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Dualization in lattices given by implicational bases [article]

Oscar Defrain, Lhouari Nourine
2020 arXiv   pre-print
It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP.  ...  In this paper, we~show that this result holds even when the premises in the implicational base are of size at most two.  ...  The dualization in lattices given by implicational bases is impossible in output-polynomial time unless P=NP, even for implicational bases of dimension two.  ... 
arXiv:1901.07503v5 fatcat:wm7iya77hjdm5mrc4nwd36d4e4

Dualization in Lattices Given by Ordered Sets of Irreducibles [article]

Mikhail A. Babin, Sergei O. Kuznetsov
2015 arXiv   pre-print
In contrast to usual dualization setting, where a lattice is given by the ordered set of its elements, dualization in this case is shown to be impossible in output polynomial time unless P = NP.  ...  In this paper we consider finite lattices given by ordered sets of their meet and join irreducibles (i.e., as a concept lattice of a formal context).  ...  The second author was supported by the Basic Research Program of the National Research University Higher School of Economics (Moscow, Russia) and Russian Foundation for Basic Research.  ... 
arXiv:1504.01145v2 fatcat:n5zsomhlpnbdviuo4urk4ewve4

Translating between the representations of a ranked convex geometry [article]

Oscar Defrain and Lhouari Nourine and Simon Vilmin
2021 arXiv   pre-print
It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements.  ...  Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem for which  ...  Let us first define the dualization problem in lattices given by implicational bases.  ... 
arXiv:1907.09433v2 fatcat:zshldvesr5d6hjoyvdf4dpxb7m

Discovery of the D-basis in binary tables based on hypergraph dualization

K. Adaricheva, J.B. Nation
2017 Theoretical Computer Science  
At the core of the method are the D-relation defined in the lattice theory framework, and the hypergraph dualization algorithm that allows us to effectively produce the set of transversals for a given  ...  The latter algorithm, first developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, artificial  ...  The results of this paper were prompted by the discussion of the hypergraph dualization algorithm with E. Boros and V. Gurvich, at the RUT-COR seminar during the first author's visit in 2011.  ... 
doi:10.1016/j.tcs.2015.11.031 fatcat:eh2sjwcy5vbmrkuips5vha5mk4

Complement-Topoi and Dual Intuitionistic Logic

Luis Estrada-González
2010 The Australasian Journal of Logic  
Mortensen studies dual intuitionistic logic by dualizing topos internal logic, but he did not study a sequent calculus.  ...  In this paper I present a sequent calculus for complement-topos logic, which throws some light on the problem of giving a dualization for LJ.  ...  In order to dualize a given formula a A b, change the antecedent by the consequent and vice versa and then replace A by −.  ... 
doi:10.26686/ajl.v9i0.1819 fatcat:x3xmpxk5dfeepnhrbsntoyatda

On Dualization over Distributive Lattices [article]

Khaled Elbassioni
2022 arXiv   pre-print
As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental  ...  and cover (by domination) the entire lattice.  ...  Given a database D, an implication base Σ, and and an integer t, a maximal t-frequent closed set, is a set X ∈ C(Σ) maximal with the property that | supp D (X)| ≥ t.  ... 
arXiv:2006.15337v3 fatcat:mcgsvasfsbezpkpiirselgqjoq

On the dualization in distributive lattices and related problems [article]

Oscar Defrain, Lhouari Nourine, Takeaki Uno
2020 arXiv   pre-print
In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem.  ...  In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal "ideal solutions" are to be generated.  ...  In [DN19] , it was shown that this result holds even when the premises in the implicational base (coding the lattice) are of size at most two.  ... 
arXiv:1902.07004v2 fatcat:w7xlm72dqvbe3e7u3ihbuuoem4

Dihypergraph Decomposition: Application to Closure System Representations

Simon Vilmin, Lhouari Nourine
2020 European Conference on Artificial Intelligence  
Among representations, dihypergraphs (or attribute implications) and meet-irreducible elements (reduced context) are widely used in the literature.  ...  From this result, we deduce an algorithm which computes meet-irreducible elements in a divide-and-conquer way and puts the light on the major role of dualization in closure systems.  ...  In this paper, we suppose that implicational bases are given as directed hypergraphs.  ... 
dblp:conf/ecai/VilminN20 fatcat:jmogxvplrvcfxc4zl4a42zz2iu

Computational aspects of monotone dualization: A brief survey

Thomas Eiter, Kazuhisa Makino, Georg Gottlob
2008 Discrete Applied Mathematics  
Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence  ...  In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan (J.  ...  Furthermore, all prime implicants and prime implicates of a monotone f are monotone, and it has a unique prime CNF, denoted CNF(f ), which is given by the conjunction of all its prime implicates.  ... 
doi:10.1016/j.dam.2007.04.017 fatcat:lgb6yd7gfnfaxmp6hrjtw4eiiu

Dualizing Clones as Models of Lawvere Theories

Sebastian Kerkhoff
2014 Electronical Notes in Theoretical Computer Science  
be beneficial for the classical problem of studying the lattice of clones on a given set.  ...  In this paper, we argue that the possibility of applying duality theory might be such a reason, and we support this claim by outlining how treating and dualizing clones as models of Lawvere theories can  ...  Vice versa, a Priestley space dualizes to the bounded distributive lattice given by its clopen increasing subsets.  ... 
doi:10.1016/j.entcs.2014.02.005 fatcat:d7eyy7ksevckhjt6sfajempv3m

Disjunctive and conjunctive normal forms of pseudo-Boolean functions

Stephan Foldes, Peter L. Hammer
2000 Discrete Applied Mathematics  
In a similar way the concepts of conjunctive normal form, implicates and prime implicates, as well as the resolution method are examined in the case of pseudo-Boolean functions. ?  ...  ), the concepts of implicants and of prime implicants are analyzed in the pseudo-Boolean context, and a consensus-type method is presented for ÿnding all the prime implicants of a pseudo-Boolean function  ...  Acknowledgements This work was partially supported by NSF grant DMS-9806389 and ONR grant N00014-92-J-1375.  ... 
doi:10.1016/s0166-218x(00)00276-6 fatcat:e4wd2a6bcbbdjp6arbtfcihxpi

The Involutive Quantaloid of Completely Distributive Lattices [article]

Luigi Santocanale
2020 arXiv   pre-print
If this is the case, then the dual tensor operation corresponds, via Raney's transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L.  ...  Let sLatt be the category of sup-lattices and join-continuous functions and let cdLatt be the full subcategory of sLatt whose objects are the completely distributive lattices.  ...  Recalling that involutive quantale structures on a given quantale are determined by the cyclic dualizing elements and that complete chains are completely distributive, the following statement from the  ... 
arXiv:1911.01085v3 fatcat:rk6ewzmn3jc7rlnb3ua6sdk7ui

Dualizability and graph algebras

Brian A. Davey, Paweł M. Idziak, William A. Lampe, George F. McNulty
2000 Discrete Mathematics  
We characterize the finite graph algebras which are dualizable.  ...  Indeed, a finite graph algebra is dualizable if and only if each connected component of the underlying graph is either complete or bipartite complete (or a single point).  ...  This method, which is used implicitly in Davey and Werner [9] to prove that the 2-element implication algebra is not dualizable and more plainly in the proof by Davey, Heindorf, and McKenzie of the Big  ... 
doi:10.1016/s0012-365x(99)00225-3 fatcat:aevgi4zhd5fatggww5p2cmes5y

Fake rings, fake modules, and duality

D.D Anderson
1977 Journal of Algebra  
A duality theory for modules over a commutative ring is developed using lattice modules. Using this duality theory several classical results for Noetherian modules are dualized to Artinian modules.  ...  F or a multiplicative lattice L, we define a fake L-module N to be a complete lattice N satisfying Our duality construction given in Section 2 carries over to fake L-modules (where L is now a multiplicative  ...  By proving theorems about modules for the larger class of fake modules, we are able to dualize the results. In Section 2 we introduce fake modules and give our basic duality construction.  ... 
doi:10.1016/0021-8693(77)90233-2 fatcat:gz2nhberyfeqzko6doesa7gmtu

Measuring the Implications of the D-Basis in Analysis of Data in Biomedical Studies [chapter]

Kira Adaricheva, J. B. Nation, Gordon Okimoto, Vyacheslav Adarichev, Adina Amanbekkyzy, Shuchismita Sarkar, Alibek Sailanbayev, Nazar Seidalin, Kenneth Alibek
2015 Lecture Notes in Computer Science  
This enables a ranking of all attributes, by relevance parameter to the same fixed attribute, and, as a consequence, reveals the implications of the basis most relevant to this attribute.  ...  Each test case requires a specialized approach in converting the real-valued data into binary data and careful analysis of the transformed data in a multi-disciplinary environment of cross-field collaboration  ...  We received the permission of Takeaki Uno, from the National Institute of Informatics in Tokyo, to implement the call to his subroutine performing the hypergraph dualization, within the structure of our  ... 
doi:10.1007/978-3-319-19545-2_3 fatcat:meyn5g3qe5c4ziycbyd7aqutzu
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