{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Handling LTL and PSL formulas\n",
    "============================="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import spot\n",
    "from IPython.display import display # not needed with recent Jupyter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For interactive use, formulas can be entered as text strings and passed to the `spot.formula` constructor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))$"
      ],
      "text/plain": [
       "spot.formula(\"p1 U (p2 R (p3 & !p4))\")"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = spot.formula('p1 U p2 R (p3 & !p4)')\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\{a \\mathbin{\\mathsf{;}} \\mathsf{first\\_match}(b^{\\star} \\mathbin{\\mathsf{;}} c^+ \\mathbin{\\mathsf{;}} d)\\}\\mathrel{\\Diamond\\kern-1.7pt\\raise.4pt\\hbox{$\\mathord{\\rightarrow}$}} (c \\land \\mathsf{G} \\mathsf{F} b)$"
      ],
      "text/plain": [
       "spot.formula(\"{a;first_match(b[*];c[+];d)}<>-> (c & GFb)\")"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = spot.formula('{a;first_match(b*;c[+];d)}<>->(GFb & c)'); g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default the parser recognizes an infix syntax, but when this fails, it tries to read the formula with the [LBT](http://www.tcs.hut.fi/Software/maria/tools/lbt/) syntax:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$c \\land (a \\lor b)$"
      ],
      "text/plain": [
       "spot.formula(\"c & (a | b)\")"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = spot.formula('& | a b c'); h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Passing a `formula` to `spot.formula` simply returns the formula."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$c \\land (a \\lor b)$"
      ],
      "text/plain": [
       "spot.formula(\"c & (a | b)\")"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.formula(h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, a formula object is presented using mathjax as above.\n",
    "When a formula is converted to string you get Spot's syntax by default:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'p1 U (p2 R (p3 & !p4))'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "str(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you prefer to print the string in another syntax, you may use the `to_str()` method, with an argument that indicates the output format to use.  The `latex` format assumes that you will the define macros such as `\\U`, `\\R` to render all operators as you wish.  On the otherhand, the `sclatex` (with `sc` for self-contained) format hard-codes the rendering of each of those operators: this is almost the same output that is used to render formulas using MathJax in a notebook. `sclatex` and `mathjax` only differ in the rendering of double-quoted atomic propositions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "spot      p1 U (p2 R (p3 & !p4))\n",
      "spin      p1 U (p2 V (p3 && !p4))\n",
      "lbt       U p1 V p2 & p3 ! p4\n",
      "wring     (p1=1) U ((p2=1) R ((p3=1) * (p4=0)))\n",
      "utf8      p1 U (p2 R (p3∧¬p4))\n",
      "latex     p_{1} \\U (p_{2} \\R (p_{3} \\land \\lnot p_{4}))\n",
      "sclatex   p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))\n",
      "mathjax   p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))\n"
     ]
    }
   ],
   "source": [
    "for i in ['spot', 'spin', 'lbt', 'wring', 'utf8', 'latex', 'sclatex', 'mathjax']:\n",
    "    print(\"%-10s%s\" % (i, f.to_str(i)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Formulas output via `format()` can also use some convenient shorthand to select the syntax:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Spin:               p1 U (p2 V (p3 && !p4))\n",
      "Spin+parentheses:   (p1) U ((p2) V ((p3) && (!(p4))))\n",
      "Spot (default):     p1 U (p2 R (p3 & !p4))\n",
      "Spot+shell quotes:  'p1 U (p2 R (p3 & !p4))'\n",
      "LBT, right aligned: ~~~~~~~~~~~~~~~~~~~~~U p1 V p2 & p3 ! p4\n",
      "LBT, no M/W/R:      U p1 U & p3 ! p4 | & & p2 p3 ! p4 G & p3 ! p4\n"
     ]
    }
   ],
   "source": [
    "print(\"\"\"\\\n",
    "Spin:               {0:s}\n",
    "Spin+parentheses:   {0:sp}\n",
    "Spot (default):     {0}\n",
    "Spot+shell quotes:  {0:q}\n",
    "LBT, right aligned: {0:l:~>40}\n",
    "LBT, no M/W/R:      {0:[MWR]l}\"\"\".format(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The specifiers that can be used with `format` are documented as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function __format__ in module spot:\n",
      "\n",
      "__format__(self, spec)\n",
      "    Format the formula according to `spec`.\n",
      "    \n",
      "    Parameters\n",
      "    ----------\n",
      "    spec : str, optional\n",
      "        a list of letters that specify how the formula\n",
      "        should be formatted.\n",
      "    \n",
      "    Supported specifiers\n",
      "    --------------------\n",
      "    \n",
      "    - 'f': use Spot's syntax (default)\n",
      "    - '8': use Spot's syntax in UTF-8 mode\n",
      "    - 's': use Spin's syntax\n",
      "    - 'l': use LBT's syntax\n",
      "    - 'w': use Wring's syntax\n",
      "    - 'x': use LaTeX output\n",
      "    - 'X': use self-contained LaTeX output\n",
      "    - 'j': use self-contained LaTeX output, adjusted for MathJax\n",
      "    \n",
      "    Add some of those letters for additional options:\n",
      "    \n",
      "    - 'p': use full parentheses\n",
      "    - 'c': escape the formula for CSV output (this will\n",
      "           enclose the formula in double quotes, and escape\n",
      "           any included double quotes)\n",
      "    - 'h': escape the formula for HTML output\n",
      "    - 'd': escape double quotes and backslash,\n",
      "           for use in C-strings (the outermost double\n",
      "           quotes are *not* added)\n",
      "    - 'q': quote and escape for shell output, using single\n",
      "           quotes or double quotes depending on the contents.\n",
      "    - '[...]': rewrite away all the operators specified in brackets,\n",
      "           using spot.unabbreviate().\n",
      "    \n",
      "    - ':spec': pass the remaining specification to the\n",
      "               formating function for strings.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(spot.formula.__format__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A `spot.formula` object has a number of built-in predicates whose value have been computed when the formula was constructed.  For instance you can check whether a formula is in negative normal form using `is_in_nenoform()`, and you can make sure it is an LTL formula (i.e. not a PSL formula) using `is_ltl_formula()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.is_in_nenoform() and f.is_ltl_formula()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.is_ltl_formula()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, `is_syntactic_stutter_invariant()` tells wether the structure of the formula guarranties it to be stutter invariant.  For LTL formula, this means the `X` operator should not be used.  For PSL formula, this function capture all formulas built using the [siPSL grammar](http://www.daxc.de/eth/paper/09atva.pdf)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.is_syntactic_stutter_invariant()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.formula('{a[*];b}<>->c').is_syntactic_stutter_invariant()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.formula('{a[+];b[*]}<>->d').is_syntactic_stutter_invariant()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`spot.relabel` renames the atomic propositions that occur in a formula, using either letters, or numbered propositions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\unicode{x201C}\\mathit{a > b}\\unicode{x201D} \\land \\unicode{x201C}\\mathit{proc[2]@init}\\unicode{x201D} \\land \\mathsf{G} \\mathsf{F} \\mathit{\\_foo\\_}$"
      ],
      "text/plain": [
       "spot.formula(\"\\\"a > b\\\" & \\\"proc[2]@init\\\" & GF_foo_\")"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gf = spot.formula('(GF_foo_) && \"a > b\" && \"proc[2]@init\"'); gf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$a \\land b \\land \\mathsf{G} \\mathsf{F} c$"
      ],
      "text/plain": [
       "spot.formula(\"a & b & GFc\")"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.relabel(gf, spot.Abc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$p_{0} \\land p_{1} \\land \\mathsf{G} \\mathsf{F} p_{2}$"
      ],
      "text/plain": [
       "spot.formula(\"p0 & p1 & GFp2\")"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.relabel(gf, spot.Pnn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The AST of any formula can be displayed with `show_ast()`.  Despite the name, this is not a tree but a DAG, because identical subtrees are merged.  Binary operators have their left and right operands denoted with `L` and `R`, while non-commutative n-ary operators have their operands numbered."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{a;first_match(b[*];c[+];d)}<>-> (c & GFb)\n"
     ]
    },
    {
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      ],
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      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(g); g.show_ast()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Any formula can also be classified in the temporal hierarchy of Manna & Pnueli"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "\n",
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       "<text x=\"160\" y=\"65\">Persistence</text>\n",
       "<text x=\"110\" y=\"125\">Obligation</text>\n",
       "<text x=\"60\" y=\"185\">Safety</text>\n",
       "<text x=\"160\" y=\"185\">Guarantee</text>\n",
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     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.show_mp_hierarchy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'recurrence'"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.mp_class(g, 'v')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (a \\land \\mathsf{X} (\\lnot a \\land b))$"
      ],
      "text/plain": [
       "spot.formula(\"F(a & X(!a & b))\")"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = spot.formula('F(a & X(!a & b))'); f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Etessami's rule for removing X (valid only in stutter-invariant formulas)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathsf{F} (a \\land ((a \\land (a \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} \\lnot a) \\lor (b \\mathbin{\\mathsf{U}} \\lnot a))) \\lor (\\lnot a \\land (\\lnot a \\mathbin{\\mathsf{U}} (a \\land \\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} a) \\lor (b \\mathbin{\\mathsf{U}} a))) \\lor (b \\land (b \\mathbin{\\mathsf{U}} (\\lnot a \\land b \\land \\lnot b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} \\lnot b) \\lor (a \\mathbin{\\mathsf{U}} \\lnot b))) \\lor (\\lnot b \\land (\\lnot b \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} b) \\lor (a \\mathbin{\\mathsf{U}} b))) \\lor (\\lnot a \\land b \\land (\\mathsf{G} \\lnot a \\lor \\mathsf{G} a) \\land (\\mathsf{G} \\lnot b \\lor \\mathsf{G} b))))$"
      ],
      "text/plain": [
       "spot.formula(\"F(a & ((a & (a U (!a & b)) & ((!b U !a) | (b U !a))) | (!a & (!a U (a & !a & b)) & ((!b U a) | (b U a))) | (b & (b U (!a & b & !b)) & ((!a U !b) | (a U !b))) | (!b & (!b U (!a & b)) & ((!a U b) | (a U b))) | (!a & b & (G!a | Ga) & (G!b | Gb))))\")"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.remove_x(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Removing abbreviated operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\bot \\mathbin{\\mathsf{R}} \\lnot (a \\leftrightarrow b)) \\rightarrow (\\top \\mathbin{\\mathsf{U}} (a \\leftrightarrow b))$"
      ],
      "text/plain": [
       "spot.formula(\"(0 R !(a <-> b)) -> (1 U (a <-> b))\")"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = spot.formula(\"G(a xor b) -> F(a <-> b)\")\n",
    "spot.unabbreviate(f, \"GF^\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\top \\mathbin{\\mathsf{U}} ((a \\land b) \\lor (\\lnot a \\land \\lnot b))) \\lor \\lnot (\\bot \\mathbin{\\mathsf{R}} ((\\lnot a \\land b) \\lor (a \\land \\lnot b)))$"
      ],
      "text/plain": [
       "spot.formula(\"(1 U ((a & b) | (!a & !b))) | !(0 R ((!a & b) | (a & !b)))\")"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spot.unabbreviate(f, \"GF^ei\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nesting level of operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U 3\n",
      "F 2\n",
      "U 3\n",
      "F 2\n",
      "FU 4\n"
     ]
    }
   ],
   "source": [
    "f = spot.formula('F(b & X(a U b U ((a W Fb) | (c U d))))')\n",
    "print(\"U\", spot.nesting_depth(f, spot.op_U))\n",
    "print(\"F\", spot.nesting_depth(f, spot.op_F))\n",
    "# These following two are syntactic sugar for the above two\n",
    "print(\"U\", spot.nesting_depth(f, \"U\"))\n",
    "print(\"F\", spot.nesting_depth(f, \"F\"))\n",
    "# If you want to consider \"U\" and \"F\" are a similar type of\n",
    "# operator, you can count both with\n",
    "print(\"FU\", spot.nesting_depth(f, \"FU\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Collecting the set of atomic propositions used by a formula:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "spot.atomic_prop_set([spot.formula(\"a\"), spot.formula(\"b\"), spot.formula(\"c\"), spot.formula(\"d\")])\n",
      "{\"a\", \"b\", \"c\", \"d\"}\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\{\\unicode{x201C}a\\unicode{x201D}, \\unicode{x201C}b\\unicode{x201D}, \\unicode{x201C}c\\unicode{x201D}, \\unicode{x201C}d\\unicode{x201D}\\}$"
      ],
      "text/plain": [
       "spot.atomic_prop_set([spot.formula(\"a\"), spot.formula(\"b\"), spot.formula(\"c\"), spot.formula(\"d\")])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ap = spot.atomic_prop_collect(f)\n",
    "print(repr(ap))  # print as an atomic_prop_set object\n",
    "print(ap)        # print as a string\n",
    "display(ap)      # LaTeX-style, for notebooks"
   ]
  }
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