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Lower Bounds on Boolean Formula Size
Fojtík, Vít ; Hrubeš, Pavel (advisor) ; Savický, Petr (referee)
The aim of this thesis is to study methods of constructing lower bounds on Boolean formula size. We focus mainly on formal complexity measures, gener- alizing the well-known Krapchenko measure to a class of graph measures, which we thereafter study. We also review one of the other main approaches, using ran- dom restrictions of Boolean functions. This approach has yielded the currently largest lower bounds. Finally, we mention a program for finding super-polynomial bounds based on the KRW conjecture. 1
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Classes of Boolean Formulae with Effectively Solvable SAT
Vlček, Václav ; Čepek, Ondřej (advisor) ; Kullmann, Oliver (referee) ; Savický, Petr (referee)
The thesis studies classes of Boolean formulae for which the well-known satisfiability problem is solvable in polynomially bounded time. It focusses on classes based on unit resolution; it describe classes of unit refutation complete formulae, unit propagation complete formulae and focuses on the class of SLUR formulae. It presents properties of SLUR formulae as well as the recently obtained results. The main result is the coNP-completness of membership testing. Finally, several hierarchies are built over the SLUR class and their properties and mutual relations are studied. Powered by TCPDF (www.tcpdf.org)
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Interval Representations of Boolean Functions
Kronus, David ; Čepek, Ondřej (advisor) ; Sgall, Jiří (referee) ; Savický, Petr (referee)
This thesis is dedicated to a research concerning representations of Boolean functions. We present the concept of a representation using intervals of integers. Boolean function f is represented by set I of intervals, if it is true just on those input vectors, which correspond to integers belonging to intervals in I, where the correspondence between vectors and integers depends on the ordering of bits determining their significancies. We define the classes of k-interval functions, which can be represented by at most k intervals with respect to a suitable ordering of variables, and we provide a full description of inclusion relations among the classes of threshold, 2-monotonic and k-interval Boolean functions (for various values of k). The possibility to recognize in polynomial time, whether a given function belongs to a specified class of Boolean functions, is another fundamental and practically important property of any class of functions. Our results concerning interval functions recognition include a proof of co-NP- hardness of the general problem and polynomial-time algorithms for several restricted variants, such as recognition of 1-interval and 2-interval positive functions. We also present an algorithm recognizing general 1-interval functions provided that their DNF representation satisfies several...
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Cut Languages in Rational Bases
Šíma, Jiří ; Savický, Petr
We introduce a so-called cut language which contains the representations of numbers in a rational base that are less than a given threshold. The cut languages can be used to refine the analysis of neural net models between integer and rational weights. We prove a necessary and sufficient condition when a cut language is regular, which is based on the concept of a quasi-periodic power series. We show that any cut language with a rational threshold is context-sensitive while examples of non-context-free cut languages are presented.
Fulltext: content.csg -  PDF
Plný tet: v1236-16 - PDF
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Experimentální srovnání triangulačních heuristik na transformovaných sítích BN2O
Vomlel, Jiří ; Savický, Petr
In this paper we present results of experimental comparisons of several triangulation heuristics on bipartite graphs. Our motivation for testing heuristics on the family of bipartite graphs is the rank-one decomposition of BN2O networks. A BN2O network is a Bayesian network having the structure of a bipartite graph with all edges directed from the top level toward the bottom level and where all conditional probability tables are noisy-or gates. After applying the rank-one decomposition, which adds an extra level of auxiliary nodes in between the top and bottom levels, and after removing simplicial nodes of the bottom level we get so called BROD graph. This is an undirected bipartite graph. It is desirable for efficiency of the inference to find a triangulation of the BROD graph having the sum of table sizes for all cliques of the triangulated graph as small as possible. From this point of view, the minfill heuristics perform in average better than other tested heuristics (minwidth, h1, and mcs).
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Methods for Multidimensional Event Classification: A Case Study
Bock, R.K. ; Chilingarian, A. ; Gaug, M. ; Hakl, František ; Hengstebeck, T. ; Jiřina, Marcel ; Klaschka, Jan ; Kotrč, Emil ; Savický, Petr ; Towers, S. ; Vaicilius, A. ; Wittek, W.
Fulltext: content.csg - PDF
Plný tet: v887-03 - PDF
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