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"# Building DTMCs\n",
"In Stormvogel, a **Discrete Time Markov Chain (DTMC)** consists of:\n",
"* states $S$,\n",
"* an initial state $s_0$,\n",
"* a successor distribution $P(s)$ for every state $s$, i.e., transitions between states $s$ and $s'$, each annotated with a probability.\n",
"* state labels $L(s)$.\n",
"\n",
"In Stormvogel, these models are explicitly stored in memory.\n",
"\n",
"In this notebook, we demonstrate how to construct a DTMC from various sources. We first demonstrate how to construct the underlying data structures explicitly and then document a programming-language like approach. In general, for smaller models, the model API is more convenient, but for bigger models the pgc API becomes more convenient.\n",
"In particular, we demonstrate the creation using a simple model of a 6-sided die.\n",
"\n",
"We would recommend at least looking at the code for the first example, using the `pgc`API in detail, as we will mainly use this in other chapters.\n",
"\n",
"For more information on how to use the [PRISM](https://www.prismmodelchecker.org/) and [stormpy APIs](https://moves-rwth.github.io/stormpy/), we refer to their respective documentations.\n",
"\n",
"**Note:** unfortunately, the visualisation of the DTMC is not always correct when it is rendered out of view. To re-center, you can simply double-click inside the window.\n",
"\n",
"## The model API\n"
]
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"source": [
"# If we use the model API, we need to create all states and transitions explicitly.\n",
"from stormvogel.show import show\n",
"from stormvogel.layout import Layout\n",
"import stormvogel.model\n",
"\n",
"# Create a new model with the name \"Die\"\n",
"die_dtmc = stormvogel.model.new_dtmc(\"Die\", create_initial_state=True)\n",
"# By default, a new model starts with an initial state, use get_initial_state to retrieve it.\n",
"init = die_dtmc.get_initial_state()\n",
"# From the initial state, add the transition to 6 new states with probability 1/6th.\n",
"# The set_transitions function takes a list of 2-tuples where the first argument is a probability and the second elment is a state.\n",
"init.set_transitions(\n",
" [(1 / 6, die_dtmc.new_state(f\"rolled{i}\")) for i in range(1, 7)]\n",
")\n",
"# Add self loops to make the model *absorbing* (i.e. there are no states where you get stuck because there are no more transitions available).\n",
"die_dtmc.add_self_loops()\n",
"vis = show(die_dtmc, layout=Layout(\"layouts/die.json\"))\n"
]
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"\n",
"\n",
"\n",
"## The pgc API\n",
"\n",
"We could also use the `pgc` API to create the exact same model. This API uses *higher order functions*, which means that you use functions in parameters for other functions. This might be difficult to understand if you are not from a computing science background, but we will try to explain."
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"source": [
"from stormvogel import pgc\n",
"from stormvogel.model import EmptyAction, ModelType\n",
"from stormvogel.show import show\n",
"from stormvogel.layout import Layout\n",
"\n",
"# Create an initial state.\n",
"# In the model API, any state that you create takes a number of keyword arguments.\n",
"# These keyword arguments store some information about the state and can be of any type.\n",
"# In this case, we have a single piece of information called 'x', and for the initial state, we give it \"\".\n",
"init = pgc.State(x=\"\")\n",
"\n",
"# This delta function takes as an argument a single state, and returns a \n",
"# list of 2-tuples where the first argument is a probability and the second elment is a state.\n",
"# Our pgc API uses this function to know where to go from a state.\n",
"def delta(s: pgc.State):\n",
" if s == init: # If we are in the initial state, roll the die, we could also test this with s.x == \"\".\n",
" return [(1 / 6, pgc.State(x=[f\"rolled{i}\"])) for i in range(1, 7)]\n",
" else: # If we are *not* in the initial state, do a self-loop with probability 1.\n",
" return [(1, s)]\n",
" \n",
"# Labels is a function that tells the pgc API to use x as the label.\n",
"# It is written as a lambda here, but it means exactly the same as\n",
"# def labels(s):\n",
"# return s.x\n",
"labels = lambda s: s.x\n",
"\n",
"pgc_die = pgc.build_pgc(\n",
" delta=delta,\n",
" initial_state_pgc=init,\n",
" labels=labels,\n",
" modeltype=ModelType.DTMC\n",
")\n",
"\n",
"vis2 = show(pgc_die, layout=Layout(\"layouts/die.json\"))"
]
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"\n",
"\n",
"For this simple model, it might not be immediately obvious why we would want such an API, but hopefully that will become clear in later examples."
]
},
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"## Via Stormpy\n",
"\n",
"We can also construct our die model in PRISM. The concepts here are similar to the pgc API. **This requires stormpy**."
]
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"source": [
"%%prism die_prism_code\n",
"dtmc\n",
"\n",
"module die\n",
"\t// The integers 0..7 represent our states, and 0 is the initial state.\n",
"\ts : [0..7] init 0;\n",
" // From s=0, we can go to 1,2,3,4,5,6 with 1/6th probability.\n",
" // The + sign can be interpreted as an 'or'\n",
" // Note that this is similar to our delta function.\n",
" [] s=0 -> 1/6 : (s'=1) + \n",
" 1/6: (s'=2) + \n",
" 1/6: (s'=3) +\n",
" 1/6: (s'=4) +\n",
" 1/6: (s'=5) + \n",
" 1/6: (s'=6);\n",
" // Self loops\n",
" [] s>0 -> (s'=s);\n",
"endmodule\n",
"\n",
"// Add the desired labels\n",
"label \"rolled1\" = s=1;\n",
"label \"rolled2\" = s=2;\n",
"label \"rolled3\" = s=3;\n",
"label \"rolled4\" = s=4;\n",
"label \"rolled5\" = s=5;\n",
"label \"rolled6\" = s=6;"
]
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"Under the hood, stormpy parses the prism model which is then converted to a stormpy model and a stormvogel model."
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"prism_die = stormvogel.from_prism(die_prism_code)\n",
"vis3 = show(prism_die, layout=Layout(\"layouts/die.json\"))"
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"Finally, we could also use the stormpy's SparseMatrix API directly. **Obviously, also requires stormpy**"
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"source": [
"import stormpy\n",
"import stormvogel.stormpy_utils.mapping as mapping\n",
"builder = stormpy.SparseMatrixBuilder(rows=0, columns=0, entries=0, force_dimensions=False, has_custom_row_grouping=False)\n",
"for s in range(1,7):\n",
" builder.add_next_value(0, s, 1/6)\n",
"for s in range(1,7):\n",
" builder.add_next_value(s, s, 1)\n",
"transition_matrix = builder.build()\n",
"state_labeling = stormpy.storage.StateLabeling(7)\n",
"\n",
"labels = {\"init\"}.union({f\"rolled{s}\" for s in range(1,7)})\n",
"for label in labels:\n",
" state_labeling.add_label(label)\n",
"state_labeling.add_label_to_state(\"init\", 0)\n",
"for s in range(1,7):\n",
" state_labeling.add_label_to_state(f\"rolled{s}\", s)\n",
"components = stormpy.SparseModelComponents(transition_matrix=transition_matrix, state_labeling=state_labeling)\n",
"stormpy_dtmc = mapping.stormpy_to_stormvogel(stormpy.storage.SparseDtmc(components))\n",
"vis4 = show(stormpy_dtmc, layout=Layout(\"layouts/die.json\"))"
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