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--- pp:l08 [2018/04/23 19:35]
sergiu
+++ pp:l08 [2020/04/12 22:34] (current)
uvlad
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-===== Laborator 08 - Programare in Prolog =====
+======= 8. The Lambda Calculus =======
+. 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''.
+. Write a function ''vars'' which returns a list of variables used in a λ-expression:
+<code haskell>
+vars :: LExpr -> [Char]
+</code>
+. 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>
+. Write a function which renames **all** occurrences of a variable with another, in a λ-expression:
+<code haskell>
+rename :: Char -> Char -> LExpr -> LExpr
+</code>
+. 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>
+. 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?
+. 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)''.
+. Write a proper display function for these new constructors.
+. Write a function ''luncurry'' which takes an uncurries λ-expression and transforms it in curry form.
+<code haskell>
+lcurry :: LExpr -> LExpr
+</code>
+. 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>
+. Write the function ''fv'' which computes the list of all **free variables** of a λ-expression.
+<code haskell>
+fv :: LExpr -> [Char]
+</code>
+. Write a function ''bv'' which computes the list of all **bound variables** of a λ-expression.
+<code haskell>
+bv :: LExpr -> [Char]
+</code>
+. 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>
+. 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>
+. Implement **normal-order** evaluation.
+. Implement **applicative** (strict) evaluation.
-==== Multimi ====
-  - Definiti predicatul ''cartesian(L1,L2,R)'' care construieste produsul cartezian al ''L1'' cu ''L2''
-  - Definiti predicatul ''union(L1,L2,R)'' care construieste reuniunea a doua multimi codificate ca liste.
-  - Definiti predicatul ''intersection(L1,L2,R)''
-  - Definiti predicatul ''diff(L1,L2,R)'' care construieste diferenta pe multimi intre ''L1'' si ''L2''
-==== Permutari, Aranjamente, Combinari ====
-  - Definiti predicatul ''pow(S,R)'' care construieste ''power-set''-ul multimii ''S''.
-  - Definiti predicatul ''perm(S,R)'' care genereaza toate permutarile lui ''S''.
-  - Definiti predicatul ''ar(K,S,R)'' care genereaza toate aranjamentele de dimensiune ''K'' cu elemente luate din ''S''
-  - Definiti predicatul ''comb(K,S,R)'' care genereaza toate combinarile de dimensiune ''K'' cu elemente luate din ''S''