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The Minimum Equivalent DNF Problem and Shortest Implicants

Christopher Umans
2001 Journal of computer and system sciences (Print)  
We prove that the Minimum Equivalent DNF problem is p 2 -complete, resolving a conjecture due to Stockmeyer.  ...  When the input is a formula, the Shortest Implicant problem is p 2complete, and p 2 -hard to approximate to within an n 1=2? factor.  ...  The problem of finding a minimum equivalent DNF then becomes exactly the problem of finding the shortest implicant core of .  ... 
doi:10.1006/jcss.2001.1775 fatcat:pckny7ewpfhwnl7brlekfkd6fm

The minimum equivalent DNF problem and shortest implicants

C. Umans
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)  
We prove that the Minimum Equivalent DNF problem is S P 2 -complete, resolving a conjecture due to Stockmeyer.  ...  We also consider the complexity and approximability of a related optimization problem in the second level of the polynomial hierarchy, that of finding shortest implicants of a Boolean function.  ...  ACKNOWLEDGMENTS We thank Christos Papadimitriou for suggesting this problem, and many useful discussions.  ... 
doi:10.1109/sfcs.1998.743506 dblp:conf/focs/Umans98 fatcat:zeaa4hhiwbcitf3mcpy2y4jzdq

On the Complexity and Inapproximability of Shortest Implicant Problems [chapter]

Christopher Umans
1999 Lecture Notes in Computer Science  
We obtain inapproximability results for the DNF and formula variants of the shortest implicant problem that show that trivial approximation algorithms are optimal for these problems, up to lower order  ...  We investigate the complexity and approximability of a basic optimization problem in the second level of the Polynomial Hierarchy, that of nding shortest implicants.  ...  in the reduction showing that the Minimum Equivalent DNF problem is p 2 -complete.  ... 
doi:10.1007/3-540-48523-6_65 fatcat:xkrdnqrrazesdm7m6475tcp37m

Boolean functions with long prime implicants

Ondřej Čepek, Petr Kučera, Stanislav Kuřík
2013 Information Processing Letters  
Moreover, in case that the answer is affirmative we present a polynomial time algorithm which outputs a shortest DNF representation of the given function.  ...  In this short note we introduce a class of Boolean functions defined by a minimum length of its prime implicants.  ...  Finding a minimum (prime) DNF is then equivalent to finding a minimum set cover in an instance where truepoints are the elements of the base set and prime implicates correspond to the system of subsets  ... 
doi:10.1016/j.ipl.2013.07.001 fatcat:jep3g5m4rnhk3lvdwei44w7aoi

Page 2057 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
2057 2003c:68099 68Q25 68Q15 Umans, Christopher (1-CA-CD; Berkeley, CA) The minimum equivalent DNF problem and shortest implicants. (English summary) Special issue on FOCS 98 (Palo Alto, CA). J.  ...  Shortest Implicant for DNF formulas is complete for the complexity class in which the guess y has O(log’) length, and TAUTOLOGY is the verification subroutine.  ... 

Bidual Horn functions and extensions

Thomas Eiter, Toshihide Ibaraki, Kazuhisa Makino
1999 Discrete Applied Mathematics  
We also give characterizations of shortest DNF expressions of a bidual Horn function f and show how to compute such an expression from a Horn DNF for f in polynomial time; for arbitrary Horn functions,  ...  Besides the extension problem, we study recognition of bidual Horn functions from Boolean formulas and properties of normal form expressions.  ...  Acknowledgements We would like to thank the participants of the Siena '96 Workshop on the Satisÿability Problem for comments on this work, and the anonymous referees for their helpful suggestions which  ... 
doi:10.1016/s0166-218x(99)00033-5 fatcat:lttktdepcjhuhanxai65akxg5i

On connected Boolean functions

Oya Ekin, Peter L. Hammer, Alexander Kogan
1999 Discrete Applied Mathematics  
This paper studies the relationships between these properties and the DNF representations of the associated Boolean functions. ? (A. Kogan) 0166-218X/99/$ -see front matter ?  ...  This concept is further strengthened to that of convexity where every shortest path connecting two points of the same kind should consist of points of the same kind.  ...  Acknowledgements The authors gratefully acknowledge the partial support provided by the O ce of Naval Research (grant N00014-92J1375) and the National Science Foundation (grant NSF-DMS-9806 389).  ... 
doi:10.1016/s0166-218x(99)00098-0 fatcat:kulhs5efyrgtpcuw6cbks4g5aa

A Boolean measure of similarity

Martin Anthony, Peter L. Hammer
2006 Discrete Applied Mathematics  
We relate the similarity measure to one based on Hamming distance and we develop from this some ways of quantifying the 'quality' of a dataset.  ...  We propose a way of measuring the similarity of a Boolean vector to a given set of Boolean vectors, motivated in part by certain data mining or machine learning problems.  ...  Martin Anthony's work is supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778.  ... 
doi:10.1016/j.dam.2006.04.011 fatcat:upgzbcrzlbajdaeholvkk3opvu

Convexity and logical analysis of data

Oya Ekin, Peter L. Hammer, Alexander Kogan
2000 Theoretical Computer Science  
We study the problem of uniqueness, and provide a polynomial algorithm for checking whether all k-convex extensions agree in a point outside the given data set.  ...  Given a set of true points and a set of false points, the central question of Logical Analysis of Data is the study of those Boolean functions whose values agree with those of the given points.  ...  The authors also express their gratitude to Endre Boros and Dan Roth for useful discussions, and to the anonymous referee for careful analysis and helpful suggestions.  ... 
doi:10.1016/s0304-3975(98)00337-5 fatcat:f4nbs24hwnhixkjklds44fk76a

FPGA HW Accelerator of the First Step of Systematic Two-Level Minimization of Single-Output Boolean Function

Branislav Madoš, Norbert Ádám, Zuzana Bilanová, Martin Chovanec
2020 Acta Polytechnica Hungarica  
normal form (DNF) and the conjunctive normal form (CNF), having defined the OFF and ON sets andalternativelyalso the DC ("don't care") set.  ...  Its advantages include an extremely short prime-implicant-generation time in the order of ns and/or tens of nsin case of Boolean functions with small amount of input variablesand the possibility to generate  ...  Table 3 3 Equivalence of decadic equivalents (DE), ternary equivalents (TE) and prime implicants (PI) for both the DNF and CNF forms DE TE DNF PI description DNF PI CNF PI description CNF PI  ... 
doi:10.12700/aph.17.7.2020.7.2 fatcat:5ig4zhe57vh47chxdfhxao5nsq

How Should We Solve Search Problems Privately?

Amos Beimel, Tal Malkin, Kobbi Nissim, Enav Weinreb
2008 Journal of Cryptology  
However, it is often the case that the parties executing a computation try to solve a search problem, where one input may have a multitude of correct answers -such as when the parties compute a shortest  ...  Picking one output arbitrarily from the solution set has significant implications on the privacy of the algorithm. Beimel et al.  ...  We thank Rafi Ostrovsky and the IPAM staff for inviting us to IPAM and making our stay pleasant and productive.  ... 
doi:10.1007/s00145-008-9032-z fatcat:cypjrmnx5zcplergc3epeu7yye

Properties of Switch-List Representations of Boolean Functions

Miloš Chromý, Ondřej Čepek
2020 The Journal of Artificial Intelligence Research  
As a by-product of this analysis, we also give a short proof of a longstanding open question proposed by Darwiche and Marquis, namely the incomparability of MODS (models) and PI (prime implicates) representations  ...  First, we compare switch-list representations with a number of standard representations (such as CNF, DNF, and OBDD) with respect to their relative succinctness.  ...  Acknowledgements The authors gratefully acknowledge a support by Czech Science Foundation (Grant 19-19463S), by SVV project number 260 575 and by Charles University project UNCE/SCI/004.  ... 
doi:10.1613/jair.1.12199 fatcat:q64uhgufdvbndpa3xzt6vaom6a

A Model of Interactive Teaching

H.David Mathias
1997 Journal of computer and system sciences (Print)  
The learner makes only equivalence queries and all hypotheses are also DNF formulas. ] 1997 Academic Press  ...  An important concept class that is not known to be learnable is DNF formulas. We demonstrate the power of our approach by providing a deterministic teacher and learner for the class of DNF formulas.  ...  We also thank the COLT '95 program committee.  ... 
doi:10.1006/jcss.1997.1491 fatcat:7dp7hsez3zhaxgcujby3dqj5pa

Lower bounds on learning decision lists and trees [chapter]

Thomas Hancock, Tao Jiang, Ming Li, John Tromp
1995 Lecture Notes in Computer Science  
ACKNOWLEDGMENTS We thank Martin Anthony and Ron Rivest for information about decision lists, Lisa Hellerstein for pointing out an omission in our proofs, and the referees for their very careful reading  ...  and pointing out of numerous small errors.  ...  The result below on DNF easily follows from a reduction from the Graph Coloring problem to the approximation of the smallest consistent DNF given in [19] (see also [12] ) and the n = non-approximability  ... 
doi:10.1007/3-540-59042-0_102 fatcat:6zrph47mhjhp7j6c4w7q4ahzpe

Lower Bounds on Learning Decision Lists and Trees

Thomas Hancock, Tao Jiang, Ming Li, John Tromp
1996 Information and Computation  
ACKNOWLEDGMENTS We thank Martin Anthony and Ron Rivest for information about decision lists, Lisa Hellerstein for pointing out an omission in our proofs, and the referees for their very careful reading  ...  and pointing out of numerous small errors.  ...  The result below on DNF easily follows from a reduction from the Graph Coloring problem to the approximation of the smallest consistent DNF given in [19] (see also [12] ) and the n = non-approximability  ... 
doi:10.1006/inco.1996.0040 fatcat:qf5rp4tg3badnlwkds3b43djl4
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