{"id":794,"date":"2014-12-10T06:00:16","date_gmt":"2014-12-10T04:00:16","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=794"},"modified":"2021-04-25T17:07:01","modified_gmt":"2021-04-25T15:07:01","slug":"normalizacion-de-expresiones-aritmetica-1","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/normalizacion-de-expresiones-aritmetica-1\/","title":{"rendered":"Normalizaci\u00f3n de expresiones aritm\u00e9ticas"},"content":{"rendered":"<h4>Enunciado<\/h4>\n<pre lang=\"text\">\n-- El siguiente tipo de dato representa expresiones construidas con\n-- variables, sumas y productos\n--    data Expr = V String\n--              | S Expr Expr\n--              | P Expr Expr\n--              deriving (Eq, Show)\n-- Por ejemplo, x*(y+z) se representa por (P (V \"x\") (S (V \"y\") (V \"z\"))) \n-- \n-- Una expresi\u00f3n es un t\u00e9rmino si es un producto de variables. Por\n-- ejemplo, x*(y*z) es un t\u00e9rmino pero x+(y*z) ni x*(y+z) lo son.\n--\n-- Una expresi\u00f3n est\u00e1 en forma normal si es una suma de t\u00e9rminos. Por\n-- ejemplo, x*(y*z) y x+(y*z) est\u00e1 en forma normal; pero x*(y+z) y\n-- (x+y)*(x+z) no lo est\u00e1n. \n-- \n-- Definir la funci\u00f3n \n--    normal :: Expr -> Expr\n-- tal que (normal e) es la forma normal de la expresi\u00f3n e obtenida\n-- aplicando, mientras que sea posible, las propiedades distributivas:\n--    (a+b)*c = a*c+b*c\n--    c*(a+b) = c*a+c*b\n-- Por ejemplo,\n--    ghci> normal (P (S (V \"x\") (V \"y\")) (V \"z\"))\n--    S (P (V \"x\") (V \"z\")) (P (V \"y\") (V \"z\"))\n--    ghci> normal (P (V \"z\") (S (V \"x\") (V \"y\")))\n--    S (P (V \"z\") (V \"x\")) (P (V \"z\") (V \"y\"))\n--    ghci> normal (P (S (V \"x\") (V \"y\")) (S (V \"u\") (V \"v\")))\n--    S (S (P (V \"x\") (V \"u\")) (P (V \"x\") (V \"v\"))) \n--      (S (P (V \"y\") (V \"u\")) (P (V \"y\") (V \"v\")))\n--    ghci> normal (S (P (V \"x\") (V \"y\")) (V \"z\"))\n--    S (P (V \"x\") (V \"y\")) (V \"z\")\n--    ghci> normal (V \"x\")\n--    V \"x\"\n--\n-- Comprobar con QuickCheck (usando la funci\u00f3n esNormal del ejercicio\n-- del 2 de diciembre) que para cualquier expresi\u00f3n e, (normal e) est\u00e1 en\n-- forma normal y que (normal (normal e)) es igual a (normal e).\n--\n-- Nota. Para la comprobaci\u00f3n se usar\u00e1 el siguiente generador de\n-- expresiones aritm\u00e9ticas:\n--    instance Arbitrary Expr where\n--        arbitrary = sized arb \n--            where\n--              arb 0          =  liftM V arbitrary\n--              arb n | n > 0  =  oneof [liftM V arbitrary,\n--                                       liftM2 S sub sub, \n--                                       liftM2 P sub sub] \n--                    where sub = arb (n `div` 2)\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\nimport Control.Monad\n\ndata Expr = V String\n          | S Expr Expr\n          | P Expr Expr\n          deriving (Eq, Show)\n\nesTermino :: Expr -> Bool\nesTermino (V _)   = True\nesTermino (S _ _) = False\nesTermino (P a b) = esTermino a && esTermino b\n\nesNormal :: Expr -> Bool\nesNormal (S a b) = esNormal a && esNormal b\nesNormal a       = esTermino a\n\nnormal :: Expr -> Expr\nnormal (V v)   = V v\nnormal (S a b) = S (normal a) (normal b)\nnormal (P a b) = p (normal a) (normal b)\n    where p (S a b) c = S (p a c) (p b c)\n          p a (S b c) = S (p a b) (p a c)\n          p a b       = P a b\n\nprop_normal :: Expr -> Bool\nprop_normal e = \n    esNormal (normal e) && normal (normal e) == normal e\n\n-- La comprobaci\u00f3n es\n--    ghci> quickCheck prop_normal\n--    +++ OK, passed 100 tests.\n\ninstance Arbitrary Expr where\n    arbitrary = sized arb \n        where\n          arb 0          =  liftM V arbitrary\n          arb n | n > 0  =  oneof [liftM V arbitrary,\n                                   liftM2 S sub sub, \n                                   liftM2 P sub sub] \n                where sub = arb (n `div` 2)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Enunciado &#8212; El siguiente tipo de dato representa expresiones construidas con &#8212; variables, sumas y productos &#8212; data Expr = V String &#8212; | S Expr Expr &#8212; | P Expr Expr &#8212; deriving (Eq, Show) &#8212; Por ejemplo, x*(y+z) se representa por (P (V \u00abx\u00bb) (S (V \u00aby\u00bb) (V \u00abz\u00bb))) &#8212; &#8212; Una expresi\u00f3n&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[7],"tags":[6,146,133],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/794"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=794"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/794\/revisions"}],"predecessor-version":[{"id":876,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/794\/revisions\/876"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}