fix: show code button parking

pages/Parking_Problem_Solver.py

@@ -5,6 +5,214 @@
 import streamlit as st
 
 st.set_page_config(page_title="Parking Problem Solver", page_icon="🚗")
+# This is automatically generated, do not modify
+if st.button('Show code'):
+    st.code('''import random
+
+import matplotlib.pyplot as plt
+import numpy as np
+import streamlit as st
+
+st.set_page_config(page_title="Parking Problem Solver", page_icon="🚗")
+
+
+problem_explanation = """
+
+In the context of Markov Decision Processes, the parking problem can be described as follows:
+
+### State Space, $E$
+The state space $E$ consists of $N$ tuples, each representing a parking slot. For a given slot $i$ (where $i = 1, ..., N$), the state can be either $(i, F)$ if the slot is free or $(i, T)$ if the slot is taken.
+
+### Action Space, $A$
+The action space depends on the current state:
+- If $x = (i, T)$, the only possible action is to *continue*.
+- If $x = (i, F)$, two actions are possible: *continue* or *park*.
+
+### Reward Function
+The reward function is defined as:
+- $r((i, F), \\text{stop}) = i$, indicating the reward for parking in slot $i$.
+- $r((i, F), \\text{continue}) = -c$, representing the choice to continue despite a free slot.
+- $r((i, T), \\text{continue}) = -c$, for continuing when encountering a taken slot.
+
+We use a constant $c \\in \\mathbb{R_+}$ to represent the cost of continuing to search for a parking spot.
+
+### Transition Probabilities
+The probability of moving to the next state, given the current state and action, is described by $\\mathbb{P}$:
+- $\\mathbb{P}[(i+1, F) | (i, T), \\text{continue}] = \\mathbb{P}[(i+1, F) | (i, F), \\text{continue}] = \\rho(i)$, where $\\rho(i)$ is a decreasing (linear) function in $i$, indicating the probability of the next slot being free.
+
+### Value Function
+The value function $V$ for the states is defined as:
+- $V((N, F)) = N$, the value of parking in the last free slot.
+- $V((N, T)) = 0$, the value of encountering the last slot taken.
+- For a free slot $i$, $V((i, F)) = \\max(i, V((i+1, F))\\rho(i) + V((i+1, T))(1 - \\rho(i)))$, the maximum value between parking now or expecting future rewards.
+- For a taken slot $i$, $V((i, T)) = \\rho(i)V((i+1, F)) + (1 - \\rho(i))V((i+1, T))$.
+
+The objective is to determine the optimal policy: when to park immediately and when to continue searching for a closer slot.
+
+### Analytical solution using backward induction
+
+The parking problem, formulated within the framework of Markov Decision Processes, finds
+a practical solution through backward induction. 
+
+**Initialization**: We start with a value function $V$ initialized for all slots, 
+extending to $N+1$ to account for a hypothetical state beyond the last parking slot. 
+This state acts as a boundary condition with $V[N+1]=0$.
+
+
+**Backward Induction Process**:
+The iterative process is as follows:
+1. For each slot $i$ from $N$ down to $1$, we evaluate the decision-making process under
+the assumption that the decision-maker is equipped with the knowledge of all subsequent
+outcomes. This reverse evaluation is essential for incorporating the future impacts of
+present decisions.
+2. For each free slot $(i, F)$, we compute the decision to either park or continue. The
+computation involves a comparison between the immediate reward of parking, $i$, and the
+expected utility of continuing, which is a function of the value function of the 
+subsequent slot $V[i+1]$ and the cost of continuing $c$. The probability $\\rho(i)$ 
+that influences the transition to subsequent states adjusts dynamically with $i$.
+3. For occupied slots $(i, T)$, the utility is derived from the value function of the
+subsequent slot minus the cost of continuing, as parking is not an option.
+
+**Recursion and Policy Determination**: The core of this process is encapsulated in the
+recursion formula:
+$$
+V_{k+1}(s) = \\max_{a \\in \\mathcal{A}(s)} \\{\\mathcal{R}(s, a) + \\gamma \\sum_{s' \\in \\mathcal{S}} \\mathcal{P}(s' | s, a)V_k(s')\\}
+$$
+where $\\gamma$ is the discount factor. In this problem, we assume $\\gamma = 1$.
+- Alongside the value function, we concurrently develop an optimal policy $\\pi^*$, 
+signifying at each slot whether to park or continue. This decision maximizes
+the expected utility based on the calculated value function, effectively guiding the 
+decision-maker towards the optimal action at each slot.
+
+**Convergence**: This iterative process progresses until we reach the first slot,
+culminating in a comprehensive policy $\\pi^*$ that covers the entire parking lot. This
+policy gives the optimal action (park or continue) for every slot, derives from
+maximizing the expected utility from that point forward.
+"""
+
+st.title("Parking Problem Solver")
+
+st.markdown(problem_explanation)
+
+
+def initialize_parking_lot(N, init_free_prop=0.9, end_free_prop=0.1):
+    slots = ["F"] * N
+    for i in range(N):
+        if random.random() > rho(i, N, init_free_prop, end_free_prop):
+            slots[i] = "T"
+    return slots
+
+
+def rho(i, N, p_start=0.9, p_end=0.1):
+    """Linearly decreasing probability of a slot being free."""
+    return p_start - ((i - 1) / (N - 1)) * (p_start - p_end)
+
+
+def backward_induction(N, c, p_start=0.9, p_end=0.1, reward_func="i"):
+    V_free = np.zeros(N + 1)
+    V_taken = np.zeros(N + 1)
+    policy = [None] * N
+
+    V_free[N] = N
+    V_taken[N] = 0
+
+    for i in range(N - 1, -1, -1):
+        prob_free_next = rho(i + 1, N, p_start, p_end)
+
+        V_continue = (
+            -c + prob_free_next * V_free[i + 1] + (1 - prob_free_next) * V_taken[i + 1]
+        )
+        if reward_func == "i":
+            V_park = i + 1
+        elif reward_func == "2i":
+            V_park = 2 * (i + 1)
+        elif reward_func == "i^2":
+            V_park = (i + 1) ** 2
+        else:
+            raise ValueError("Invalid reward function")
+
+        V_free[i] = max(V_park, V_continue)
+        policy[i] = 1 if V_park >= V_continue else 0
+
+        V_taken[i] = (
+            -c + prob_free_next * V_free[i + 1] + (1 - prob_free_next) * V_taken[i + 1]
+        )
+
+    return V_free[:-1], policy
+
+
+def plot_parking_lot(parking_lot, policy):
+    fig, ax = plt.subplots(figsize=(10, 2))
+    N = len(parking_lot)
+    ax.bar(
+        range(N),
+        [1] * N,
+        color=["green" if s == "F" else "red" for s in parking_lot],
+        edgecolor="black",
+    )
+    ax.set_xticks(range(N))
+    x_labels = []
+    first_park_index = 0
+    for i in range(N):
+        if policy[i]:
+            x_labels.append(f"Park")
+            first_park_index = i
+            break
+        else:
+            x_labels.append(f"Continue")
+    for i in range(first_park_index + 1, N):
+        x_labels.append(f"")
+
+    ax.set_xticklabels(
+        x_labels,
+        rotation=90,
+    )
+    ax.set_yticks([])
+    ax.set_title("Parking Lot Status and Optimal Policy")
+    st.pyplot(fig)
+
+
+N = st.slider("Number of Parking Slots", 1, 100, 20)
+p_start = st.slider("Initial Probability of Free Slot", 0.0, 1.0, 0.9)
+p_end = st.slider("Final Probability of Free Slot", 0.0, p_start, 0.1)
+reward_func = st.selectbox(
+    "Reward Function",
+    ["i", "2i", "i^2"],
+    format_func=lambda x: "Reward = " + x,
+)
+cost_of_continuing = st.slider("Cost of Continuing", 0.0, 10.0, 0.0, step=0.01)
+
+if st.button("Generate Random Parking Lot"):
+    parking_lot = initialize_parking_lot(N, init_free_prop=p_start, end_free_prop=p_end)
+    # save the parking_lot in the session state
+    st.session_state.parking_lot = parking_lot
+    # Display the parking lot
+    fig, ax = plt.subplots(figsize=(10, 2))
+    ax.bar(
+        range(N),
+        [1] * N,
+        color=["green" if s == "F" else "red" for s in parking_lot],
+        edgecolor="black",
+    )
+    ax.set_xticks(range(N))
+    ax.set_xticklabels(
+        [f"{i+1}" for i in range(N)],
+        rotation=90,
+    )
+    ax.set_yticks([])
+    ax.set_title("Parking Lot Status")
+    st.pyplot(fig)
+
+if st.button("Solve Parking Problem"):
+    if "parking_lot" not in st.session_state:
+        st.error("Please generate a parking lot first.")
+        st.stop()
+    values, policy = backward_induction(
+        N, cost_of_continuing, p_start, p_end, reward_func
+    )
+    plot_parking_lot(st.session_state.parking_lot, policy)
+''')
+
 
 
 problem_explanation = """