{
"cells": [
{
"cell_type": "markdown",
"id": "3396690e",
"metadata": {},
"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": "9a28976b",
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"execution": {
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"source": [
"from stormvogel import parametric"
]
},
{
"cell_type": "markdown",
"id": "c72114b8",
"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": "5132c171",
"metadata": {
"execution": {
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"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": "b9784951",
"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": 3,
"id": "d7b8dc31",
"metadata": {
"execution": {
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"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
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"text/plain": [
""
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},
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"output_type": "display_data"
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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",
"\n",
"# we build the knuth yao dice using the bird model builder\n",
"def delta(\n",
" s: bird.State,\n",
") -> list[tuple[float | parametric.Polynomial, bird.State]] | None:\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",
"\n",
"def labels(s: bird.State):\n",
" if s.s == 7:\n",
" return f\"rolled{str(s.d)}\"\n",
"\n",
"\n",
"knuth_yao_pmc = 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_pmc)"
]
},
{
"cell_type": "markdown",
"id": "007908f1",
"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."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "a1d86378",
"metadata": {
"execution": {
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"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
],
"text/plain": [
""
]
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}
],
"source": [
"p = 1 / 2\n",
"\n",
"\n",
"eval_knuth_yao_pmc = knuth_yao_pmc.parameter_valuation({\"x\": p})\n",
"vis = show(eval_knuth_yao_pmc)"
]
},
{
"cell_type": "markdown",
"id": "289206e7",
"metadata": {},
"source": [
"# Interval Models\n",
"We can also set an interval between two values x and y as transition value, meaning that we don't know the probability precisely, but we know it is between x and y. We represent intervals using a class in model.py, where we have two attributes: bottom and top. Both of these should be an element of type Number, i.e., int, float or fraction."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "23607e4b",
"metadata": {
"execution": {
"iopub.execute_input": "2026-03-26T10:47:20.251793Z",
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"shell.execute_reply": "2026-03-26T10:47:20.254251Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[0.3333333333333333,0.6666666666666666]\n"
]
}
],
"source": [
"interval = model.Interval(1 / 3, 2 / 3)\n",
"print(interval)"
]
},
{
"cell_type": "markdown",
"id": "9cb7e964",
"metadata": {},
"source": [
"Similar to parametric models, creating an interval model is as straightforward as just setting some interval objects as transition values."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "9d24618e",
"metadata": {
"execution": {
"iopub.execute_input": "2026-03-26T10:47:20.256426Z",
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" \n",
"\n"
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],
"source": [
"from stormvogel import bird\n",
"from stormvogel.show import show\n",
"\n",
"# We create our interval values\n",
"interval = model.Interval(2 / 7, 6 / 7)\n",
"inv_interval = model.Interval(1 / 7, 5 / 7)\n",
"\n",
"\n",
"# we build the knuth yao dice using the bird model builder\n",
"def delta(s: bird.State) -> list[tuple[float | model.Interval, bird.State]] | None:\n",
" match s.s:\n",
" case 0:\n",
" return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=2))]\n",
" case 1:\n",
" return [(interval, bird.State(s=3)), (inv_interval, bird.State(s=4))]\n",
" case 2:\n",
" return [(interval, bird.State(s=5)), (inv_interval, bird.State(s=6))]\n",
" case 3:\n",
" return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=7, d=1))]\n",
" case 4:\n",
" return [\n",
" (interval, bird.State(s=7, d=2)),\n",
" (invx, bird.State(s=7, d=3)),\n",
" ]\n",
" case 5:\n",
" return [\n",
" (interval, bird.State(s=7, d=4)),\n",
" (invx, bird.State(s=7, d=5)),\n",
" ]\n",
" case 6:\n",
" return [(interval, bird.State(s=2)), (inv_interval, bird.State(s=7, d=6))]\n",
" case 7:\n",
" return [(1, s)]\n",
"\n",
"\n",
"def labels(s: bird.State):\n",
" if s.s == 7:\n",
" return f\"rolled{str(s.d)}\"\n",
"\n",
"\n",
"knuth_yao_imc = 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_imc)"
]
},
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