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from Part XVII - Modularity
Published online by Cambridge University Press: 05 February 2013
Modularity is the most important technique for controlling the complexity of programs. Programs are decomposed into separate components with precisely specified, and tightly controlled, interactions. The pathways for interaction among components determine dependencies that constrain the process by which the components are integrated, or linked, to form a complete system. Different systems may use the same components, and a single system may use multiple instances of a single component. Sharing of components amortizes the cost of their development across systems and helps limit errors by limiting coding effort.
Modularity is not limited to programming languages. In mathematics the proof of a theorem is broken down into a collection of definitions and lemmas. Cross references among lemmas determine a dependency structure that constrains their integration to form a complete proof of the main theorem. Of course, one person's theorem is another person's lemma; there is no intrinsic limit on the depth and complexity of the hierarchies of results in mathematics. Mathematical structures are themselves composed of separable parts, as, for example, a Lie group is a group structure on a manifold.
Modularity arises from the structural properties of the hypothetical and general judgments. Dependencies among components are expressed by free variables whose typing assumptions state the presumed properties of the component. Linking consists of substitution and discharge of the hypothesis.
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