L05.ipynb

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    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "from liblet import Derivation, Grammar, animate_derivation, ProductionGraph, side_by_side, Tree"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Dalle derivazioni agli alberi di parsing"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "def derivation_to_parsetree(d):\n",
        "\n",
        "    # questa variabile si riferià all'albero di derivazione di d\n",
        "    # inizialmente contiene l'albero annotato col simbolo di partenza\n",
        "    tree = Tree({'Symbol': d.G.S, 'nprod': None})\n",
        "\n",
        "    # all'inizio la forma sentenziale è data da tale albero\n",
        "    frontier = [tree, ] \n",
        "\n",
        "    for nprod, pos in d.steps():\n",
        "\n",
        "        # i figli sono dati dal lato destro d.G.P[prod].rhs\n",
        "        children = [Tree({'Symbol': X, 'nprod': None}) for X in d.G.P[nprod].rhs]\n",
        "\n",
        "        # l'albero è dato dalla posizione in cui è applicata la produzione\n",
        "        curr = frontier[pos]\n",
        "\n",
        "        # si aggiornano l'albero e la forma sentenziale\n",
        "        curr.root['nprod'] = nprod\n",
        "        curr.children = children\n",
        "        frontier = frontier[:pos] + children + frontier[pos + 1:]\n",
        "\n",
        "    return tree"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Le ε-produzioni\n",
        "\n",
        "In alcuni casi sono comode produzioni della forma $A\\to \\epsilon$ (chiamate ε-produzioni). Nella costruzione dell'albero dalla derivazione richiedono però un po' di attenzione…\n",
        "\n",
        "Consideriamo ad esempio il caso del [linguaggio di Dyck](https://en.wikipedia.org/wiki/Dyck_language)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
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      "source": [
        "G_dyck = Grammar.from_string('S -> S S | ( S ) | ε')"
      ]
    },
    {
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              "S -> S S -> ( S ) S -> ( ( S ) ) S -> ( ( ) ) S -> ( ( ) )"
            ]
          },
          "execution_count": 4,
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      ],
      "source": [
        "# una derivazione leftmost di '(())'\n",
        "\n",
        "dyck = 0, 1, 1, 2, 2\n",
        "\n",
        "deriv = Derivation(G_dyck).leftmost(dyck)\n",
        "deriv"
      ]
    },
    {
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              "'(())'"
            ]
          },
          "execution_count": 5,
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        }
      ],
      "source": [
        "# la parola \n",
        "''.join(deriv.sentential_form())"
      ]
    },
    {
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              "ProductionGraph(S -> S S -> ( S ) S -> ( ( S ) ) S -> ( ( ) ) S -> ( ( ) ))"
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      "source": [
        "# l'albero \n",
        "\n",
        "ProductionGraph(deriv)"
      ]
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      "source": [
        "Applicando la funzione `derivation_to_parsetree` si nota però un errore: sembra che da `)` derivi ε!"
      ]
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      "source": [
        "derivation_to_parsetree(deriv) "
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    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Si lascia come esercizio correggere la funzione `derivation_to_parsetree` in modo che funzioni correttamente anche nel caso di ε-produzioni."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Ambiguità\n",
        "\n",
        "\n",
        "## È sempre rilevante? \n",
        "\n",
        "Pensiamo a grammatiche della forma `E -> E op E | n` dove `op` sia un operatore binario, definiamo una funzione ricorsiva che, dato un albero di parsing e il valore di `n`, valuti l'espressione aritmetica corrispondente."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "from operator import add, sub, mul\n",
        "\n",
        "DT = {\n",
        "  '+': add,\n",
        "  '-': sub,\n",
        "  '*': mul\n",
        "}\n",
        "\n",
        "def eval_tree(tree, n):\n",
        "  root, *children = tree\n",
        "  if root['Symbol'] == 'n': return n\n",
        "  if root['Symbol'] == 'E' and len(children) == 3:\n",
        "    left, op, right = children\n",
        "    return DT[op.root['Symbol']](eval_tree(left, n), eval_tree(right, n))\n",
        "  else: return eval_tree(children[0], n)"
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      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Il caso \"spurio\""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Costruzione di una derivazione leftomst\n",
        "\n",
        "def lmd(G, leftmost_steps):\n",
        "  return Derivation(G).leftmost(leftmost_steps)"
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      "execution_count": null,
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      "source": [
        "G_as = Grammar.from_string('E -> E + E | n')\n",
        "\n",
        "# due derivazioni di n + n + n\n",
        "\n",
        "lmd0, lmd1 = lmd(G_as, [0, 1, 0, 1, 1]), lmd(G_as, [0, 0, 1, 1, 1])\n",
        "\n",
        "side_by_side(ProductionGraph(lmd0), ProductionGraph(lmd1))"
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    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(3, 3)"
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          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
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      ],
      "source": [
        "# 1 + 1 + 1, comunque parsato, vale 3\n",
        "\n",
        "eval_tree(derivation_to_parsetree(lmd0), 1), eval_tree(derivation_to_parsetree(lmd1), 1)"
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    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### I casi \"essenziali\"\n",
        "\n",
        "Iniziamo dall'associatività/commutatività"
      ]
    },
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      "cell_type": "code",
      "execution_count": null,
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          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "G_ae = Grammar.from_string('E -> E - E | n')\n",
        "\n",
        "# due derivazioni di n - n - n\n",
        "\n",
        "lmd0, lmd1 = lmd(G_ae, [0, 1, 0, 1, 1]), lmd(G_ae, [0, 0, 1, 1, 1])\n",
        "\n",
        "side_by_side(ProductionGraph(lmd0), ProductionGraph(lmd1))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(1, -1)"
            ]
          },
          "execution_count": 13,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# 1 - (1 - 1) e (1 - 1) - 1 hanno valori diversi!\n",
        "\n",
        "eval_tree(derivation_to_parsetree(lmd0), 1), eval_tree(derivation_to_parsetree(lmd1), 1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Consideriamo la *precedenza* "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
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          "execution_count": 14,
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      ],
      "source": [
        "G_perr = Grammar.from_string(\"\"\"\n",
        "E -> E + E | E * E | n\n",
        "\"\"\")\n",
        "\n",
        "# n + n * n\n",
        "\n",
        "lmd0, lmd1 = lmd(G_perr, [1, 0, 2, 2, 2]), lmd(G_perr, [0, 2, 1, 2, 2])\n",
        "\n",
        "side_by_side(ProductionGraph(lmd0), ProductionGraph(lmd1))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(8, 6)"
            ]
          },
          "execution_count": 15,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# ((2 + 2) * 2) e 2 + (2 * 2) hanno valori diversi!\n",
        "\n",
        "eval_tree(derivation_to_parsetree(lmd0), 2), eval_tree(derivation_to_parsetree(lmd1), 2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Per finire, un esempio non collegato alle *espressioni*, ma al *dangling else*"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "G_iferr = Grammar.from_string(\"\"\"\n",
        "S -> if C then S | if C then S else S | stm\n",
        "C -> cond\n",
        "\"\"\")"
      ]
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    {
      "cell_type": "code",
      "execution_count": null,
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      "source": [
        "# if cond the if cont then stm else stm\n",
        "\n",
        "lmd0, lmd1 = lmd(G_iferr, [0, 3, 1, 3, 2, 2]), lmd(G_iferr, [1, 3, 0, 3, 2, 2])\n",
        "\n",
        "side_by_side(ProductionGraph(lmd0), ProductionGraph(lmd1))"
      ]
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    {
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      "metadata": {},
      "source": [
        "## Possibili soluzioni\n",
        "\n",
        "Un simbolo in più, ricorsione *sinistra* o *destra* e relazione con l'*associatività*. Iniziamo con l'associatibità a sinistra."
      ]
    },
    {
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      "execution_count": null,
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        {
          "data": {
            "text/plain": [
              "Grammar(N={E, E'}, T={+, n}, P=(E -> E + E', E -> E', E' -> n), S=E)"
            ]
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          "execution_count": 18,
          "metadata": {},
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      ],
      "source": [
        "G_la = Grammar.from_string(\"\"\"\n",
        "E -> E + E' | E'\n",
        "E' -> n \n",
        "\"\"\")\n",
        "G_la"
      ]
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      "cell_type": "code",
      "execution_count": null,
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            "text/plain": [
              "ProductionGraph(E -> E + E' -> E + E' + E' -> E' + E' + E' -> n + E' + E' -> n + n + E' -> n + n + n)"
            ]
          },
          "execution_count": 19,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "ProductionGraph(lmd(G_la, [0, 0, 1, 2, 2, 2]))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "E quindi consideriamo quella a destra."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "Grammar(N={E, E'}, T={^, n}, P=(E -> E' ^ E, E -> E', E' -> n), S=E)"
            ]
          },
          "execution_count": 20,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "G_ra = Grammar.from_string(\"\"\"\n",
        "E -> E' ^ E | E'\n",
        "E' -> n \n",
        "\"\"\")\n",
        "G_ra"
      ]
    },
    {
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      "execution_count": null,
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              "ProductionGraph(E -> E' ^ E -> n ^ E -> n ^ E' ^ E -> n ^ n ^ E -> n ^ n ^ E' -> n ^ n ^ n)"
            ]
          },
          "execution_count": 21,
          "metadata": {},
          "output_type": "execute_result"
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      ],
      "source": [
        "ProductionGraph(lmd(G_ra, [0, 2, 0, 2, 1, 2]))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Per la *precedenza* occorre aggiungere più di un simbolo…"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "G_p = Grammar.from_string(\"\"\"\n",
        "E -> E + P | P\n",
        "P -> P * F | F\n",
        "F -> n\n",
        "\"\"\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
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              "ProductionGraph(E -> E + P -> P + P -> F + P -> n + P -> n + P * F -> n + F * F -> n + n * F -> n + n * n)"
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          "execution_count": 23,
          "metadata": {},
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        "ProductionGraph(lmd(G_p, [0, 1, 3, 4, 2, 3, 4, 4]))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Il caso del *dangling else*"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "G_if = Grammar.from_string(\"\"\"\n",
        "S -> M | U\n",
        "M -> if C then M else M | stm\n",
        "U -> if C then M else U | if C then S\n",
        "C -> cond\n",
        "\"\"\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [
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            "text/plain": [
              "ProductionGraph(S -> U -> if C then S -> if cond then S -> if cond then M -> if cond then if C then M else M -> if cond then if cond then M else M -> if cond then if cond then stm else M -> if cond then if cond then stm else stm)"
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          },
          "execution_count": 25,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "ProductionGraph(lmd(G_if, [1, 5, 6, 0, 2, 6, 3, 3]))"
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    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Parsing\n",
        "\n",
        "## Top down"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "w = 'aabbcc'\n",
        "\n",
        "G = Grammar.from_string(\"\"\"\n",
        "S -> a S Q | a b c\n",
        "b Q c -> b b c c  \n",
        "c Q -> Q c\n",
        "\"\"\", False)"
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        "## Bottom up\n",
        "\n",
        "Nel caso *bottom up* si può procedere con una grammatica ottenuta rovesciando le produzioni (e aggiungendo una produzione `Inizio`$\\to \\omega$ dove `Inizio` è il nuovo simbolo di partenza (e volendo una produzione $S\\to $`Fine` dove `Fine` è un simbolo usato per indicare la fine della derivazione)."
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