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pp:l08 [2020/04/02 19:12] pdmatei |
pp:l08 [2020/04/12 22:34] (current) uvlad |
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| - | ======= Lab 8 - The Lambda Calculus ======= | + | ======= 8. The Lambda Calculus ======= |
| 1. Consider the following datatype which encodes λ-expressions: | 1. Consider the following datatype which encodes λ-expressions: | ||
| Line 29: | Line 29: | ||
| </code> | </code> | ||
| - | 4. Write a function which takes a λ-expression of the form ''(λx.<body> <arg>)'' and **reduces it in a SINGLE step**. | + | 6. Write a function which takes a λ-expression of the form ''(λx.<body> <arg>)'' and **reduces it in a SINGLE step**. |
| - What should ''(λx.(x x) y)'' produce? | - What should ''(λx.(x x) y)'' produce? | ||
| - What should ''(λx.λx.(x x) y)'' produce? | - What should ''(λx.λx.(x x) y)'' produce? | ||
| - | 5. Add two data constructors to the type ''LExpr'' so that we can also model functions and applications in uncurry form. | + | 7. Add two data constructors to the type ''LExpr'' so that we can also model functions and applications in uncurry form. |
| Examples: ''(λx y z.<body>)'', ''(f x y z)''. | Examples: ''(λx y z.<body>)'', ''(f x y z)''. | ||
| - | 6. Write a proper display function for these new constructors. | + | 8. Write a proper display function for these new constructors. |
| - | 7. Write a function ''lcurry'' which takes a curried λ-expression and transforms it in uncurry form. | + | 9. Write a function ''luncurry'' which takes an uncurries λ-expression and transforms it in curry form. |
| <code haskell> | <code haskell> | ||
| lcurry :: LExpr -> LExpr | lcurry :: LExpr -> LExpr | ||
| </code> | </code> | ||
| - | 8. Write a function ''luncurry'' which takes an uncurries λ-expression and transforms it in curry form. | + | 10. Write a function ''lcurry'' which takes a curried λ-expression and transforms it in uncurry form. |
| - | Examples: ''( ((f x) y) z)'' becomes ''(f x y z)''. | + | <code> |
| + | (((f x) y) z) becomes (f x y z) | ||
| + | (((f ((g a) b)) y) ((h u) v)) becomes (f (g a b) y (h u v)) | ||
| + | </code> | ||
| + | |||
| + | 11. Write the function ''fv'' which computes the list of all **free variables** of a λ-expression. | ||
| + | <code haskell> | ||
| + | fv :: LExpr -> [Char] | ||
| + | </code> | ||
| + | |||
| + | 12. Write a function ''bv'' which computes the list of all **bound variables** of a λ-expression. | ||
| + | <code haskell> | ||
| + | bv :: LExpr -> [Char] | ||
| + | </code> | ||
| + | |||
| + | 13. Write a function ''subst'' which computes the //textual substitution of all free occurrences of some variable x by e in e'//, according to the lecture definition: | ||
| + | <code haskell> | ||
| + | subst :: Char -> LExpr -> LExpr -> LExpr | ||
| + | </code> | ||
| + | |||
| + | 14. Implement a function which reduces a **reducible** λ-expression to an irreducible one. (According to the lecture definition, what happens with λx.(λx.x x) ? | ||
| + | <code haskell> | ||
| + | reduce :: LExpr -> LExpr | ||
| + | </code> | ||
| + | |||
| + | 15. Implement **normal-order** evaluation. | ||
| + | |||
| + | 16. Implement **applicative** (strict) evaluation. | ||
| - | Also, ''( ((f ((g a) b)) y) ((h u) v))'' becomes ''(f (g a b) y (h u v))'' | ||