spot/wrap/python/tests/formulas.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Handling LTL and PSL formulas\n",
      "============================="
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import spot"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "f = spot.formula('p1 U p2 R (p3 & !p4)')\n",
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$p_{1} \\mathbin{\\mathsf{U}} (p_{2} \\mathbin{\\mathsf{R}} (p_{3} \\land \\lnot p_{4}))$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "p1 U (p2 R (p3 & !p4))"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "g = spot.formula('{a;b*;c[+]}<>->GFb'); g"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$\\{a \\mathbin{\\mathsf{;}} b^{\\star} \\mathbin{\\mathsf{;}} c^+\\}\\mathrel{\\Diamond\\kern-1.7pt\\raise.4pt\\hbox{$\\mathord{\\rightarrow}$}} \\mathsf{G} \\mathsf{F} b$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "{a;b[*];c[+]}<>-> GFb"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "h = spot.formula('& | a b c'); h"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$c \\land (a \\lor b)$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "c & (a | b)"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "str(f)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "'p1 U (p2 R (p3 & !p4))'"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "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 typically the output that is used to render formulas using MathJax in a notebook. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "for i in ['spot', 'spin', 'lbt', 'wring', 'utf8', 'latex', 'sclatex']:\n",
      "    print(\"%-10s%s\" % (i, f.to_str(i)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "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\u2227\u00acp4))\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"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "f.is_in_nenoform() and f.is_ltl_formula()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "g.is_ltl_formula()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "f.is_syntactic_stutter_invariant()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.formula('{a[*];b}<>->c').is_syntactic_stutter_invariant()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.formula('{a[+];b[*]}<>->d').is_syntactic_stutter_invariant()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "gf = spot.formula('(GF_foo_) && \"a > b\" && \"proc[2]@init\"'); gf"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$``\\mathit{a > b}\\textrm{''} \\land ``\\mathit{proc[2]@init}\\textrm{''} \\land \\mathsf{G} \\mathsf{F} \\mathit{\\_foo\\_}$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "\"a > b\" & \"proc[2]@init\" & GF_foo_"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.relabel(gf, spot.Abc)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$a \\land b \\land \\mathsf{G} \\mathsf{F} c$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "a & b & GFc"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.relabel(gf, spot.Pnn)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$p_{0} \\land p_{1} \\land \\mathsf{G} \\mathsf{F} p_{2}$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "p0 & p1 & GFp2"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "print(g); g.show_ast()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "{a;b[*];c[+]}<>-> GFb\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "svg": [
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     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = spot.formula('F(a & X(!a & b))'); f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$\\mathsf{F} (a \\land \\mathsf{X} (\\lnot a \\land b))$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "F(a & X(!a & b))"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Etessami's rule for removing X (valid only in stutter-invariant formulas)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.remove_x(f)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$\\mathsf{F} (a \\land ((a \\land (a \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} \\lnot a) \\lor (b \\mathbin{\\mathsf{U}} \\lnot a))) \\lor (\\lnot a \\land (\\lnot a \\mathbin{\\mathsf{U}} (a \\land \\lnot a \\land b)) \\land ((\\lnot b \\mathbin{\\mathsf{U}} a) \\lor (b \\mathbin{\\mathsf{U}} a))) \\lor (b \\land (b \\mathbin{\\mathsf{U}} (\\lnot a \\land b \\land \\lnot b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} \\lnot b) \\lor (a \\mathbin{\\mathsf{U}} \\lnot b))) \\lor (\\lnot b \\land (\\lnot b \\mathbin{\\mathsf{U}} (\\lnot a \\land b)) \\land ((\\lnot a \\mathbin{\\mathsf{U}} b) \\lor (a \\mathbin{\\mathsf{U}} b))) \\lor (\\lnot a \\land b \\land (\\mathsf{G} \\lnot a \\lor \\mathsf{G} a) \\land (\\mathsf{G} \\lnot b \\lor \\mathsf{G} b))))$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "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))))"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Removing abbreviated operators"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = spot.formula(\"G(a xor b) -> F(a <-> b)\")\n",
      "spot.unabbreviate(f, \"GF^\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$(\\bot \\mathbin{\\mathsf{R}} \\lnot (a \\leftrightarrow b)) \\rightarrow (\\top \\mathbin{\\mathsf{U}} (a \\leftrightarrow b))$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "(0 R !(a <-> b)) -> (1 U (a <-> b))"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spot.unabbreviate(f, \"GF^ei\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$(\\top \\mathbin{\\mathsf{U}} ((a \\land b) \\lor (\\lnot a \\land \\lnot b))) \\lor \\lnot (\\bot \\mathbin{\\mathsf{R}} ((\\lnot a \\land b) \\lor (a \\land \\lnot b)))$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "(1 U ((a & b) | (!a & !b))) | !(0 R ((!a & b) | (a & !b)))"
       ]
      }
     ],
     "prompt_number": 19
    }
   ],
   "metadata": {}
  }
 ]
}