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Abstract : |
It is often claimed that the essence of functional programming is the use of functions as values, i.e., of higher order functions, and many interesting examples have been given showing the power of this approach. Unfortunately, the logic of higher order functions is difficult, and in particular, higher order unification is undecidable. Moreover (and closely related), higher order expressions are notoriously difficult for humans to read and write correctly. However, this paper shows that typical higher order programming examples can be captured with just first order functions, by the systematic use of parameterized modules, in a style that we call parameterized programming. This has the advantages that correctness proofs can be done entirely within first order logic, and that interpreters and compilers can be simpler and more efficient. Moreover, it is natural to impose semantic requirements on modules, and hence on functions. A more subtle point is that higher order logic does not always mix well with subsorts, which can nonetheless be very useful in functional programming by supporting the clean and rigorous treatment of partially defined functions, exceptions, overloading, multiple representation, and coercion. Although higher order logic cannot always be avoided in specification and verification, it should be avoided wherever possible, for the same reasons as in programming. This paper contains several examples, including one in hardware verification. An appendix shows how to extend standard equational logic with quantification over functions, and justifies a perhaps surprising technique for proving such equations using only ground term reduction. 1, |
