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Von Neumann was a founding figure in computer science. He and Stanisław Ulam developed simulations on von Neumann's digital computers for hydrodynamic computations. He contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers.

While consulting for the Moore School of Electrical Engineering at the University of Pennsylvania on the EDVAC project, von Neumann wrote an incomplete First Draft of a Report on the EDVAC. The paper, which was widely distributed, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space. John von Neumann also consulted for the ENIAC project, when ENIAC was being modified to contain a stored program. Since the modified ENIAC was fully functional by 1948 and the EDVAC wasn't delivered to Ballistics Research Laboratory until 1949, one could argue that ENIAC was the first computer to use a stored program. John von Neumann also designed the instruction set for the modified ENIAC, and he should be given credit for this. The stored program version of ENIAC ran 6 times slower, but it was still entirely I/O bound, and programs could be developed and debugged in days rather than weeks, which is one of the advantages of having stored programs. This architecture is to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. The single-memory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper.

Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953. However, the theory could not be implemented until advances in computing of the 1960s.

Von Neumann also created the field of cellular automata without the aid of computers, constructing the first self-replicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his posthumous work Theory of Self Reproducing Automata. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon or asteroid belt would be by using self-replicating machines, taking advantage of their exponential growth.

Beginning in 1949, Von Neumann's design for a self-reproducing computer program is considered the world's first computer virus, and he is considered to be the theoretical father of computer virology.

Donald Knuth cites von Neumann as the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged together.

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Q: Why is John Neumann so important to the study of computer science?
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