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http://hdl.handle.net/11320/6275
Mon, 23 Nov 2020 16:09:44 GMT2020-11-23T16:09:44ZFormal Introduction to Fuzzy Implications
http://hdl.handle.net/11320/6298
Tytuł: Formal Introduction to Fuzzy Implications
Autorzy: Grabowski, Adam
Abstrakt: SummaryIn the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized previously).Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11320/62982017-01-01T00:00:00ZIntegral of Non Positive Functions
http://hdl.handle.net/11320/6297
Tytuł: Integral of Non Positive Functions
Autorzy: Endou, Noboru
Abstrakt: SummaryIn this article, we formalize in the Mizar system [1, 7] the Lebesgue type integral and convergence theorems for non positive functions [8],[2]. Many theorems are based on our previous results [5], [6].Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11320/62972017-01-01T00:00:00ZGauge Integral
http://hdl.handle.net/11320/6296
Tytuł: Gauge Integral
Autorzy: Coghetto, Roland
Abstrakt: SummarySome authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korniłowicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12].A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in [10] and [4].Using the Mizar system [1], we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval [a, b] (see [2], [3], [15], [14], [11]). In the next section we formalize that the Henstock-Kurzweil integral is linear.In the last section, we verified that a real-valued bounded integrable (in sense Darboux/Riemann [6, 7, 8]) function over a interval a, b is Gauge integrable.Note that, in accordance with the possibilities of the MML [9], we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in [7] (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11320/62962017-01-01T00:00:00ZSimple-Named Complex-Valued Nominative Data – Definition and Basic Operations
http://hdl.handle.net/11320/6295
Tytuł: Simple-Named Complex-Valued Nominative Data – Definition and Basic Operations
Autorzy: Ivanov, Ievgen; Nikitchenko, Mykola; Kryvolap, Andrii; Korniłowicz, Artur
Abstrakt: SummaryIn this paper we give a formal definition of the notion of nominative data with simple names and complex values [15, 16, 19] and formal definitions of the basic operations on such data, including naming, denaming and overlapping, following the work [19].The notion of nominative data plays an important role in the composition-nominative approach to program formalization [15, 16] which is a development of composition programming [18]. Both approaches are compared in [14].The composition-nominative approach considers mathematical models of computer software and data on various levels of abstraction and generality and provides mathematical tools for reasoning about their properties. In particular, nominative data are mathematical models of data which are stored and processed in computer systems. The composition-nominative approach considers different types [14, 19] of nominative data, but all of them are based on the name-value relation. One powerful type of nominative data, which is suitable for representing many kinds of data commonly used in programming like lists, multidimensional arrays, trees, tables, etc. is the type of nominative data with simple (abstract) names and complex (structured) values. The set of nominative data of given type together with a number of basic operations on them like naming, denaming and overlapping [19] form an algebra which is called data algebra.In the composition-nominative approach computer programs which process data are modeled as partial functions which map nominative data from the carrier of a given data algebra (input data) to nominative data (output data). Such functions are also called binominative functions. Programs which evaluate conditions are modeled as partial predicates on nominative data (nominative predicates). Programming language constructs like sequential execution, branching, cycle, etc. which construct programs from the existing programs are modeled as operations which take binominative functions and predicates and produce binominative functions. Such operations are called compositions. A set of binominative functions and a set of predicates together with appropriate compositions form an algebra which is called program algebra. This algebra serves as a semantic model of a programming language.For functions over nominative data a special computability called abstract computability is introduces and complete classes of computable functions are specified [16].For reasoning about properties of programs modeled as binominative functions a Floyd-Hoare style logic [1, 2] is introduced and applied [12, 13, 8, 11, 9, 10]. One advantage of this approach to reasoning about programs is that it naturally handles programs which process complex data structures (which can be quite straightforwardly represented as nominative data). Also, unlike classical Floyd-Hoare logic, the mentioned logic allows reasoning about assertions which include partial pre- and post-conditions [11].Besides modeling data processed by programs, nominative data can be also applied to modeling data processed by signal processing systems in the context of the mathematical systems theory [4, 6, 7, 5, 3].Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11320/62952017-01-01T00:00:00Z