# Mathematical Foundation of the Synchronous/Reactive Model

The Synchronous/Reactive Model is built upon three mathematical concepts:
• complete partial order (CPO): "a set with an abstract notion of the amount of 'information' in each element." [1]
see Definitions 1-4 in chapter 3 of [1]
• monotonic function: "applying a monotonic function to an element of CPO always increases the amount of information, unless it is a fixed point, in which case the element is unchanged." [1]
see Definition 6 in chapter 3 of [1]
• fixed point: given the vector x in which each element of x belongs to a CPO, then x is a fixed point for a function f if f(x) = x.
see Definition 8 in chapter 3 of [1]
Theorem 2 on page 48 of [1] proves that "an SR system always has a unique behavior, and its proof contains the fundamental idea used to evaluate the systems". The proof relies on Proposition 6 which states that a continuous function is monotonic.

The Synchronous/Reactive model of computation requires that the blocks (actors) be monotonic functions. Pages 50-52 of [1] show that the Synchronous/Reactive model is deterministic because an SR graph always has a least fixed point.

Note that in Stephen's thesis, he uses the symbol

```|  |
|  |
--
```
to mean the least upper bound. He also uses poset as an abbreviation for a partially-ordered set.

# References

1. Stephen Anthony Edwards, The Specification and Execution of Synchronous Reactive Systems, Ph.D. Thesis, University of California, Berkeley, 1997, Available as UCB/ERL M97/31.

Updated 04/22/02.