CS442 Module 4: Types
CS442 Module 4: Types University of Waterloo Winter 2021 1 Getting Stuck In Module 2, we studied the λ-calculus as a model for programming. We showed how many of the elements of “real-world” programming languages can be built from λ-expressions. But, because these “real-world” constructs were all modelled as λ-expressions, we can combine them in any way we like without regard for “what makes sense”. When we added semantics, in Module 3, we also added direct (“primitive”) semantic implementations of particular patterns, such as booleans, rather than expressing them directly as λ-expressions. That change introduces a similar problem: what happens when you try to use a primitive in a way that makes no sense? For instance, what happens if we try to add two lists as if they were numbers? Well, let’s work out an example that tries to do that, using the semantics of AOE, plus numbers and lists, from Module 3, and the expression (+ (cons 1 (cons 2 empty)) (cons 3 (cons 4 empty))): • The only semantic rule for + that matches is to reduce the first argument. After several steps, it reduces to [1, 2], so our expression is now (+ [1, 2] (cons 3 (cons 4 empty))). • The rule for + to reduce the first argument no longer matches, because the first argument is not reducible. But, the rule to reduce the second argument doesn’t match, because the first argument is not a number1. No rules match, so we can reduce no further. What happens next? The answer is that nothing happens next! There’s no semantic rule that matches the current state of our program, so we can’t take another step.
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