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"source": [
"# Parametric models\n",
"Instead of setting numeric values as transition probabilities, we may also use parameters, polynomials or even rational functions in multiple variables."
]
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"from stormvogel import parametric"
]
},
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"source": [
"Polynomials are represented as dictionaries where the keys are the exponents and the values are coefficients. In addition, we must also supply a list of variable names. Rational functions are then represented as a pair of two polynomials (numerator and denominator)."
]
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"text": [
"x^2 + y^2\n",
"2.0*1 + z + 6.0*z^3\n",
"(x^2 + y^2)/(2.0*1 + z + 6.0*z^3)\n"
]
}
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"source": [
"polynomial1 = parametric.Polynomial([\"x\",\"y\"])\n",
"polynomial1.add_term((2,0),1)\n",
"polynomial1.add_term((0,2),1)\n",
"\n",
"print(polynomial1)\n",
"\n",
"polynomial2 = parametric.Polynomial([\"z\"])\n",
"polynomial2.add_term((0,),2)\n",
"polynomial2.add_term((1,),1)\n",
"polynomial2.add_term((3,),6)\n",
"\n",
"print(polynomial2)\n",
"\n",
"rational_function = parametric.RationalFunction(polynomial1, polynomial2)\n",
"\n",
"print(rational_function)"
]
},
{
"cell_type": "markdown",
"id": "7ce9b612",
"metadata": {},
"source": [
"To create a parametric model (e.g. pmc or pmdp) we simply have to set such a value as a transition probability. As an example, we provide the knuth yao dice, but with parameters instead of concrete probabilities."
]
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"source": [
"from stormvogel import model, bird\n",
"from stormvogel.show import show\n",
"\n",
"#we first make polynomials 'x' and '1-x'\n",
"x = parametric.Polynomial([\"x\"])\n",
"x.add_term((1,),1)\n",
"\n",
"invx = parametric.Polynomial([\"x\"])\n",
"invx.add_term((1,),-1)\n",
"invx.add_term((0,),1)\n",
"\n",
"#we build the knuth yao dice using the bird model builder\n",
"def delta(s: bird.State):\n",
" match s.s:\n",
" case 0:\n",
" return [(x, bird.State(s=1)), (invx, bird.State(s=2))]\n",
" case 1:\n",
" return [(x, bird.State(s=3)), (invx, bird.State(s=4))]\n",
" case 2:\n",
" return [(x, bird.State(s=5)), (invx, bird.State(s=6))]\n",
" case 3:\n",
" return [(x, bird.State(s=1)), (invx, bird.State(s=7, d=1))]\n",
" case 4:\n",
" return [\n",
" (x, bird.State(s=7, d=2)),\n",
" (invx, bird.State(s=7, d=3)),\n",
" ]\n",
" case 5:\n",
" return [\n",
" (x, bird.State(s=7, d=4)),\n",
" (invx, bird.State(s=7, d=5)),\n",
" ]\n",
" case 6:\n",
" return [(x, bird.State(s=2)), (invx, bird.State(s=7, d=6))]\n",
" case 7:\n",
" return [(1, s)]\n",
"\n",
"def labels(s: bird.State):\n",
" if s.s == 7:\n",
" return f\"rolled{str(s.d)}\"\n",
"\n",
"knuth_yao = bird.build_bird(\n",
" delta=delta,\n",
" init=bird.State(s=0),\n",
" labels=labels,\n",
" modeltype=model.ModelType.DTMC,\n",
")\n",
"\n",
"vis = show(knuth_yao)"
]
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"We can now evaluate the model by assigning the variable x to any concrete value. This induces a regular dtmc with fixed probabilities."
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"p = 1/2\n",
"\n",
"eval_knuth_yao = knuth_yao.parameter_valuation({\"x\":p})\n",
"vis = show(eval_knuth_yao)"
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