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Introduction to Software Engineering
Introduction to Software Engineering

... Java exceptions are either checked or unchecked. Checked exceptions have to be handled by the programmer: • the programmer either has to catch them or declare them • if this is not done, the compilation will fail • all IOExceptions are checked exceptions Unchecked exceptions are different: • the pro ...
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... – f (t1 , . . . , tn ) ∈ TA for any function symbol f : A1 , . . . , An → A, and terms ti ∈ TAi with Ai  Ai for i = 1, . . . , n, – (A)t ∈ TA for any term t ∈ TA where A is an arbitrary type. We write the static type of t as σ(t) := A for any term t ∈ TA . A term (A)t is a type coercion, also c ...
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... “object-oriented, functional” language How do the “paradigms” actually fit together? Some features, such as case classes, are more obviously “object-oriented” versions of “functional” constructs Until now, we have pretended pairs, λ-abstractions, etc. are primitives in Scala They aren’t; and they ne ...
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Testing an Optimising Compiler by Generating Random Lambda

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Sexy types in action - Indiana University Computer Science
Sexy types in action - Indiana University Computer Science

... (This is a variant of the standard foldr function.) Then reduce ◦ build is equal to the identity function in a parametric model. This law can be used as a rewriting rule in the left-to-right direction to eliminate intermediate trees in an optimizing functional compiler, in other words, to deforest ( ...
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... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...
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... signifying addition, subtraction, multiplication and division. As usual, we can also use in front of an expression to negate its value, as in -(x+y). In addition, the function div and mod signify integer division and remainder, respectively. So div 3 2 is 1, div 7 3 is 2, . . . while mod 10 6 is 4, ...
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Covariance and contravariance (computer science)

Many programming language type systems support subtyping. For instance, if Cat is subtype of Animal, then an expression of type Cat can be used whenever an expression of type Animal could. Variance refers to how subtyping between more complex types (list of Cats versus list of Animals, function returning Cat versus function returning Animal, ...) relates to subtyping between their components. Depending on the variance of the type constructor, the subtyping relation may be either preserved, reversed, or ignored. For example, in C#: IEnumerable is a subtype of IEnumerable. The subtyping is preserved because IEnumerable is covariant on T. Action is a subtype of Action. The subtyping is reversed because Action is contravariant on T. Neither IList nor IList is a subtype of the other, because IList is invariant on T.The variance of a C# interface is determined by in/out annotations on its type parameters; the above interfaces are declared as IEnumerable, Action, and IList. Types with more than one type parameter may specify different variances on each type parameter. For example, the delegate type Func represents a function with a contravariant input parameter of type T and a covariant return value of type TResult.The typing rules for interface variance ensure type safety. For example, an Action represents a first-class function expecting an argument of type T, and a function that can handle any type of animal can always be used instead of one that can only handle cats.A programming language designer will consider variance when devising typing rules for e.g. arrays, inheritance, and generic datatypes. By making type constructors covariant or contravariant instead of invariant, more programs will be accepted as well-typed. On the other hand, programmers often find contravariance unintuitive, and accurately tracking variance to avoid runtime type errors can lead to complex typing rules. In order to keep the type system simple and allow useful programs, a language may treat a type constructor as invariant even if it would be safe to consider it variant, or treat it as covariant even when that can violate type safety.
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