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{
"cell_type": "markdown",
"id": "aa79e617-6b6b-4620-8c4f-808d3fca1566",
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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."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "386ffe0b-99fb-4bc4-ad93-7d5bebe3b206",
"metadata": {},
"outputs": [],
"source": [
"from stormvogel import parametric"
]
},
{
"cell_type": "markdown",
"id": "e16876b3-7b6a-4555-9dd3-25015453836b",
"metadata": {},
"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)."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "497f79e8-f2e1-4fe8-a64b-fba502301549",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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"
]
}
],
"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": "424d1c58-805f-4f52-8601-4caafa06a6ea",
"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."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "9ac1f65e-ff50-49d9-9064-e7ab71b4036b",
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"Output()"
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"Output()"
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"source": [
"from stormvogel import model, pgc\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 pgc model builder\n",
"def delta(s: pgc.State):\n",
" match s.s:\n",
" case 0:\n",
" return [(x, pgc.State(s=1)), (invx, pgc.State(s=2))]\n",
" case 1:\n",
" return [(x, pgc.State(s=3)), (invx, pgc.State(s=4))]\n",
" case 2:\n",
" return [(x, pgc.State(s=5)), (invx, pgc.State(s=6))]\n",
" case 3:\n",
" return [(x, pgc.State(s=1)), (invx, pgc.State(s=7, d=1))]\n",
" case 4:\n",
" return [\n",
" (x, pgc.State(s=7, d=2)),\n",
" (invx, pgc.State(s=7, d=3)),\n",
" ]\n",
" case 5:\n",
" return [\n",
" (x, pgc.State(s=7, d=4)),\n",
" (invx, pgc.State(s=7, d=5)),\n",
" ]\n",
" case 6:\n",
" return [(x, pgc.State(s=2)), (invx, pgc.State(s=7, d=6))]\n",
" case 7:\n",
" return [(1, s)]\n",
"\n",
"def labels(s: pgc.State):\n",
" if s.s == 7:\n",
" return f\"rolled{str(s.d)}\"\n",
"\n",
"knuth_yao = pgc.build_pgc(\n",
" delta=delta,\n",
" initial_state_pgc=pgc.State(s=0),\n",
" labels=labels,\n",
" modeltype=model.ModelType.DTMC,\n",
")\n",
"\n",
"vis = show(knuth_yao)"
]
},
{
"cell_type": "markdown",
"id": "0d03e392-6724-44f8-b97f-3fa1f9c48218",
"metadata": {},
"source": [
"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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"Output()"
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"source": [
"p = 1/2\n",
"\n",
"eval_knuth_yao = knuth_yao.parameter_valuation({\"x\":p})\n",
"vis = show(eval_knuth_yao)"
]
}
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