http://hdl.handle.net/11320/21361 2026-10-30T08:30:03Z http://hdl.handle.net/11320/21377 Tytuł: String Rewriting Systems Autorzy: Trybulec, Michał Abstrakt: Basing on the definitions from [15], semi-Thue systems, Thue systems, and direct derivations are introduced. Next, the standard reduction relation is defined that, in turn, is used to introduce derivations using the theory from [1]. Finally, languages generated by rewriting systems are defined as all strings reachable from an initial word. This is followed by the introduction of the equivalence of semi-Thue systems with respect to the initial word. 2007-01-01T00:00:00Z http://hdl.handle.net/11320/21376 Tytuł: Definition and some Properties of Information Entropy Autorzy: Zhang, Bo; Nakamura, Yatsuka Abstrakt: In this article we mainly define the information entropy [3], [11] and prove some its basic properties. First, we discuss some properties on four kinds of transformation functions between vector and matrix. The transformation functions are LineVec2Mx, ColVec2Mx, Vec2DiagMx and Mx2FinS. Mx2FinS is a horizontal concatenation operator for a given matrix, treating rows of the given matrix as finite sequences, yielding a new finite sequence by horizontally joining each row of the given matrix in order to index. Then we define each concept of information entropy for a probability sequence and two kinds of probability matrices, joint and conditional, that are defined in article [25]. Further, we discuss some properties of information entropy including Shannon's lemma, maximum property, additivity and super-additivity properties. 2007-01-01T00:00:00Z http://hdl.handle.net/11320/21370 Tytuł: Mizar Analysis of Algorithms: Preliminaries Autorzy: Bancerek, Grzegorz Abstrakt: Algorithms and its parts - instructions - are formalized as elements of if-while algebras. An if-while algebra is a (1-sorted) universal algebra which has 4 operations: a constant - the empty instruction, a binary catenation of instructions, a ternary conditional instruction, and a binary while instruction. An execution function is defined on pairs (s, I), where s is a state (an element of certain set of states) and I is an instruction, and results in states. The execution function obeys control structures using the set of distinguished true states, i.e. a condition instruction is executed and the continuation of execution depends on if the resulting state is in true states or not. Termination is also defined for pairs (s, I) and depends on the execution function. The existence of execution function determined on elementary instructions and its uniqueness for terminating instructions are shown. 2007-01-01T00:00:00Z http://hdl.handle.net/11320/21367 Tytuł: The Product Space of Real Normed Spaces and its Properties Autorzy: Endou, Noboru; Shidama, Yasunari; Miyajima, Keiichi Abstrakt: In this article, we define the product space of real linear spaces and real normed spaces. We also describe properties of these spaces. 2007-01-01T00:00:00Z