Simple-Named Complex-Valued Nominative Data – Definition and Basic Operations

Abstract

Summary In 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].

Partial Correctness of a Power Algorithm

“…A and loc /1 ,loc /2 ,loc /3 ,loc /4 ,loc /5 are mutually different. Then PP-composition(A, loc, b 0, n 0 ), power-main-loop (A, loc), Equality(A, loc /1 , loc /4 )∧PP-composition(A, loc, b 0 , n 0 ) is an SF-HT of ND SC (V, A).The theorem is a consequence of (5) and(6). (8) Suppose Vis not empty and A is complex containing and V is without nonatomic nominative data w.r.t.…”

Formalization of the Algebra of Nominative Data in Mizar

“…We implemented our library in the form of a Mizar paper entitled NOMIN_1.MIZ [39], in which we formalized the carrier set and the operations of the algebra of nominative data of the type T N D SC .…”

Partial Correctness of an Algorithm Computing Lucas Sequences

“…And a 1 , a 2 , a 3 , a 4 , a 5 , a 6 (3) reduces to a 3 and a 1 , a 2 , a 3 , a 4 , a 5 , a 6 (4) reduces to a 4 and a 1 , a 2 , a 3 , a 4 , a 5 , a 6 (5) reduces to a 5 and a 1 , a 2 , a 3 , a 4 , a 5 , a 6 (6) reduces to a 6 .…”