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Статті в журналах з теми "Prime implicants"

Автор: Grafiati

Опубліковано: 30 листопада 2024

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1

Prasad, V. C. "Quality of Minimal Sets of Prime Implicants of Boolean Functions." International Journal of Electronics and Telecommunications 63, no. 2 (June 27, 2017): 165–69. http://dx.doi.org/10.1515/eletel-2017-0022.

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Анотація:

Abstract Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

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2

Bienvenu, M. "Prime Implicates and Prime Implicants: From Propositional to Modal Logic." Journal of Artificial Intelligence Research 36 (October 20, 2009): 71–128. http://dx.doi.org/10.1613/jair.2754.

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Prime implicates and prime implicants have proven relevant to a number of areas of artificial intelligence, most notably abductive reasoning and knowledge compilation. The purpose of this paper is to examine how these notions might be appropriately extended from propositional logic to the modal logic K. We begin the paper by considering a number of potential definitions of clauses and terms for K. The different definitions are evaluated with respect to a set of syntactic, semantic, and complexity-theoretic properties characteristic of the propositional definition. We then compare the definitions with respect to the properties of the notions of prime implicates and prime implicants that they induce. While there is no definition that perfectly generalizes the propositional notions, we show that there does exist one definition which satisfies many of the desirable properties of the propositional case. In the second half of the paper, we consider the computational properties of the selected definition. To this end, we provide sound and complete algorithms for generating and recognizing prime implicates, and we show the prime implicate recognition task to be PSPACE-complete. We also prove upper and lower bounds on the size and number of prime implicates. While the paper focuses on the logic K, all of our results hold equally well for multi-modal K and for concept expressions in the description logic ALC.

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3

Kean, Alex, and George Tsiknis. "An incremental method for generating prime implicants/implicates." Journal of Symbolic Computation 9, no. 2 (February 1990): 185–206. http://dx.doi.org/10.1016/s0747-7171(08)80029-6.

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4

Qi, Guilin, and Kewen Wang. "Conflict-Based Belief Revision Operators in Possibilistic Logic." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (September 20, 2021): 800–806. http://dx.doi.org/10.1609/aaai.v26i1.8226.

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In this paper, we investigate belief revision in possibilistic logic, which is a weighted logic proposed to deal with incomplete and uncertain information. Existing revision operators in possibilistic logic are restricted in the sense that the input information can only be a formula instead of a possibilistic knowledge base which is a set of weighted formulas. To break this restriction, we consider weighted prime implicants of a possibilistic knowledge base and use them to define novel revision operators in possibilistic logic. Intuitively, a weighted prime implicant of a possibilistic knowledge base is a logically weakest possibilistic term (i.e., a set of weighted literals) that can entail the knowledge base. We first show that the existing definition of a weighted prime implicant is problematic and need a modification. To define a revision operator using weighted prime implicants, we face two problems. The first problem is that we need to define the notion of a conflict set between two weighted prime implicants of two possibilistic knowledge bases to achieve minimal change. The second problem is that we need to define the disjunction of possibilistic terms. We solve these problems and define two conflict-based revision operators in possibilistic logic. We then adapt the well-known postulates for revision proposed by Katsuno and Mendelzon and show that our revision operators satisfy four of the basic adapted postulates and satisfy two others in some special cases.

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5

Al-Wardi, Zaid, and Osama Al-Wardi. "RECURSIVE TERNARY-BASED ALGORITHM FOR COMPUTING PRIME IMPLICANTS OF MULTI-OUTPUT BOOLEAN FUNCTIONS." Journal of Engineering and Sustainable Development 27, no. 3 (May 1, 2023): 308–16. http://dx.doi.org/10.31272/jeasd.27.3.2.

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The problem of computing the set of prime implicants to represent a Boolean function is a classical problem that is still considered a running problem for research because all known approaches have limitations. The article reviews existing methods for computing prime implicants and highlights their limitations, particularly for multi-output functions and limited scalability due to the growth in memory required to complete the computation. Then it proposes a recursive ternary-based minimization algorithm to compute the prime implicants of multi-output Boolean functions. The algorithm is based on the concept of Programmable Logic Array (PLA) tables, which provide a structured and efficient representation of Boolean functions. The algorithm takes advantage of the ternary logic system to efficiently compute the prime implicants while maintaining scalability for large and complex functions, which has significant implications for digital circuit design and optimization.

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6

Singh, Arindama. "Computing prime implicants via transversal clauses." International Journal of Computer Mathematics 70, no. 3 (January 1999): 417–27. http://dx.doi.org/10.1080/00207169908804764.

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7

Tyrväinen, Tero. "Prime implicants in dynamic reliability analysis." Reliability Engineering & System Safety 146 (February 2016): 39–46. http://dx.doi.org/10.1016/j.ress.2015.10.007.

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8

Čepek, Ondřej, Petr Kučera, and Stanislav Kuřík. "Boolean functions with long prime implicants." Information Processing Letters 113, no. 19-21 (September 2013): 698–703. http://dx.doi.org/10.1016/j.ipl.2013.07.001.

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9

Audemard, Gilles, Steve Bellart, Louènas Bounia, Frédéric Koriche, Jean-Marie Lagniez, and Pierre Marquis. "Trading Complexity for Sparsity in Random Forest Explanations." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 5 (June 28, 2022): 5461–69. http://dx.doi.org/10.1609/aaai.v36i5.20484.

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Random forests have long been considered as powerful model ensembles in machine learning. By training multiple decision trees, whose diversity is fostered through data and feature subsampling, the resulting random forest can lead to more stable and reliable predictions than a single decision tree. This however comes at the cost of decreased interpretability: while decision trees are often easily interpretable, the predictions made by random forests are much more difficult to understand, as they involve a majority vote over multiple decision trees. In this paper, we examine different types of reasons that explain "why" an input instance is classified as positive or negative by a Boolean random forest. Notably, as an alternative to prime-implicant explanations taking the form of subset-minimal implicants of the random forest, we introduce majoritary reasons which are subset-minimal implicants of a strict majority of decision trees. For these abductive explanations, the tractability of the generation problem (finding one reason) and the optimization problem (finding one minimum-sized reason) are investigated. Unlike prime-implicant explanations, majoritary reasons may contain redundant features. However, in practice, prime-implicant explanations - for which the identification problem is DP-complete - are slightly larger than majoritary reasons that can be generated using a simple linear-time greedy algorithm. They are also significantly larger than minimum-sized majoritary reasons which can be approached using an anytime Partial MaxSAT algorithm.

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10

Michalak, Marcin. "Searching for Continuous n-Clusters with Boolean Reasoning." Symmetry 16, no. 10 (October 1, 2024): 1286. http://dx.doi.org/10.3390/sym16101286.

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A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper.

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11

Palopoli, Luigi, Fiora Pirri, and Clara Pizzuti. "Algorithms for selective enumeration of prime implicants." Artificial Intelligence 111, no. 1-2 (July 1999): 41–72. http://dx.doi.org/10.1016/s0004-3702(99)00035-1.

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12

Rawat, Saurabh, and Anushree Sah. "Prime and Essential Prime Implicants of Boolean Functions through Cubical Representation." International Journal of Computer Applications 70, no. 23 (May 31, 2013): 1–6. http://dx.doi.org/10.5120/12204-7638.

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13

Dieu, Phan Dinh, and Phan Hong Giang. "Interval –valued probabilistic logic for logic programs." Journal of Computer Science and Cybernetics 10, no. 3 (April 15, 2016): 1–13. http://dx.doi.org/10.15625/1813-9663/10/3/8193.

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Анотація:

This paper presents an approximate method for probabilistic entailment problem in knowledge bases where a portion of knowledge is given by a sentence in propositional logic accompanied with an interval presenting its truth probalibity. This method reduces the entailment problem to one of finding “prime implicants” of the target sentence expressed through sentences in the given knowledge base. It is shown that in the case of probabilistic logic programs the set of such prime implicants can be found by using the SLD-resolution method for usual definte logic programs.

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14

Garribba, S., E. Guagnini, and P. Mussio. "Multiple-Valued Logic Trees: Meaning and Prime Implicants." IEEE Transactions on Reliability R-34, no. 5 (December 1985): 463–72. http://dx.doi.org/10.1109/tr.1985.5222234.

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15

Di Maio, Francesco, Samuele Baronchelli, and Enrico Zio. "A Visual Interactive Method for Prime Implicants Identification." IEEE Transactions on Reliability 64, no. 2 (June 2015): 539–49. http://dx.doi.org/10.1109/tr.2014.2371015.

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16

Strzemecki, Tadeusz. "Polynomial-time algorithms for generation of prime implicants." Journal of Complexity 8, no. 1 (March 1992): 37–63. http://dx.doi.org/10.1016/0885-064x(92)90033-8.

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17

Shiny, A. "Computation of prime implicants using matrix and paths." Journal of Logic and Computation 8, no. 2 (April 1, 1998): 135–45. http://dx.doi.org/10.1093/logcom/8.2.135.

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18

SACALUGA, DAVID LUQUE. "FRACTALS IN LOGIC." International Journal of Computational Methods 02, no. 01 (March 2005): 1–13. http://dx.doi.org/10.1142/s0219876205000326.

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Анотація:

The main target of this document is to propose a new graphical method to find prime implicants, necessary prime implicants and minimum sum-of-product expressions for digital systems with any number of variables. Moreover, a simple example of application of the same method is included to be compared with Karnaugh's method. As a consequence of this proposed method, a few conditions are established and a fractal image is created, which explains why it is not possible to have more than one binary variable changing simultaneously and why the change between two non adjacent states has to be realized always going through an intermediate state.

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19

Raut, Manoj K., and Arindama Singh. "Prime implicants of first order formulas via transversal clauses." International Journal of Computer Mathematics 81, no. 2 (February 2004): 157–67. http://dx.doi.org/10.1080/00207160310001650071.

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20

Hayase, K. "OBDDs of a Monotone Function and Its Prime Implicants." Theory of Computing Systems 31, no. 5 (September 1, 1998): 579–91. http://dx.doi.org/10.1007/s002240000104.

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21

Wiśniewski, R., Ł. Stefanowicz, G. Bazydło, and M. Węgrzyn. "Application of Hypergraphs in the Prime Implicants Selection Process." IFAC-PapersOnLine 48, no. 4 (2015): 302–5. http://dx.doi.org/10.1016/j.ifacol.2015.07.051.

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22

Chukhrov, I. P. "Properties of Boolean Functions with Extremal Number of Prime Implicants." Journal of Applied and Industrial Mathematics 16, no. 1 (February 2022): 8–21. http://dx.doi.org/10.1134/s1990478922010021.

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23

McMullen and Shearer. "Prime Implicants, Minimum Covers, and the Complexity of Logic Simplification." IEEE Transactions on Computers C-35, no. 8 (August 1986): 761–62. http://dx.doi.org/10.1109/tc.1986.1676828.

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24

Chukhrov, I. P. "Properties of Boolean functions with the extremal number of prime implicants." Diskretnyi analiz i issledovanie operatsii 29, no. 1 (March 6, 2022): 74–93. http://dx.doi.org/10.33048/daio.2022.29.725.

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25

Chukhrov, I. P. "Connected Boolean Functions with a Locally Extremal Number of Prime Implicants." Journal of Applied and Industrial Mathematics 15, no. 1 (February 2021): 17–38. http://dx.doi.org/10.1134/s1990478921010038.

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26

Chukhrov, I. P. "Connected Boolean functions with a locally extremal number of prime implicants." Diskretnyi analiz i issledovanie operatsii 28, no. 1 (March 6, 2021): 68–96. http://dx.doi.org/10.33048/daio.2021.28.699.

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27

Sloan, Robert H., Balázs Szörényi, and György Turán. "On k-Term DNF with the Largest Number of Prime Implicants." SIAM Journal on Discrete Mathematics 21, no. 4 (January 2008): 987–98. http://dx.doi.org/10.1137/050632026.

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28

Maio, Francesco Di, Samuele Baronchelli, and Enrico Zio. "A Computational Framework for Prime Implicants Identification in Noncoherent Dynamic Systems." Risk Analysis 35, no. 1 (July 14, 2014): 142–56. http://dx.doi.org/10.1111/risa.12251.

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29

Gorji, Niku, and Sasha Rubin. "Sufficient Reasons for Classifier Decisions in the Presence of Domain Constraints." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 5 (June 28, 2022): 5660–67. http://dx.doi.org/10.1609/aaai.v36i5.20507.

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Анотація:

Recent work has unveiled a theory for reasoning about the decisions made by binary classifiers: a classifier describes a Boolean function, and the reasons behind an instance being classified as positive are the prime-implicants of the function that are satisfied by the instance. One drawback of these works is that they do not explicitly treat scenarios where the underlying data is known to be constrained, e.g., certain combinations of features may not exist, may not be observable, or may be required to be disregarded. We propose a more general theory, also based on prime-implicants, tailored to taking constraints into account. The main idea is to view classifiers as describing partial Boolean functions that are undefined on instances that do not satisfy the constraints. We prove that this simple idea results in more parsimonious reasons. That is, not taking constraints into account (e.g., ignoring, or taking them as negative instances) results in reasons that are subsumed by reasons that do take constraints into account. We illustrate this improved succinctness on synthetic classifiers and classifiers learnt from real data.

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30

Prasad, V. C. "Efficient Minimisation of Boolean Functions." International Journal of Electrical Engineering & Education 45, no. 4 (October 2008): 321–26. http://dx.doi.org/10.7227/ijeee.45.4.5.

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Quine Mc Cluskey's (QM) method is a popular method for minimisation of Boolean functions. This method is widely taught at undergraduate level. In this paper simple modifications are suggested to make it more efficient. They allow us to avoid repetitions in the QM method. Further, a minimal set of prime implicants is easily obtained.

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31

Prescott, Rasa Remenyte, and John D. Andrews. "Prime implicants for modularised non-coherent fault trees using binary decision diagrams." International Journal of Reliability and Safety 1, no. 4 (2007): 446. http://dx.doi.org/10.1504/ijrs.2007.016259.

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32

Boufkhad, Y., and O. Dubois. "Length of prime implicants and number of solutions of random CNF formulae." Theoretical Computer Science 215, no. 1-2 (February 1999): 1–30. http://dx.doi.org/10.1016/s0304-3975(95)00184-0.

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33

Yau, Michael, George Apostolakis, and Sergio Guarro. "The use of prime implicants in dependability analysis of software controlled systems." Reliability Engineering & System Safety 62, no. 1-2 (October 1998): 23–32. http://dx.doi.org/10.1016/s0951-8320(98)00002-7.

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34

Talebanfard, Navid. "On the structure and the number of prime implicants of 2-CNFs." Discrete Applied Mathematics 200 (February 2016): 1–4. http://dx.doi.org/10.1016/j.dam.2015.06.036.

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35

Lee, Sang-Un. "A Selection-Deletion of Prime Implicants Algorithm Based on Frequency for Circuit Minimization." Journal of the Korea Society of Computer and Information 20, no. 4 (April 30, 2015): 95–102. http://dx.doi.org/10.9708/jksci.2015.20.4.095.

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36

Darwiche, Adnan, and Chunxi Ji. "On the Computation of Necessary and Sufficient Explanations." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 5 (June 28, 2022): 5582–91. http://dx.doi.org/10.1609/aaai.v36i5.20498.

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The complete reason behind a decision is a Boolean formula that characterizes why the decision was made. This recently introduced notion has a number of applications, which include generating explanations, detecting decision bias and evaluating counterfactual queries. Prime implicants of the complete reason are known as sufficient reasons for the decision and they correspond to what is known as PI explanations and abductive explanations. In this paper, we refer to the prime implicates of a complete reason as necessary reasons for the decision. We justify this terminology semantically and show that necessary reasons correspond to what is known as contrastive explanations. We also study the computation of complete reasons for multi-class decision trees and graphs with nominal and numeric features for which we derive efficient, closed-form complete reasons. We further investigate the computation of shortest necessary and sufficient reasons for a broad class of complete reasons, which include the derived closed forms and the complete reasons for Sentential Decision Diagrams (SDDs). We provide an algorithm which can enumerate their shortest necessary reasons in output polynomial time. Enumerating shortest sufficient reasons for this class of complete reasons is hard even for a single reason. For this problem, we provide an algorithm that appears to be quite efficient as we show empirically.

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37

Rauzy, Antoine, and Yves Dutuit. "Exact and truncated computations of prime implicants of coherent and non-coherent fault trees within Aralia." Reliability Engineering & System Safety 58, no. 2 (November 1997): 127–44. http://dx.doi.org/10.1016/s0951-8320(97)00034-3.

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38

Di Maio, Francesco, Samuele Baronchelli, Matteo Vagnoli, and Enrico Zio. "Determination of prime implicants by differential evolution for the dynamic reliability analysis of non-coherent nuclear systems." Annals of Nuclear Energy 102 (April 2017): 91–105. http://dx.doi.org/10.1016/j.anucene.2016.12.018.

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39

Borgonovo, E. "The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions." European Journal of Operational Research 204, no. 3 (August 2010): 485–95. http://dx.doi.org/10.1016/j.ejor.2009.10.021.

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40

Başçiftçi, Fatih, and Şirzat Kahramanli. "Fast computation of the prime implicants by exact direct-cover algorithm based on the new partial ordering operation rule." Advances in Engineering Software 42, no. 6 (June 2011): 316–21. http://dx.doi.org/10.1016/j.advengsoft.2011.02.009.

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41

Weber, Christopher, and Daniel Bryce. "Planning and Acting in Incomplete Domains." Proceedings of the International Conference on Automated Planning and Scheduling 21 (March 22, 2011): 274–81. http://dx.doi.org/10.1609/icaps.v21i1.13463.

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Анотація:

Engineering complete planning domain descriptions is often very costly because of human error or lack of domain knowl- edge. Learning complete domain descriptions is also very challenging because many features are irrelevant to achieving the goals and data may be scarce. We present a planner and agent that respectively plan and act in incomplete domains by i) synthesizing plans to avoid execution failure due to ignorance of the domain model, and ii) passively learning about the domain model during execution to improve later re-planning attempts. Our planner DeFault is the first to reason about a domain’s incompleteness to avoid potential plan failure. DeFault computes failure explanations for each action and state in the plan and counts the number of interpretations of the incomplete domain where failure will occur. We show that DeFault performs best by counting prime implicants (failure diagnoses) rather than propositional models. Our agent Goalie learns about the preconditions and effects of incompletely-specified actions while monitoring its state and, in conjunction with DeFault plan failure explanations, can diagnose past and future action failures. We show that by reasoning about incompleteness (as opposed to ignoring it) Goalie fails and re-plans less and executes fewer actions.

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42

Di Maio, Francesco, Matteo Vagnoli, and Enrico Zio. "Risk-Based Clustering for Near Misses Identification in Integrated Deterministic and Probabilistic Safety Analysis." Science and Technology of Nuclear Installations 2015 (2015): 1–29. http://dx.doi.org/10.1155/2015/693891.

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Анотація:

In integrated deterministic and probabilistic safety analysis (IDPSA), safe scenarios and prime implicants (PIs) are generated by simulation. In this paper, we propose a novel postprocessing method, which resorts to a risk-based clustering method for identifying Near Misses among the safe scenarios. This is important because the possibility of recovering these combinations of failures within a tolerable grace time allows avoiding deviations to accident and, thus, reducing the downtime (and the risk) of the system. The postprocessing risk-significant features for the clustering are extracted from the following: (i) the probability of a scenario to develop into an accidental scenario, (ii) the severity of the consequences that the developing scenario would cause to the system, and (iii) the combination of (i) and (ii) into the overall risk of the developing scenario. The optimal selection of the extracted features is done by a wrapper approach, whereby a modified binary differential evolution (MBDE) embeds aK-means clustering algorithm. The characteristics of the Near Misses scenarios are identified solving a multiobjective optimization problem, using the Hamming distance as a measure of similarity. The feasibility of the analysis is shown with respect to fault scenarios in a dynamic steam generator (SG) of a nuclear power plant (NPP).

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43

Xu, Wei, Xunhong Zhu, Liping Zhang, and Jun Gao. "The Iterative Exclusion of Compatible Samples Workflow for Multi-SNP Analysis in Complex Diseases." Algorithms 16, no. 10 (October 16, 2023): 480. http://dx.doi.org/10.3390/a16100480.

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Complex diseases are affected by various factors, and single-nucleotide polymorphisms (SNPs) are the basis for their susceptibility by affecting protein structure and gene expression. Complex diseases often arise from the interactions of multiple SNPs and are investigated using epistasis detection algorithms. Nevertheless, the computational burden associated with the “combination explosion” hinders these algorithms’ ability to detect these interactions. To perform multi-SNP analysis in complex diseases, the iterative exclusion of compatible samples (IECS) workflow is proposed in this work. In the IECS workflow, qualitative comparative analysis (QCA) is firstly employed as the calculation engine to calculate the solution; secondly, the pattern is extracted from the prime implicants with the greatest raw coverage in the solution; then, the pattern is tested with the chi-square test in the source dataset; finally, all compatible samples are excluded from the current dataset. This process is repeated until the QCA calculation has no solution or reaches the iteration threshold. The workflow was applied to analyze simulated datasets and the Alzheimer’s disease dataset, and its performance was compared with that of the BOOST and MDR algorithms. The findings illustrated that IECS exhibits greater power with less computation and can be applied to perform multi-SNP analysis in complex diseases.

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44

Wei, Yu-Chung. "BLogic: A Bayesian Model Combination Approach in Logic Regression." Mathematics 11, no. 20 (October 19, 2023): 4353. http://dx.doi.org/10.3390/math11204353.

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With the increasing complexity and dimensionality of datasets in statistical research, traditional methods of identifying interactions are often more challenging to apply due to the limitations of model assumptions. Logic regression has emerged as an effective tool, leveraging Boolean combinations of binary explanatory variables. However, the prevalent simulated annealing approach in logic regression sometimes faces stability issues. This study introduces the BLogic algorithm, a novel approach that amalgamates multiple runs of simulated annealing on a dataset and synthesizes the results via the Bayesian model combination technique. This algorithm not only facilitates predicting response variables using binary explanatory ones but also offers a score computation for prime implicants, elucidating key variables and their interactions within the data. In simulations with identical parameters, conventional logic regression, when executed with a single instance of simulated annealing, exhibits reduced predictive and interpretative capabilities as soon as the ratio of explanatory variables to sample size surpasses 10. In contrast, the BLogic algorithm maintains its effectiveness until this ratio approaches 50. This underscores its heightened resilience against challenges in high-dimensional settings, especially the large p, small n problem. Moreover, employing real-world data from the UK10K Project, we also showcase the practical performance of the BLogic algorithm.

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45

Di Maio, Francesco, Roberta Rossetti, and Enrico Zio. "Postprocessing of Accidental Scenarios by Semi-Supervised Self-Organizing Maps." Science and Technology of Nuclear Installations 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/2709109.

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Integrated Deterministic and Probabilistic Safety Analysis (IDPSA) of dynamic systems calls for the development of efficient methods for accidental scenarios generation. The necessary consideration of failure events timing and sequencing along the scenarios requires the number of scenarios to be generated to increase with respect to conventional PSA. Consequently, their postprocessing for retrieving safety relevant information regarding the system behavior is challenged because of the large amount of generated scenarios that makes the computational cost for scenario postprocessing enormous and the retrieved information difficult to interpret. In the context of IDPSA, the interpretation consists in the classification of the generated scenarios as safe, failed, Near Misses (NMs), and Prime Implicants (PIs). To address this issue, in this paper we propose the use of an ensemble of Semi-Supervised Self-Organizing Maps (SSSOMs) whose outcomes are combined by a locally weighted aggregation according to two strategies: a locally weighted aggregation and a decision tree based aggregation. In the former, we resort to the Local Fusion (LF) principle for accounting the classification reliability of the different SSSOM classifiers, whereas in the latter we build a classification scheme to select the appropriate classifier (or ensemble of classifiers), for the type of scenario to be classified. The two strategies are applied for the postprocessing of the accidental scenarios of a dynamic U-Tube Steam Generator (UTSG).

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46

Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati, Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati. "Using Variable-Entered Karnaugh Maps in Determining Dependent and Independent Sets of Boolean Functions." journal of king abdulaziz university computing and information technology sciences 1, no. 2 (January 15, 2012): 45–67. http://dx.doi.org/10.4197/comp.1-2.3.

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An important class for Boolean reasoning problems involves interdependence among the members of a set T of Boolean functions. Two notable problems among this class are (a) to establish whether a given subset of T is dependent, and (b) to produce economical representations for the complementary families of all dependent subsets and independent subsets of T. This paper solves these two problems via a powerful manual pictorial tool, namely, the variableentered Karnaugh map (VEKM). The VEKM is utilized in executing a Label-and-Eliminate procedure for producing certain prime implicants or consequents used in tackling the two aforementioned problems. The VEKM procedure is a time-saving short cut indeed, since it efficiently handles the three basic tasks demanded by the solution procedure, which are: (a) To combine several Boolean relations into a single one, (b) to compute conjunctive eliminants of a Boolean function, and (c) to derive the complete sum (CS) of a Boolean function. The VEKM procedure significantly reduces the complexities of these tasks by introducing useful shortcuts and allowing simultaneous processing. The VEKM procedure is described in detail, and then demonstrated via two illustrative examples, which previously had only black-box computer solutions as they were thought to be not amenable to manual solution. The first example deals with switching or bivalent functions while the second handles 'big' Boolean functions. Both examples indicate that the VEKM procedure proposed herein enjoys the merits of insightfulness, simplicity and efficiency

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47

Echenim, Mnacho, Nicolas Peltier, and Sophie Tourret. "Prime Implicate Generation in Equational Logic." Journal of Artificial Intelligence Research 60 (December 18, 2017): 827–80. http://dx.doi.org/10.1613/jair.5481.

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We present an algorithm for the generation of prime implicates in equational logic, that is, of the most general consequences of formulæ containing equations and disequations between first-order terms. This algorithm is defined by a calculus that is proved to be correct and complete. We then focus on the case where the considered clause set is ground, i.e., contains no variables, and devise a specialized tree data structure that is designed to efficiently detect and delete redundant implicates. The corresponding algorithms are presented along with their termination and correctness proofs. Finally, an experimental evaluation of this prime implicate generation method is conducted in the ground case, including a comparison with state-of-the-art propositional and first-order prime implicate generation tools.

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48

Schrag, Robert, and James M. Crawford. "Implicates and prime implicates in Random 3-SAT." Artificial Intelligence 81, no. 1-2 (March 1996): 199–222. http://dx.doi.org/10.1016/0004-3702(95)00053-4.

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49

Van De Putte, F. "Prime implicates and relevant belief revision." Journal of Logic and Computation 23, no. 1 (November 7, 2011): 109–19. http://dx.doi.org/10.1093/logcom/exr040.

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50

Rymon, Ron. "An SE-tree-based prime implicant generation algorithm." Annals of Mathematics and Artificial Intelligence 11, no. 1-4 (March 1994): 351–65. http://dx.doi.org/10.1007/bf01530750.

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