New PDF release: Introduction to functional programming

By Harrison J.

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Prove Newman’s Lemma, which states that if a relation is weakly Church-Rosser and obeys strong normalization, then −→∗ is Church-Rosser. )10 9 10 See Mairson (1991) for more on this question. If you get stuck, there is a nice proof in Huet (1980). Chapter 5 A taste of ML In the last few chapters, we have started with pure lambda calculus and systematically extended it with new primitive features. For example we added the ‘let’ construct as primitive for the sake of making the polymorphic typing more useful, and a recursion operation in order to restore the computational power that was lost when adding types.

T in (λx. y) t. Of course, eager evaluation means that some expressions may loop that would terminate in a lazy regime. But this is considered acceptable, since these instances are generally easy to avoid in practice. In any case, an eager evaluation strategy is the standard in many programming languages like C, where it is referred to as call by value. ML adopts eager evaluation, for two main reasons. Choreographing the reductions and sharings that occur in lazy evaluation is quite tricky, and implementations tend to be relatively inefficient and complicated.

F p (λg. g q (λh. h r s)) where in the last line we have performed an alpha-conversion for the sake of clarity. Although tuples are built up in a flat manner, it is easy to create arbitrary finitely-branching tree structures by using tupling repeatedly. Finally, if one prefers conventional functions over Cartesian products to our ‘curried’ functions, one can convert between the two using the following: CURRY f = λx y. f (x, y) UNCURRY g = λp. 1. REPRESENTING DATA IN LAMBDA CALCULUS 29 These special operations for pairs can easily be generalized to arbitrary ntuples.

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