**Section 3**

**Algorithms and Logic Programming** 

96 Semantics – Advances in Theories and Mathematical Models

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588–593 (2002).

*Softw. Eng. 5*, 10 (Oct.), 168–175.

International Multitopic Conference, INMIC 2008, IEEE, Karachi, Pakistan,

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**A Common Mathematical Framework for**

**Asymptotic Complexity Analysis and**

**Denotational Semantics for Recursive**

Salvador Romaguera1 and Oscar Valero2

*Universitat Politècnica de València*

<sup>1</sup>*Instituto Universitario de Matemática Pura y Aplicada,*

**Programs Based on Complexity Spaces**

<sup>2</sup>*Departamento de Ciencias Matemáticas e Informática, Universidad de las Islas Baleares*

In Denotational Semantics one of the aims consists of giving mathematical models of programming languages so that the meaning of a recursive algorithm can be obtained as an

Most programming languages allow to construct recursive algorithms by means of a recursive definition expressing the meaning of such a definition in terms of its own meaning. In order to analyze the correctness of such recursive definitions, D.S. Scott developed a mathematical theory of computation which is based on ideas from order theory and topology (Davey & Priestley, 1990; Scott, 1970; 1972). From the Scott theory viewpoint, the meaning of such a denotational specification is obtained as the fixed point of a nonrecursive mapping, induced by the denotational specification, which is at the same time the topological limit of successive iterations of the nonrecursive mapping acting on a distinguished element of the model. Moreover, the order of Scott's model represents some notion of information so that each iteration of the nonrecursive mapping, which models each step of the program computation, is identified with an element of the mathematical model which is greater than (or equal to) the other ones associated with the preceding iterations (preceding steps of the program computation) because each iteration gives more information about the meaning than those computed before. Hence the aforesaid meaning of the recursive denotational specification is modeled as the fixed point of the nonrecursive mapping which is obtained as the limit, with respect to the so-called Scott topology, of the increasing sequence of successive iterations. Consequently, the fixed point captures the amount of information defined by the increasing sequence, i.e. the fixed point yields the total information about the meaning provided by the elements of the increasing sequence, and it does not contain more information than can be

A typical and illustrative example of such recursive definitions is given by those recursive algorithms that compute the factorial of a nonnegative integer number by means of the

**1. Introduction**

element of the constructed model.

obtained from the elements of such a sequence.

following recursive denotational specification:
