and ElGamal cryptographic algorithms by carrying out a survey of works done so far by researchers. This study has therefore examined some of the results of the studies already done and highlighted which of the RSA and ElGamal algorithms performed better under given parameters. It is expected that this study would spur further investigation of the behaviour of cryptographic structures in order to ascertain their complexity and impact on the field of theoretical computer science. The experimental results of many of the papers reviewed showed that RSA cryptographic algorithm performs better as regards to energy usage, time complexity and space complexity of text, image and audio data during encryption process while some studies showed that ElGamal performs better in terms of time complexity during decryption process.