✎ pp:l08 [books]

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======= 8. The Lambda Calculus =======

1. Consider the following datatype which encodes λ-expressions:

<code haskell>
data LExpr = Var Char | Lambda Char LExpr | App LExpr LExpr 
</code>

Enroll ''LExpr'' in class ''Show''.

2. Write a function ''vars'' which returns a list of variables used in a λ-expression:
<code haskell>
vars :: LExpr -> [Char]
</code>

3. Write a function ''reducible'' which tests if an expression can be reduced to another. **Write tests first!** What are the cases when an expression is reducible?
<code haskell>
reducible :: LExpr -> Bool
</code>

4. Write a function which renames **all** occurrences of a variable with another, in a λ-expression:
<code haskell>
rename :: Char -> Char -> LExpr -> LExpr
</code>

5. Write a function which **replaces all** occurrences of a variable with a λ-expression, in a λ-expression:
<code haskell>
replace :: Char -> LExpr -> LExpr -> LExpr
</code>

4. 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.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.
Examples: ''(λx y z.<body>)'', ''(f x y z)''.

6. Write a proper display function for these new constructors.

7. Write a function ''luncurry'' which takes an uncurries λ-expression and transforms it in curry form.
<code haskell>
lcurry :: LExpr -> LExpr
</code>

8. Write a function ''lcurry'' which takes a curried λ-expression and transforms it in uncurry form.
<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> 

9. Write the function ''fv'' which computes the list of all **free variables** of a λ-expression.
<code haskell>
fv :: LExpr -> [Char]
</code>

10. Write a function ''bv'' which computes the list of all **bound variables** of a λ-expression.
<code haskell>
bv :: LExpr -> [Char]
</code>

11. 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>

12. 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>

13. Implement **normal-order** evaluation.

14. Implement **applicative** (strict) evaluation.