AiS 27: Mapping, Scope, and Partial Order

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On the SSG oven software project, Raymond and I used several mathematical truths to prove that the design met the stated requirements. Three of those truths are mapping, scope, and partial order.

A mapping is a function from one set, called the domain, to another set, called the range. The range is not larger than the domain. If the mapping is one-to-one, then a reverse mapping exists and the range is exactly the same size as the domain. A hash is a many-to-one mapping, so the range is smaller than the domain. Therefore, no reverse mapping exists.

A scope is a context in which simpler objects reside. You can abstract the relationships among the enclosed objects as relationships among the scopes in which they live.

A partial order is a transitive relationship between objects in a set. It is not as restrictive as a full order, but allows for some independence. A partial order, because it is transitive, must be acyclic. Replacing a full order with a partial order can relax the constraints on a design, and give you room for optimization.

Your homework is to design an Integrated Circuit designer. Allow the user to build a component out of other components. Prove that your design does not allow the user to create a recursive cycle.

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