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Boolean algebras and first order theories.
Cepák, Jiří ; Pajas, Petr (referee) ; Mlček, Josef (advisor)
We will study Lindenbaum algebras and algebras of definable subsets of selected first order theories: constants theory for a, Presburger, Robinson, Peano and standard arithmetic, successor theory, successor theory with zero, theory of dense linear orders without endpoints, theory of discrete linear orders, random graph theory and theory of algebraically closed fields. For finite algebras we will determine their cardinality, for countable algebras we will determine whether they are atomic or atomless and for some of them we will carry out classification up to isomorphism using algebras FA, ASA and CA. For this purpose we will prove several general theorems.
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SIEMENS LOGO! RCE - posibility to use in informatics learning - Elementary functions
LOUCKÝ, Václav
Aim of this bachelor thesis is to clarify issues of PLC automat Siemens LOGO! RCE, in practical part I'll focus on the work with sensors and home using of this automat. In theoretical part I focus on elementary functions with controlling of this device and creation of learning materials. Siemens LOGO! RCE presents universal logical module. It is used in ordinary life and it could be used in households (doors opening, blinds, etc...) even in commer-cial world (controlling strips, timing of shifts, etc). In practical part I'm going to deal with water heating using LOGO! and solar panels. Program will provide proper finding of operating point using optocouplers and proper impedance adaptation in certain automat. In theoretical part there will be description of graphical interface design, description of simple logical elements (OR, AND, NOR, NAND, XOR, NOT), flip-flops, Boolean logic, Boolean algebra. This thesis will find ways, how to attain higher series with relevant programming. As a last one will be created e-learning materials and worksheets.
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Orthocomplemented difference lattices
Havlík, František ; Matoušek, Milan (referee) ; Pták, Pavel (advisor)
The theme of this thesis is the investigation of a binary operator, 4, that models the standard symmetric di erence of sets. This operator is studied both separately (in Chapter II) and with the supplementary lattice structure (Chapter III and the rest). The class ODL is introduced and some of its basic properties are investigated. Then there is exhibited the class HOR. The class HOR is a subclass of ODL which is closely related to the class of Boolean algebras. In the last Chapter there is described the construction of free orthocomplemented di erence lattice with two generators. 3
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Theories and algebras of formulas
Garlík, Michal ; Mlček, Josef (advisor) ; Glivický, Petr (referee)
In the present work we study first-order theories and their Lindenbaum alge- bras by analyzing the properties of the chain BnT n<ω, called B-chain, where BnT is the subalgebra of the Lindenbaum algebra given by formulas with up to n free variables. We enrich the structure of Lindenbaum algebra in order to cap- ture some differences between theories with term-by-term isomorphic B-chains. Several examples of theories and calculations of their B-chains are given. We also construct a model of Robinson arithmetic, whose n-th algebras of definable sets are isomorphic to the Cartesian product of the countable atomic saturated Boolean algebra and the countable atomless Boolean algebra. 1
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Kerdockovy kódy a okolí
Teplá, Kateřina ; Drápal, Aleš (advisor) ; Šťovíček, Jan (referee)
Title: Kerdock codes and around Author: Kateřina Teplá Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc., Department of algebra Abstract: Kerdock codes form a family of nonlinear codes, that contains more codewords than any known linear code with the same parameters. The main goal of this thesis is a connection of Kerdock codes with other areas of mathematics, mainly orthogonal geometry, combinatorics and cryptogra- phy. It describes theory of symplectic and quadratic forms on vector spaces of characteristic 2 and its relationship to Kerdock codes. Then it is pro- ven, that codewords of Kerdock code of constant weight form combinatorial 3-design. Finally usage of Kerdock codes in construction of Boolean bent functions and t-resilient functions, that are basis of many cryptographic pri- mitives, is analysed. Keywords: Kerdock code, Kerdock set, t-design, resilient function 1
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Fast algebraic attacks
Hána, Martin ; Hojsík, Michal (referee) ; Holub, Štěpán (advisor)
In the present work we study algebraic attacks and cascading fast algebraic attack on stream ciphers using in their construction linear feedback shift registers. For deeper understanding of attacks we present some facts which are needed to know from theory of linear recurrence sequences in rst chapter. We show their connection to formalized description of construction we attack. In second chapter we show algebraic attacks on both ciphers using memory or memoryless. We introduce denitions of annihilator and algebraic immunity of Boolean function and show their main properties. In third chapter we use knowledge from rst two chapters and show process and principle of fast algebraic attack.
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Minimisation of Logical Functions
Horký, Miroslav ; Davidová, Olga (referee) ; Šeda, Miloš (advisor)
For minimisation of logical functions, laws of the Boolean algebra and the Karnaugh maps are mostly used. However, use of Karnaugh's maps is based on visual recognition of adjacent cells for functions with no more than 6 variables and, therefore, the method is not suitable for automated processing on computers. A direct application of the Boolean algebra laws is not restricted in this way, but there is no general algorithm defining the sequence of their application and thus this approach is not suitable for computer implementation either. The well-known method usable on computers is the algorithm proposed by E. J. McCluskey and W. Orman Quine.
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