{"id":8031,"date":"2023-04-14T06:00:44","date_gmt":"2023-04-14T04:00:44","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8031"},"modified":"2023-04-14T08:13:15","modified_gmt":"2023-04-14T06:13:15","slug":"14-abr-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/14-abr-23\/","title":{"rendered":"Clausura transitiva"},"content":{"rendered":"<p>Usando el <a href=\"https:\/\/bit.ly\/3IVVqOT\">tipo de las relaciones binarias<\/a>, definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   clausuraTransitiva :: Eq a => Rel a -> Rel a\n<\/pre>\n<p>tal que <code>clausuraTransitiva r<\/code> es la clausura transitiva de <code>r<\/code>; es decir, la menor relaci\u00f3n transitiva que contiene a r. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> clausuraTransitiva (R ([1..6],[(1,2),(2,5),(5,6)]))\n   R ([1,2,3,4,5,6],[(1,2),(2,5),(5,6),(1,5),(2,6),(1,6)])\n<\/pre>\n<p>Comprobar con QuickCheck que clausuraTransitiva es transitiva.<\/p>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nimport Relaciones_binarias (Rel(R))\nimport Relaciones_transitivas (transitiva)\nimport Data.List (union)\nimport Test.QuickCheck\n\n--  1\u00aa soluci\u00f3n\n--  ===========\n\nclausuraTransitiva :: Ord a => Rel a -> Rel a\nclausuraTransitiva (R (u,g)) = R (u, aux g)\n  where\n    aux u' | cerradoTr u' = u'\n           | otherwise    = aux (u' `union` comp u' u')\n    cerradoTr r       = subconjunto (comp r r) r\n    comp r s          = [(x,z) | (x,y) <- r, (y',z) <- s, y == y']\n    subconjunto xs ys = all (`elem` ys) xs\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nclausuraTransitiva2 :: Ord a => Rel a -> Rel a\nclausuraTransitiva2 (R (u,g)) =\n  R (u, until cerradoTr (\\r -> r `union` comp r r) g)\n  where\n    cerradoTr r       = subconjunto (comp r r) r\n    comp r s          = [(x,z) | (x,y) <- r, (y',z) <- s, y == y']\n    subconjunto xs ys = all (`elem` ys) xs\n\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_clausuraTransitiva :: Rel Int -> Bool\nprop_clausuraTransitiva r =\n  clausuraTransitiva r == clausuraTransitiva2 r\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_clausuraTransitiva\n--    +++ OK, passed 100 tests.\n\n-- Propiedad\n-- =========\n\n-- La propiedad es\nprop_clausuraTransitivaEsTransitiva :: Rel Int -> Bool\nprop_clausuraTransitivaEsTransitiva r =\n  transitiva (clausuraTransitiva r)\n\n-- La funci\u00f3n transitiva est\u00e1 definida en el ejercicio\n-- \"Relaciones transitivas\" que se encuentra en\n-- https:\/\/bit.ly\/42WRPJv\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_clausuraTransitivaEsTransitiva\n--    +++ OK, passed 100 tests.\n<\/pre>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom typing import TypeVar\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nfrom src.Relaciones_binarias import Rel, relacionArbitraria\nfrom src.Relaciones_transitivas import transitiva\n\nA = TypeVar('A')\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef clausuraTransitiva(r: Rel[A]) -> Rel[A]:\n    (u, g) = r\n\n    def subconjunto(xs: list[tuple[A, A]], ys: list[tuple[A, A]]) -> bool:\n        return set(xs) <= set(ys)\n\n    def comp(r: list[tuple[A, A]], s: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        return list({(x, z) for (x, y) in r for (y1, z) in s if y == y1})\n\n    def cerradoTr(r: list[tuple[A, A]]) -> bool:\n        return subconjunto(comp(r, r), r)\n\n    def union(xs: list[tuple[A, A]], ys: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        return xs + [y for y in ys if y not in xs]\n\n    def aux(u1: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        if cerradoTr(u1):\n            return u1\n        return aux(union(u1, comp(u1, u1)))\n\n    return (u, aux(g))\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef clausuraTransitiva2(r: Rel[A]) -> Rel[A]:\n    (u, g) = r\n\n    def subconjunto(xs: list[tuple[A, A]], ys: list[tuple[A, A]]) -> bool:\n        return set(xs) <= set(ys)\n\n    def comp(r: list[tuple[A, A]], s: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        return list({(x, z) for (x, y) in r for (y1, z) in s if y == y1})\n\n    def cerradoTr(r: list[tuple[A, A]]) -> bool:\n        return subconjunto(comp(r, r), r)\n\n    def union(xs: list[tuple[A, A]], ys: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        return xs + [y for y in ys if y not in xs]\n\n    def aux(u1: list[tuple[A, A]]) -> list[tuple[A, A]]:\n        if cerradoTr(u1):\n            return u1\n        return aux(union(u1, comp(u1, u1)))\n\n    g1: list[tuple[A, A]] = g\n    while not cerradoTr(g1):\n        g1 = union(g1, comp(g1, g1))\n    return (u, g1)\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=0, max_value=10))\ndef test_clausuraTransitiva(n: int) -> None:\n    r = relacionArbitraria(n)\n    assert clausuraTransitiva(r) == clausuraTransitiva2(r)\n\n# Propiedad\n# =========\n\n# La propiedad es\n@given(st.integers(min_value=0, max_value=10))\ndef test_cla(n: int) -> None:\n    r = relacionArbitraria(n)\n    assert transitiva(clausuraTransitiva(r))\n\n# La funci\u00f3n transitiva est\u00e1 definida en el ejercicio\n# \"Relaciones transitivas\" que se encuentra en\n# https:\/\/bit.ly\/42WRPJv\n\n# La comprobaci\u00f3n es\n#    > poetry run pytest -q Clausura_transitiva.py\n#    2 passed in 0.16s\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Usando el tipo de las relaciones binarias, definir la funci\u00f3n clausuraTransitiva :: Eq a => Rel a -> Rel a tal que clausuraTransitiva r es la clausura transitiva de r; es decir, la menor relaci\u00f3n transitiva que contiene a r. Por ejemplo, \u03bb> clausuraTransitiva (R ([1..6],[(1,2),(2,5),(5,6)])) R ([1,2,3,4,5,6],[(1,2),(2,5),(5,6),(1,5),(2,6),(1,6)]) Comprobar con QuickCheck que clausuraTransitiva es transitiva&#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":[581],"tags":[576],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8031"}],"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=8031"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8031\/revisions"}],"predecessor-version":[{"id":8050,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8031\/revisions\/8050"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}