{
"cells": [
{
"cell_type": "markdown",
"id": "4454f460",
"metadata": {},
"source": [
"# Parametric and Interval Models\n",
"In this notebook, we will show how to create and work with parametric and\n",
"interval models.\n",
"## Parametric Models\n",
"\n",
"Stormvogel represents parametric transition values as **sympy expressions**.\n",
"Any polynomial or rational function over `sympy.Symbol` parameters is a valid\n",
"transition value, which means you can write them using ordinary Python\n",
"arithmetic."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "765f814d",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:48.890832Z",
"iopub.status.busy": "2026-07-01T08:29:48.890621Z",
"iopub.status.idle": "2026-07-01T08:29:49.127275Z",
"shell.execute_reply": "2026-07-01T08:29:49.126559Z"
}
},
"outputs": [],
"source": [
"import sympy as sp"
]
},
{
"cell_type": "markdown",
"id": "433bc69e",
"metadata": {},
"source": [
"For example, a polynomial in two variables:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "b9d9b0f1",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.129688Z",
"iopub.status.busy": "2026-07-01T08:29:49.129465Z",
"iopub.status.idle": "2026-07-01T08:29:49.159881Z",
"shell.execute_reply": "2026-07-01T08:29:49.159240Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x**2 + y**2\n",
"6*z**3 + z + 2\n",
"(x**2 + y**2)/(6*z**3 + z + 2)\n"
]
}
],
"source": [
"x, y, z = sp.symbols(\"x y z\")\n",
"\n",
"polynomial1 = x**2 + y**2\n",
"print(polynomial1)\n",
"\n",
"polynomial2 = 6 * z**3 + z + 2\n",
"print(polynomial2)\n",
"\n",
"rational_function = polynomial1 / polynomial2\n",
"print(rational_function)"
]
},
{
"cell_type": "markdown",
"id": "da94b095",
"metadata": {},
"source": [
"To create a parametric model (e.g. pMC or pMDP) we simply use these sympy\n",
"expressions as transition probabilities. As an example, we build a\n",
"parametric Knuth–Yao dice: a coin with success probability `x` is flipped\n",
"repeatedly to simulate a six-sided die."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "ad92adcb",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.161858Z",
"iopub.status.busy": "2026-07-01T08:29:49.161685Z",
"iopub.status.idle": "2026-07-01T08:29:49.758020Z",
"shell.execute_reply": "2026-07-01T08:29:49.757365Z"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/tmp/ipykernel_1255/1846297749.py:10: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" s: bird.State,\n",
"/tmp/ipykernel_1255/1846297749.py:11: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" ) -> list[tuple[float | sp.Expr, bird.State]] | None:\n",
"/tmp/ipykernel_1255/1846297749.py:37: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" def labels(s: bird.State):\n",
"/tmp/ipykernel_1255/1846297749.py:44: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" init=bird.State(s=0),\n",
"/tmp/ipykernel_1255/1846297749.py:14: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(x, bird.State(s=1)), (1 - x, bird.State(s=2))]\n",
"/tmp/ipykernel_1255/1846297749.py:16: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(x, bird.State(s=3)), (1 - x, bird.State(s=4))]\n",
"/tmp/ipykernel_1255/1846297749.py:18: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(x, bird.State(s=5)), (1 - x, bird.State(s=6))]\n",
"/tmp/ipykernel_1255/1846297749.py:20: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(x, bird.State(s=1)), (1 - x, bird.State(s=7, d=1))]\n",
"/tmp/ipykernel_1255/1846297749.py:23: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (x, bird.State(s=7, d=2)),\n",
"/tmp/ipykernel_1255/1846297749.py:24: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (1 - x, bird.State(s=7, d=3)),\n",
"/tmp/ipykernel_1255/1846297749.py:28: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (x, bird.State(s=7, d=4)),\n",
"/tmp/ipykernel_1255/1846297749.py:29: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (1 - x, bird.State(s=7, d=5)),\n",
"/tmp/ipykernel_1255/1846297749.py:32: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(x, bird.State(s=2)), (1 - x, bird.State(s=7, d=6))]\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
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"text/plain": [
""
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"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/plain": [
""
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from stormvogel import model, bird\n",
"from stormvogel.show import show\n",
"\n",
"# A single symbolic parameter for the coin bias.\n",
"x = sp.Symbol(\"x\")\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 | sp.Expr, bird.State]] | None:\n",
" match s.s:\n",
" case 0:\n",
" return [(x, bird.State(s=1)), (1 - x, bird.State(s=2))]\n",
" case 1:\n",
" return [(x, bird.State(s=3)), (1 - x, bird.State(s=4))]\n",
" case 2:\n",
" return [(x, bird.State(s=5)), (1 - x, bird.State(s=6))]\n",
" case 3:\n",
" return [(x, bird.State(s=1)), (1 - x, bird.State(s=7, d=1))]\n",
" case 4:\n",
" return [\n",
" (x, bird.State(s=7, d=2)),\n",
" (1 - x, bird.State(s=7, d=3)),\n",
" ]\n",
" case 5:\n",
" return [\n",
" (x, bird.State(s=7, d=4)),\n",
" (1 - x, bird.State(s=7, d=5)),\n",
" ]\n",
" case 6:\n",
" return [(x, bird.State(s=2)), (1 - x, 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",
"show(knuth_yao_pmc)"
]
},
{
"cell_type": "markdown",
"id": "078bf74f",
"metadata": {},
"source": [
"The symbol `x` was auto-declared on the model as soon as it appeared in a\n",
"transition. You can inspect the parameters any time:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "869f7c2a",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.766074Z",
"iopub.status.busy": "2026-07-01T08:29:49.765689Z",
"iopub.status.idle": "2026-07-01T08:29:49.769639Z",
"shell.execute_reply": "2026-07-01T08:29:49.769119Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Parameters: ('x',)\n"
]
}
],
"source": [
"print(\"Parameters:\", knuth_yao_pmc.parameters)"
]
},
{
"cell_type": "markdown",
"id": "67323b60",
"metadata": {},
"source": [
"We can now evaluate the model by assigning the parameter `x` to any concrete\n",
"value. This induces a regular DTMC with fixed probabilities."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "316609d8",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.771381Z",
"iopub.status.busy": "2026-07-01T08:29:49.771210Z",
"iopub.status.idle": "2026-07-01T08:29:49.802574Z",
"shell.execute_reply": "2026-07-01T08:29:49.801923Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
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"data": {
"text/plain": [
""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = 1 / 2\n",
"\n",
"eval_knuth_yao_pmc = knuth_yao_pmc.get_instantiated_model({\"x\": p})\n",
"show(eval_knuth_yao_pmc)"
]
},
{
"cell_type": "markdown",
"id": "370ac4d5",
"metadata": {},
"source": [
"## Interval Models\n",
"We can also set an interval between two values x and y as transition value,\n",
"meaning that we don't know the probability precisely, but we know it is\n",
"between x and y. We represent intervals using `model.Interval`, where we\n",
"have two attributes: `lower` and `upper`. Both of these should be an\n",
"element of type `Number`, i.e., `int`, `float` or `Fraction`."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "14bb49df",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.810759Z",
"iopub.status.busy": "2026-07-01T08:29:49.810526Z",
"iopub.status.idle": "2026-07-01T08:29:49.814139Z",
"shell.execute_reply": "2026-07-01T08:29:49.813639Z"
}
},
"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": "6c8cf991",
"metadata": {},
"source": [
"Similar to parametric models, creating an interval model is as\n",
"straightforward as just setting some interval objects as transition values."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "e41debcb",
"metadata": {
"execution": {
"iopub.execute_input": "2026-07-01T08:29:49.816047Z",
"iopub.status.busy": "2026-07-01T08:29:49.815884Z",
"iopub.status.idle": "2026-07-01T08:29:49.851400Z",
"shell.execute_reply": "2026-07-01T08:29:49.850735Z"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/tmp/ipykernel_1255/1271625296.py:10: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" def delta(s: bird.State) -> list[tuple[float | model.Interval, bird.State]] | None:\n",
"/tmp/ipykernel_1255/1271625296.py:36: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" def labels(s: bird.State):\n",
"/tmp/ipykernel_1255/1271625296.py:43: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" init=bird.State(s=0),\n",
"/tmp/ipykernel_1255/1271625296.py:13: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=2))]\n",
"/tmp/ipykernel_1255/1271625296.py:15: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(interval, bird.State(s=3)), (inv_interval, bird.State(s=4))]\n",
"/tmp/ipykernel_1255/1271625296.py:17: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(interval, bird.State(s=5)), (inv_interval, bird.State(s=6))]\n",
"/tmp/ipykernel_1255/1271625296.py:19: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(interval, bird.State(s=1)), (inv_interval, bird.State(s=7, d=1))]\n",
"/tmp/ipykernel_1255/1271625296.py:22: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (interval, bird.State(s=7, d=2)),\n",
"/tmp/ipykernel_1255/1271625296.py:23: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (inv_interval, bird.State(s=7, d=3)),\n",
"/tmp/ipykernel_1255/1271625296.py:27: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (interval, bird.State(s=7, d=4)),\n",
"/tmp/ipykernel_1255/1271625296.py:28: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" (inv_interval, bird.State(s=7, d=5)),\n",
"/tmp/ipykernel_1255/1271625296.py:31: DeprecationWarning: bird.State is deprecated, use bird.BirdState instead.\n",
" return [(interval, bird.State(s=2)), (inv_interval, bird.State(s=7, d=6))]\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
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" \n",
" \n",
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" \n",
" \n",
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"\n"
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""
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"text/plain": [
""
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"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",
" (inv_interval, bird.State(s=7, d=3)),\n",
" ]\n",
" case 5:\n",
" return [\n",
" (interval, bird.State(s=7, d=4)),\n",
" (inv_interval, 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",
"show(knuth_yao_imc)"
]
},
{
"cell_type": "code",
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"id": "f7a20df4",
"metadata": {},
"outputs": [],
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