Foundations of Functional Programming/Comparison of λ-calculi
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This page compares λ-calculi according to various interesting properties they have. We give a table which concisely lays out all of the information for the calculi we discuss, followed by an explanation of each property featured in the table.
System | Implicit types? | Polymorphic types? | Type constructors? | Dependent types? | property? | Turing complete? | Normalizing? | Consistent? | Type checking decidable? | Typability decidable? | Types unique? | Subject reduction theorem? |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Untyped λ-calculus | Yes | No | No | No | No | Yes | No | No | Yes | Yes | Yes | Yes |
Implicitly simply typed λ-calculus | Yes | No | No | No | No | No | Yes | Yes | Yes | Yes | No | Yes |
Explicitly simply typed λ-calculus (λ→) | No | No | No | No | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
Implicitly typed second-order λ-calculus | Yes | Yes | No | No | No | No | Yes | Yes | No | No | No | Yes |
Explicitly typed second-order λ-calculus (λ2) | No | Yes | No | No | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
Hindley-Milner type theory | Yes | Yes | No | No | No | Yes | No | No | Yes | Yes | No | Yes |
λP | No | No | No | Yes | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
λP2 | No | Yes | No | Yes | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
λω | No | No | Yes | No | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
λω | No | Yes | Yes | No | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
λPω | No | No | Yes | Yes | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
Calculus of constructions (λPω) | No | Yes | Yes | Yes | No | No | Yes | Yes | Yes | Yes | Yes | Yes |
Naïve type theory (λ*) | No | Yes | Yes | Yes | Yes | ? | No | No | No | No | Yes | Yes |
Explanations of properties[edit | edit source]
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