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Implementations that follow the functional programming paradigm are being used in more and more domains. As functional programming paradigm has mathematical reference transparency, refinement to functional programs contributes to improving the reliability of the transformation process and simplifying the refinement steps. However, it is a challenge to generate functional programs from specifications. Most existing transformation methods refine specifications into abstract algorithm-level programs based on loop invariants rather than functional programs. This paper proposes a novel functional program generation method based on the partition-recursion refinement rule. It establishes a novel program refinement framework based on functional theory for the first time. This is the first study to regard the whole program refinement process as a composition of abstract functions. This paper designs a recurrence-based algorithm design language (Radl+) and implements a software prototype to map Radl+ algorithms into executable Haskell programs. To prove the feasibility and efficiency of this method, this paper transforms the polynomial multiplication problem from a specification into an executable Haskell program. This case shows that compared with existing approaches, the proposed method can simplify the transformation steps and reduce the number of lines of generated code from 38 to 10. • Novel refinement framework provides a new approach to generating a functional program. • The composition of abstract functions explains the program refinement process. • Substitution rule and Recursion rule have none of the side effects. • Software prototype transforms the polynomial multiplication problem into Haskell program. [ABSTRACT FROM AUTHOR]

Aggregation theory is devoted to the fusing of several values into a unique output that summarizes the given information. Typically, the aggregation process is formalized in terms of an increasing mathematical function that maps the input values to the result, fulfilling some boundary conditions. However, this formalization can be too restrictive for some scenarios. In some cases, the inputs can be seen as observations of random variables, the aggregation result being also a random variable. In others, the aggregation process can be identified as a program that performs the aggregation rather than a mathematical function. In this direction, the concepts of aggregation of random variables and computable aggregation have been defined in the literature. This paper is devoted to the definition of computable aggregation of random variables, which are computer programs, not functions, that aggregate random variables, not numbers. Special attention is given to different possible alternatives to modelize random variables and monotonicity. The implementation of some examples is also provided. • Different possible representations of random variables in a computation framework are considered. • The definition of computable aggregations of random variables is stated. • The boundary and monotonicity conditions are studied. • The implementation of some remarkable examples is provided. [ABSTRACT FROM AUTHOR]