{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Formulas & Automata generators\n",
    "\n",
    "The `spot.gen` package contains the functions used to generate the patterns produced by [`genltl`](https://spot.lrde.epita.fr/genltl.html) and [`genaut`](https://spot.lrde.epita.fr/genaut.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import spot\n",
    "import spot.gen as sg\n",
    "spot.setup()\n",
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LTL patterns\n",
    "\n",
    "Generation of LTL formulas is done via the `ltl_pattern()` function.  This takes two arguments: a pattern id, and a pattern size (or index if the id refers to a list)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} p_{1} \\land \\mathsf{G} \\mathsf{F} p_{2} \\land \\mathsf{G} \\mathsf{F} p_{3}$"
      ],
      "text/plain": [
       "spot.formula(\"GFp1 & GFp2 & GFp3\")"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.ltl_pattern(sg.LTL_AND_GF, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (p \\land \\mathsf{X} p \\land \\mathsf{X} \\mathsf{X} p \\land \\mathsf{X} \\mathsf{X} \\mathsf{X} p) \\land \\mathsf{F} (q \\land \\mathsf{X} q \\land \\mathsf{X} \\mathsf{X} q \\land \\mathsf{X} \\mathsf{X} \\mathsf{X} q)$"
      ],
      "text/plain": [
       "spot.formula(\"F(p & Xp & XXp & XXXp) & F(q & Xq & XXq & XXXq)\")"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg.ltl_pattern(sg.LTL_CCJ_BETA_PRIME, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To see the list of supported patterns, the easiest way is to look at the `--help` output of `genltl`.  The above pattern for instance is attached to option `--ccj-beta-prime`.  The name of the pattern identifier is the same using capital letters, underscores, and an `LTL_` prefix.  If a pattern has multiple aliased options in `genltl`, the first one used for the identifier (e.g., `genltl` accept both `--dac-patterns` and `--spec-patterns` as synonyms to denote the patterns of `spot.gen.LTL_DAC_PATTERNS`).\n",
    "\n",
    "Multiple patterns can be generated using the `ltl_patterns()` function.  It's arguments should be either can be:\n",
    "  - pairs of the form `(id, n)`: in this case the pattern `id` with size/index `n` is returned,\n",
    "  - triplets of the form `(id, min, max)`: in this case the patterns are output for all `n` between `min` and `max` included,\n",
    "  - an integer `id`: then this is equivalent to `(id, 1, 10)` if the pattern has now upper bound, or `(id, 1, upper)` if the patter `id` has an upper bound `upper`.  This is mostly used when the pattern id correspond to a hard-coded list of formulas.\n",
    "\n",
    "Here is an example showing these three types of arguments:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\mathsf{G} \\mathsf{F} p_{1} \\lor \\mathsf{F} \\mathsf{G} p_{2}) \\land (\\mathsf{G} \\mathsf{F} p_{2} \\lor \\mathsf{F} \\mathsf{G} p_{3}) \\land (\\mathsf{G} \\mathsf{F} p_{3} \\lor \\mathsf{F} \\mathsf{G} p_{4})$"
      ],
      "text/plain": [
       "spot.formula(\"(GFp1 | FGp2) & (GFp2 | FGp3) & (GFp3 | FGp4)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} \\mathsf{G} p_{1}$"
      ],
      "text/plain": [
       "spot.formula(\"FGp1\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} \\mathsf{G} p_{1} \\land \\mathsf{F} \\mathsf{G} p_{2}$"
      ],
      "text/plain": [
       "spot.formula(\"FGp1 & FGp2\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} \\mathsf{G} p_{1} \\land \\mathsf{F} \\mathsf{G} p_{2} \\land \\mathsf{F} \\mathsf{G} p_{3}$"
      ],
      "text/plain": [
       "spot.formula(\"FGp1 & FGp2 & FGp3\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$p_{0} \\mathbin{\\mathsf{U}} (p_{1} \\land \\mathsf{G} p_{2})$"
      ],
      "text/plain": [
       "spot.formula(\"p0 U (p1 & Gp2)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$p_{0} \\mathbin{\\mathsf{U}} (p_{1} \\land \\mathsf{X} (p_{2} \\mathbin{\\mathsf{U}} p_{3}))$"
      ],
      "text/plain": [
       "spot.formula(\"p0 U (p1 & X(p2 U p3))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$p_{0} \\mathbin{\\mathsf{U}} (p_{1} \\land \\mathsf{X} (p_{2} \\land \\mathsf{F} (p_{3} \\land \\mathsf{X} \\mathsf{F} (p_{4} \\land \\mathsf{X} \\mathsf{F} (p_{5} \\land \\mathsf{X} \\mathsf{F} p_{6})))))$"
      ],
      "text/plain": [
       "spot.formula(\"p0 U (p1 & X(p2 & F(p3 & XF(p4 & XF(p5 & XFp6)))))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (p_{0} \\land \\mathsf{X} \\mathsf{G} p_{1})$"
      ],
      "text/plain": [
       "spot.formula(\"F(p0 & XGp1)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (p_{0} \\land \\mathsf{X} (p_{1} \\land \\mathsf{X} \\mathsf{F} p_{2}))$"
      ],
      "text/plain": [
       "spot.formula(\"F(p0 & X(p1 & XFp2))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (p_{0} \\land \\mathsf{X} (p_{1} \\mathbin{\\mathsf{U}} p_{2}))$"
      ],
      "text/plain": [
       "spot.formula(\"F(p0 & X(p1 U p2))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} p_{0} \\lor \\mathsf{F} \\mathsf{G} p_{1}$"
      ],
      "text/plain": [
       "spot.formula(\"GFp0 | FGp1\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} (p_{0} \\rightarrow (p_{1} \\mathbin{\\mathsf{U}} p_{2}))$"
      ],
      "text/plain": [
       "spot.formula(\"G(p0 -> (p1 U p2))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} (p_{0} \\land \\mathsf{X} \\mathsf{F} (p_{1} \\land \\mathsf{X} \\mathsf{F} (p_{2} \\land \\mathsf{X} \\mathsf{F} p_{3})))$"
      ],
      "text/plain": [
       "spot.formula(\"G(p0 & XF(p1 & XF(p2 & XFp3)))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} p_{1} \\land \\mathsf{G} \\mathsf{F} p_{2} \\land \\mathsf{G} \\mathsf{F} p_{3} \\land \\mathsf{G} \\mathsf{F} p_{0} \\land \\mathsf{G} \\mathsf{F} p_{4}$"
      ],
      "text/plain": [
       "spot.formula(\"GFp1 & GFp2 & GFp3 & GFp0 & GFp4\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$(p_{0} \\mathbin{\\mathsf{U}} (p_{1} \\mathbin{\\mathsf{U}} p_{2})) \\lor (p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{U}} p_{0})) \\lor (p_{2} \\mathbin{\\mathsf{U}} (p_{0} \\mathbin{\\mathsf{U}} p_{1}))$"
      ],
      "text/plain": [
       "spot.formula(\"(p0 U (p1 U p2)) | (p1 U (p2 U p0)) | (p2 U (p0 U p1))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} (p_{0} \\rightarrow (p_{1} \\mathbin{\\mathsf{U}} (\\mathsf{G} p_{2} \\lor \\mathsf{G} p_{3})))$"
      ],
      "text/plain": [
       "spot.formula(\"G(p0 -> (p1 U (Gp2 | Gp3)))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for f in sg.ltl_patterns((sg.LTL_GH_R, 3), (sg.LTL_AND_FG, 1, 3), sg.LTL_EH_PATTERNS):\n",
    "    display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some LTL patterns take two arguments.  In this case, the arguments passed to `ltl_pattern()` should be one of:\n",
    "- `(id, n, m)` giving the value of both arguments,\n",
    "- `(id, min1, max1, min2, max2)` specifying a range for both arguments,\n",
    "- `(id, n)` equivalent to `(id, n, n)`,\n",
    "- `id` equivalent to `(id, 1, 3, 1, 3)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} b)$"
      ],
      "text/plain": [
       "spot.formula(\"GFa1 U G(GFa0 U Xb)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b)$"
      ],
      "text/plain": [
       "spot.formula(\"GFa1 U G(GFa0 U XXb)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} \\mathsf{X} b)$"
      ],
      "text/plain": [
       "spot.formula(\"GFa1 U G(GFa0 U XXXb)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U Xb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U XXb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U XXXb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{3} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} b)))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa3 U G(GFa2 U G(GFa1 U G(GFa0 U Xb)))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{3} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b)))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa3 U G(GFa2 U G(GFa1 U G(GFa0 U XXb)))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{3} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} \\mathsf{X} b)))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa3 U G(GFa2 U G(GFa1 U G(GFa0 U XXXb)))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for f in sg.ltl_patterns(sg.LTL_SEJK_F):\n",
    "    display(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U XXXb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U XXb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} b)$"
      ],
      "text/plain": [
       "spot.formula(\"GFa1 U G(GFa0 U Xb)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b)$"
      ],
      "text/plain": [
       "spot.formula(\"GFa1 U G(GFa0 U XXb)\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U Xb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathsf{G} \\mathsf{F} a_{2} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{1} \\mathbin{\\mathsf{U}} \\mathsf{G} (\\mathsf{G} \\mathsf{F} a_{0} \\mathbin{\\mathsf{U}} \\mathsf{X} \\mathsf{X} b))$"
      ],
      "text/plain": [
       "spot.formula(\"GFa2 U G(GFa1 U G(GFa0 U XXb))\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for f in sg.ltl_patterns((sg.LTL_SEJK_F, 2, 3), (sg.LTL_SEJK_F, 2), (sg.LTL_SEJK_F, 1, 2, 1, 2)):\n",
    "    display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Automata patterns\n",
    "\n",
    "We currently have only a couple of generators of automata:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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     "metadata": {},
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    }
   ],
   "source": [
    "display(sg.aut_pattern(sg.AUT_KS_NCA, 3).show('.a'),\n",
    "        sg.aut_pattern(sg.AUT_L_DSA, 3).show('.a'),\n",
    "        sg.aut_pattern(sg.AUT_L_NBA, 3).show('.a'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiple automata can be generated using the `aut_patterns()` function, which works similarly to `ltl_patterns()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "5\n",
      "7\n",
      "9\n",
      "11\n",
      "13\n",
      "15\n",
      "17\n",
      "19\n",
      "21\n"
     ]
    }
   ],
   "source": [
    "for aut in sg.aut_patterns(sg.AUT_KS_NCA):\n",
    "    print(aut.num_states())"
   ]
  }
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