Higher Order Functions Considered Unnecessary for Higher Order Programming

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 diicult, and in particular, higher order uniication is undecidable. Moreover (and closely related), higher order expressions are notoriously diicult for humans to read and write correctly. However, this paper shows that typical higher order programming examples can be captured with just rst 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 rst order logic, and that interpreters and compilers can be simpler and more eecient. 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 deened functions, exceptions, overloading, multiple representation, and coercion. Although higher order logic cannot always be avoided in speciication and veriication, it should be avoided wherever possible, for the same reasons as in programming. This paper contains several examples, including one in hardware veriication. An appendix shows how to extend standard equational logic with quantiication over functions, and justiies a perhaps surprising technique for proving such equations using only ground term reduction.