{"id":3266,"date":"2017-04-26T06:00:31","date_gmt":"2017-04-26T04:00:31","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3266"},"modified":"2017-05-10T07:32:41","modified_gmt":"2017-05-10T05:32:41","slug":"normalizacion-de-expresiones-aritmeticas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/normalizacion-de-expresiones-aritmeticas\/","title":{"rendered":"Normalizaci\u00f3n de expresiones aritm\u00e9ticas"},"content":{"rendered":"

El siguiente tipo de dato representa expresiones construidas con variables, sumas y productos<\/p>\n

\n   data Expr = Var String\n             | S Expr Expr\n             | P Expr Expre\n             deriving (Eq, Show)\n<\/pre>\n
Por ejemplo, x*(y+z)<\/code> se representa por (P (V \"x\") (S (V \"y\") (V \"z\")))<\/code><\/p>\n
Una expresi\u00f3n es un t\u00e9rmino si es un producto de variables. Por ejemplo, x*(y*z)<\/code> es un t\u00e9rmino pero x+(y*z)<\/code> ni x*(y+z)<\/code> lo son.<\/p>\n
Una expresi\u00f3n est\u00e1 en forma normal si es una suma de t\u00e9rminos. Por ejemplo, x*(y*z)<\/code> y x+(y*z)<\/code> est\u00e1n en forma normal; pero x*(y+z)<\/code> y (x+y)*(x+z)<\/code> no lo est\u00e1n.<\/p>\n
Definir la funci\u00f3n<\/p>\n
\n   esTermino :: Expr -> Bool\n   esTermino :: Expr -> Bool\n   normal    :: Expr -> Expr\n<\/pre>\n
tales que<\/p>\n
    \n
  • (esTermino a) se verifica si a es un t\u00e9rmino. Por ejemplo,<\/li>\n<\/ul>\n
    \n     esTermino (V \"x\")                         == True\n     esTermino (P (V \"x\") (P (V \"y\") (V \"z\"))) == True\n     esTermino (P (V \"x\") (S (V \"y\") (V \"z\"))) == False\n     esTermino (S (V \"x\") (P (V \"y\") (V \"z\"))) == False\n<\/pre>\n
      \n
    • (esNormal a) se verifica si a est\u00e1 en forma normal. Por ejemplo,<\/li>\n<\/ul>\n
      \n     esNormal (V \"x\")                                     == True\n     esNormal (P (V \"x\") (P (V \"y\") (V \"z\")))             == True\n     esNormal (S (V \"x\") (P (V \"y\") (V \"z\")))             == True\n     esNormal (P (V \"x\") (S (V \"y\") (V \"z\")))             == False\n     esNormal (P (S (V \"x\") (V \"y\")) (S (V \"y\") (V \"z\"))) == False\n     esNormal (S (P (V \"x\") (V \"y\")) (S (V \"z\") (V \"x\"))) == True\n<\/pre>\n
        \n
      • (normal e) es la forma normal de la expresi\u00f3n e obtenida aplicando, mientras que sea posible, las propiedades distributivas:<\/li>\n<\/ul>\n
        \n     (a+b)*c = a*c+b*c\n     c*(a+b) = c*a+c*b\n<\/pre>\n
        Por ejemplo,<\/p>\n
        \n     \u03bb> normal (P (S (V \"x\") (V \"y\")) (V \"z\"))\n     S (P (V \"x\") (V \"z\")) (P (V \"y\") (V \"z\"))\n     \u03bb> normal (P (V \"z\") (S (V \"x\") (V \"y\")))\n     S (P (V \"z\") (V \"x\")) (P (V \"z\") (V \"y\"))\n     \u03bb> 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     \u03bb> normal (S (P (V \"x\") (V \"y\")) (V \"z\"))\n     S (P (V \"x\") (V \"y\")) (V \"z\")\n     \u03bb> normal (V \"x\")\n     V \"x\"\n<\/pre>\n
        Comprobar con QuickCheck que para cualquier expresi\u00f3n e, (normal e) est\u00e1 en forma normal y que (normal (normal e)) es igual a (normal e).<\/p>\n
        Nota<\/strong>. Para la comprobaci\u00f3n se usar\u00e1 el siguiente generador de expresiones aritm\u00e9ticas<\/p>\n
        \n   import Test.QuickCheck\n   import Control.Monad\n   \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
        Soluciones<\/h4>\n
        \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)\n  && normal (normal e) == normal e\n\n-- La comprobaci\u00f3n es\n--    \u03bb> 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":"
        El siguiente tipo de dato representa expresiones construidas con variables, sumas y productos data Expr = Var String | S Expr Expr | P Expr Expre deriving (Eq, Show) Por ejemplo, x*(y+z) se representa por (P (V \u00abx\u00bb) (S (V \u00aby\u00bb) (V \u00abz\u00bb))) Una expresi\u00f3n es un t\u00e9rmino si es un producto de variables. Por…<\/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\/3266"}],"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=3266"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3266\/revisions"}],"predecessor-version":[{"id":3294,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3266\/revisions\/3294"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3266"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3266"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3266"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}