From Kleene’s Model to the Parallel Abstract Machine Mihaela Malit ¸a and Gheorghe M. S ¸tefan Abstract Parallel computation is mathematically defined by Kleene’s model of partial recursive functions. The paper proves that only the compo- sition rule is independent. Therefore, it is used to define the Kleene Machine, KM, and the Universal Kleene Machine, UKM, as the mathematical model for parallel computation. The Map-Reduce parallel abstract machine model is defined starting from UKM. Key words and phrases : mathematical model of computation, abstract parallel machine, parallel computation, Kleene’s recursive functions Mathematics Subject Classification (2000) : 03D75, 03D80. 1 Introduction The historical evolution of the mono-core computation teaches us about the right way to be followed in developing multi- and many-core computation. The main steps in the history of mono-cores are: 1936: four equivalent mathematical models are published [10] [4] [7] [8] (all seem to be triggered by G¨odel’s paper on non-decidability) out of which the Turing Machine offered the most appropriate technologically suggestion for future developments leading eventually to mono-core computing 1944-45: MARK 1 computer, built by IBM for Harvard University, con- secrated the Harvard abstract model, while von Neumann’s report [11] introduced the von Neumann abstract model 1953: IBM launched IBM 701, the first electronic computer manufac- tured in quantity 1964: the concept of computer architecture is introduced [3] to allow the independent development for the two different aspects of computer de- sign with different rates of evolution: software and hardware. Thus, there are now on the market few stable and successful mono-core architectures, such as x86, ARM. The emergence of multi- and many-core computing, under the market criteria pressure, had its logical evolution dis- torted (in 1962, Burroughs started the manufacturing in quantity; only in 1965, Edsger W. Dijkstra raised architectural issues [6]; in 1974, the first 1