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State is fully determined by (row, col) position. """ def __init__(self, rows: int, cols: int): self.rows = rows self.cols = cols # Total number of states self.n_states = rows * cols def state_to_idx(self, row: int, col: int) -> int: """Convert (row, col) to a single state index.""" return row * self.cols + col def idx_to_state(self, idx: int) -> Tuple[int, int]: """Convert state index to (row, col).""" return idx // self.cols, idx % self.cols def is_valid_state(self, row: int, col: int) -> bool: """Check if a position is within the grid.""" return 0 <= row < self.rows and 0 <= col < self.cols # ================================================ # Example 2: Continuous State Space - Pendulum # ================================================ @dataclass class PendulumState: """ State representation for inverted pendulum. Continuous state: (angle, angular_velocity) """ theta: float # Angle from vertical (radians), θ ∈ [-π, π] theta_dot: float # Angular velocity, θ̇ ∈ [-8, 8] @property def as_vector(self) -> np.ndarray: """Return state as numpy array for function approximation.""" return np.array([self.theta, self.theta_dot]) @classmethod def from_vector(cls, vec: np.ndarray) -> 'PendulumState': """Construct state from numpy array.""" return cls(theta=vec[0], theta_dot=vec[1]) # ================================================ # Example 3: Hybrid State Space - Robot with Battery # ================================================ class RoomType(Enum): HALLWAY = 0 OFFICE = 1 KITCHEN = 2 CHARGING_STATION = 3 @dataclass class RobotState: """ Hybrid state: discrete room + continuous battery. The state space is the Cartesian product: S = {HALLWAY, OFFICE, KITCHEN, CHARGING_STATION} × [0, 100] """ room: RoomType # Discrete: which room battery_level: float # Continuous: battery percentage def is_critical_battery(self) -> bool: """Check if battery is critically low.""" return self.battery_level < 15.0 def can_perform_task(self) -> bool: """Check if robot has enough battery for a task.""" return self.battery_level >= 20.073:Td8a,import numpy as np from typing import List, Tuple from dataclasses import dataclass # ================================================ # Example 1: Discrete Action Space # ================================================ class DiscreteActionSpace: """ Discrete action space with finite actions. Used in: Gridworld, Atari, Board Games """ def __init__(self, actions: List[str]): self.actions = actions self.n = len(actions) self.action_to_idx = {a: i for i, a in enumerate(actions)} self.idx_to_action = {i: a for i, a in enumerate(actions)} def sample(self) -> int: """Sample a random action (for exploration).""" return np.random.randint(0, self.n) def contains(self, action: int) -> bool: """Check if action is valid.""" return 0 <= action < self.n # Usage gridworld_actions = DiscreteActionSpace(["UP", "DOWN", "LEFT", "RIGHT"]) print(f"Number of actions: {gridworld_actions.n}") # 4 print(f"Random action: {gridworld_actions.sample()}") # ================================================ # Example 2: Continuous Action Space (Box) # ================================================ @dataclass class ContinuousActionSpace: """ Box-shaped continuous action space. Action is a vector a ∈ [low, high] ⊂ ℝ^d Used in: Robotics, Control, Physics Simulation """ low: np.ndarray # Lower bounds for each dimension high: np.ndarray # Upper bounds for each dimension @property def shape(self) -> Tuple[int, ...]: return self.low.shape @property def dim(self) -> int: return self.low.shape[0] def sample(self) -> np.ndarray: """Sample uniformly from the action space.""" return np.random.uniform(self.low, self.high) def clip(self, action: np.ndarray) -> np.ndarray: """Clip action to lie within bounds.""" return np.clip(action, self.low, self.high) def contains(self, action: np.ndarray) -> bool: """Check if action is within bounds.""" return np.all(action >= self.low) and np.all(action <= self.high) # Usage: Robot arm with 7 torque outputs robot_actions = ContinuousActionSpace( low=np.array([-2.0] * 7), # -2 Nm minimum torque high=np.array([2.0] * 7) # +2 Nm maximum torque ) print(f"Action dimension: {robot_actions.dim}") # 7 print(f"Sample action: {robot_actions.sample()}") # ================================================ # Example 3: State-Dependent Action Space # ================================================ class ChessActionSpace: """ Action space that depends on current state. In chess, legal moves depend on piece positions. """ def get_legal_actions(self, board_state: 'ChessBoardState') -> List[str]: """ Return legal moves in algebraic notation. This would call a chess engine in practice. """ # Placeholder - real implementation uses chess rules # The key point: A(s) varies with s if board_state.is_check: # Fewer legal moves when in check return self._get_check_escapes(board_state) return self._get_all_legal_moves(board_state) def _get_check_escapes(self, state) -> List[str]: # Return only moves that escape check pass def _get_all_legal_moves(self, state) -> List[str]: # Return all legal moves pass74:T419,The defining characteristic of MDPs is the **Markov property** (also called the Markov assumption or memorylessness). Formally: $$P(S_{t+1} | S_t, A_t, S_{t-1}, A_{t-1}, \ldots, S_0, A_0) = P(S_{t+1} | S_t, A_t)$$ In words: the future is **conditionally independent** of the past, given the present state. The current state $S_t$ is a *sufficient statistic* for predicting everything about the future—no historical information beyond $S_t$ adds predictive power. **Why is this property so important?** 1. **Computational tractability**: Without the Markov property, we'd need to consider exponentially many histories 2. **Recursive structure**: Enables the Bellman equations, which decompose long-term value into one-step look-aheads 3. **Algorithm design**: Allows value functions $V(s)$ and $Q(s, a)$ to be well-defined If your problem violates the Markov property, you either need to **augment the state** to make it Markov, or use **Partially Observable MDPs (POMDPs)** which explicitly model the agent's uncertainty about the true state.75:T15ef,import numpy as np from typing import Dict, Tuple, List from enum import IntEnum from dataclasses import dataclass """ Complete MDP Specification: Frozen Lake The 4x4 grid: S F F F F H F H F F F H H F F G S = Start, F = Frozen (safe), H = Hole (terminal, reward=0), G = Goal (terminal, reward=1) """ # ================================================ # STATE SPACE: S = {0, 1, 2, ..., 15} # ================================================ # Each cell is a state, indexed row-major: # 0 1 2 3 # 4 5 6 7 # 8 9 10 11 # 12 13 14 15 N_STATES = 16 GRID_SIZE = 4 # Terminal states HOLE_STATES = {5, 7, 11, 12} # States where episode ends (failure) GOAL_STATE = 15 # State where episode ends (success) TERMINAL_STATES = HOLE_STATES | {GOAL_STATE} # ================================================ # ACTION SPACE: A = {0, 1, 2, 3} = {LEFT, DOWN, RIGHT, UP} # ================================================ class Action(IntEnum): LEFT = 0 DOWN = 1 RIGHT = 2 UP = 3 N_ACTIONS = 4 ACTIONS = list(Action) # ================================================ # TRANSITION FUNCTION: P(s' | s, a) # ================================================ # Due to slippery ice: # - P(intended direction | s, a) = 1/3 # - P(slip left | s, a) = 1/3 # - P(slip right | s, a) = 1/3 def get_next_state(state: int, action: Action) -> int: """Compute next state given current state and action (deterministic step).""" if state in TERMINAL_STATES: return state # Terminal states are absorbing row, col = state // GRID_SIZE, state % GRID_SIZE if action == Action.LEFT: col = max(0, col - 1) elif action == Action.RIGHT: col = min(GRID_SIZE - 1, col + 1) elif action == Action.UP: row = max(0, row - 1) elif action == Action.DOWN: row = min(GRID_SIZE - 1, row + 1) return row * GRID_SIZE + col def get_slip_actions(action: Action) -> List[Action]: """Get the three possible outcomes: intended, slip-left, slip-right.""" # Perpendicular actions based on intended direction slip_map = { Action.LEFT: [Action.LEFT, Action.UP, Action.DOWN], Action.RIGHT: [Action.RIGHT, Action.UP, Action.DOWN], Action.UP: [Action.UP, Action.LEFT, Action.RIGHT], Action.DOWN: [Action.DOWN, Action.LEFT, Action.RIGHT], } return slip_map[action] def build_transition_matrix() -> Dict[Tuple[int, int, int], float]: """ Build the complete transition probability table. Returns P[s, a, s'] = P(s' | s, a) for all valid transitions. """ P = {} # (s, a, s') -> probability for s in range(N_STATES): for a in ACTIONS: if s in TERMINAL_STATES: # Terminal states are absorbing (stay with prob 1) P[(s, a, s)] = 1.0 else: # Slippery: 1/3 each for intended and two perpendicular slip_actions = get_slip_actions(a) for slip_a in slip_actions: s_prime = get_next_state(s, slip_a) key = (s, a, s_prime) P[key] = P.get(key, 0.0) + 1/3 return P # ================================================ # REWARD FUNCTION: R(s, a, s') # ================================================ def reward_function(s: int, a: int, s_prime: int) -> float: """ Reward function R(s, a, s'). Reward structure: - Reaching goal (s' = 15): R = 1.0 - Falling in hole: R = 0.0 - All other transitions: R = 0.0 """ if s_prime == GOAL_STATE: return 1.0 return 0.0 # Alternative: Step penalty to encourage faster solutions def reward_with_step_penalty(s: int, a: int, s_prime: int) -> float: """Variant with step penalty to discourage long paths.""" if s_prime == GOAL_STATE: return 1.0 elif s_prime in HOLE_STATES: return -1.0 else: return -0.01 # Small penalty per step # ================================================ # DISCOUNT FACTOR: γ = 0.99 # ================================================ GAMMA = 0.99 # ================================================ # COMPLETE MDP OBJECT # ================================================ @dataclass class FrozenLakeMDP: """Complete MDP specification as a data object.""" n_states: int = N_STATES n_actions: int = N_ACTIONS transition_probs: Dict = None # Built lazily gamma: float = GAMMA terminal_states: set = None def __post_init__(self): self.terminal_states = TERMINAL_STATES self.transition_probs = build_transition_matrix() def P(self, s: int, a: int, s_prime: int) -> float: """Transition probability P(s' | s, a).""" return self.transition_probs.get((s, a, s_prime), 0.0) def R(self, s: int, a: int, s_prime: int) -> float: """Reward R(s, a, s').""" return reward_function(s, a, s_prime) def is_terminal(self, s: int) -> bool: """Check if state is terminal.""" return s in self.terminal_states # Create the MDP frozen_lake = FrozenLakeMDP() # Verify: probabilities from state 0 with action RIGHT print("Transitions from s=0, a=RIGHT:") for s_prime in range(N_STATES): p = frozen_lake.P(0, Action.RIGHT, s_prime) if p > 0: print(f" P(s'={s_prime} | s=0, a=RIGHT) = {p:.4f}") # Output: # Transitions from s=0, a=RIGHT: # P(s'=0 | s=0, a=RIGHT) = 0.3333 (slip UP, hit wall) # P(s'=1 | s=0, a=RIGHT) = 0.3333 (intended direction) # P(s'=4 | s=0, a=RIGHT) = 0.3333 (slip 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Consider the following practical questions that arise when designing RL systems:\n\n- How do we know if an algorithm will converge to the optimal solution?\n- Under what conditions is the optimal solution even well-defined?\n- How can we compare different RL algorithms on a level playing field?\n- How do we identify when a problem is too hard to solve efficiently?\n\nWithout a formal framework, these questions remain unanswerable. We would be reduced to empirical trial-and-error, with no principled way to debug failures or predict behavior on new problems.\n\n**The MDP formalism provides:**\n\n1. **Precise problem specification** — A language to unambiguously describe decision-making problems\n2. **Optimality criteria** — Clear definitions of what it means to \"solve\" a problem\n3. **Algorithmic foundation** — The mathematical structure that algorithms exploit\n4. **Theoretical guarantees** — Conditions under which algorithms provably work"},"type":"text"},{"data":{"icon":"BookOpen","title":"Historical Context","content":"Markov Decision Processes were formalized by Richard Bellman in the 1950s as part of his work on dynamic programming at RAND Corporation. The name honors Andrey Markov's foundational work on stochastic processes. Remarkably, Bellman's framework remains the primary lens through which we understand modern deep RL—a testament to its mathematical elegance and generality.","variant":"note"},"type":"callout"}],"titleType":"number","titleNumber":1},{"id":"mdp-five-tuple","title":"The MDP 5-Tuple: Formal Definition","content":[{"data":{"content":"A **Markov Decision Process** is formally defined as a 5-tuple:\n\n$$\\mathcal{M} = (\\mathcal{S}, \\mathcal{A}, P, R, \\gamma)$$\n\nwhere:\n\n- $\\mathcal{S}$ is the **state space** — the set of all possible states the environment can be in\n- $\\mathcal{A}$ is the **action space** — the set of all possible actions the agent can take\n- $P: \\mathcal{S} \\times \\mathcal{A} \\times \\mathcal{S} \\rightarrow [0, 1]$ is the **transition probability function** (or transition dynamics)\n- $R: \\mathcal{S} \\times \\mathcal{A} \\times \\mathcal{S} \\rightarrow \\mathbb{R}$ is the **reward function**\n- $\\gamma \\in [0, 1]$ is the **discount factor**\n\nLet's examine each component with the rigor it deserves."},"type":"text"},{"data":{"rows":[["State Space","$$\\mathcal{S}$","Set (finite or infinite)","All possible configurations of the environment"],["Action Space","$$\\mathcal{A}$","Set (finite or infinite)","All possible actions available to the agent"],["Transition Function","$$P(s'|s, a)$","Probability distribution","Probability of reaching state $s'$ from state $s$ after action $a$"],["Reward Function","$$R(s, a, s')$","Real-valued function","Immediate reward received for a transition"],["Discount Factor","$$\\gamma$","Scalar in $[0, 1]$","Weight given to future rewards relative to immediate rewards"]],"caption":"MDP Components at a Glance","headers":["Component","Symbol","Type","Description"]},"type":"table"},{"data":{"icon":"AlertCircle","title":"Notation Variations","content":"Different textbooks use different notations. Some write $R(s, a)$ for expected reward, some include the initial state distribution $\\mu_0$ making it a 6-tuple, and some use $T$ instead of $P$ for transitions. The core concepts remain identical—always check the notation conventions of your source.","variant":"warning"},"type":"callout"}],"titleType":"number","titleNumber":2},{"id":"state-space-deep-dive","title":"The State Space: Capturing World Configurations","content":[{"data":{"content":"The **state space** $\\mathcal{S}$ represents all possible configurations of the environment. Each state $s \\in \\mathcal{S}$ is a complete description of the environment at a given moment—containing all information relevant to predicting future behavior.\n\n**Types of State Spaces:**\n\n**Finite state spaces** contain a countable number of states. Examples:\n- Chess: Each board configuration is a state (approximately $10^{43}$ legal positions)\n- Gridworld: Each cell position is a state (rows × columns states)\n- Inventory management: Integer inventory levels\n\n**Continuous state spaces** contain uncountably many states. Examples:\n- Robot arm: Joint angles and velocities in $\\mathbb{R}^{14}$\n- Autonomous driving: Position, velocity, and sensor readings\n- Financial trading: Portfolio allocations and market prices\n\n**Hybrid state spaces** mix discrete and continuous components:\n- Mobile robot with discrete locations and continuous battery level\n- Game with discrete levels but continuous player stats"},"type":"text"},{"data":{"code":"$72","fileName":"state_space_examples.py","languages":[{"label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"},{"data":{"icon":"Lightbulb","title":"State Design Principle","content":"The state must satisfy the **Markov property**: it must contain all information needed to predict the future, independent of the past. If your state representation violates this—e.g., the agent needs to remember what happened 10 steps ago—you need to either augment the state (e.g., stack frames) or use architectures designed for partial observability (POMDPs, recurrent networks).","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":3},{"id":"action-space-deep-dive","title":"The Action Space: Agent Capabilities","content":[{"data":{"content":"The **action space** $\\mathcal{A}$ defines all actions available to the agent. In general MDPs, the action space may depend on the current state: $\\mathcal{A}(s)$ denotes actions available in state $s$.\n\n**Discrete Action Spaces:**\n\n$$\\mathcal{A} = \\{a_1, a_2, \\ldots, a_n\\}$$\n\nExamples:\n- Atari games: 18 discrete button combinations\n- Board games: Legal moves from current position\n- Recommendation: Select from $n$ items to recommend\n\n**Continuous Action Spaces:**\n\n$$\\mathcal{A} \\subseteq \\mathbb{R}^d$$\n\nExamples:\n- Robot control: Joint torques in $\\mathbb{R}^7$\n- Autonomous driving: Steering angle and throttle in $\\mathbb{R}^2$\n- Portfolio management: Asset allocation weights\n\n**Multi-Discrete and Mixed Spaces:**\n\nSome problems have factored action spaces—multiple discrete choices or combinations of discrete and continuous components."},"type":"text"},{"data":{"left":[{"data":{"items":["**Finite choices**: Navigate {N, S, E, W}","**Natural for games**: Button presses, card plays","**Easier exploration**: Enumerate all possibilities","**Q-learning friendly**: Q(s, a) table feasible","**Argmax tractable**: Find best action exactly"],"title":{"icon":"Target","text":"Discrete Actions","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"}],"right":[{"data":{"items":["**Infinite choices**: Torque ∈ [-2, 2] Nm","**Natural for control**: Physics-based systems","**Exploration challenge**: Can't enumerate","**Requires function approx**: No tabular Q","**Argmax hard**: Often use gradient-based optimization"],"title":{"icon":"Zap","text":"Continuous Actions","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"}]},"type":"two-column"},{"data":{"code":"$73","fileName":"action_space_examples.py","languages":[{"label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"}],"titleType":"number","titleNumber":4},{"id":"markov-property","title":"The Markov Property: Memory-Free Dynamics","content":[{"data":{"content":"$74"},"type":"text"},{"data":{"input":"state = [frame_t-3, frame_t-2, frame_t-1, frame_t]","title":"Why Pong Needs Frame Stacking","output":"Ball position AND velocity inferrable from state sequence","description":"Consider the Atari game Pong. A single frame shows the ball's position but not its velocity. Without velocity, we can't predict where the ball will be next—it might be moving left or right! Two identical frames could lead to completely different next frames depending on the hidden velocity.\n\n**The solution**: Stack 4 consecutive frames as the state. Now the state contains positional information across time, from which velocity can be inferred. The stacked state satisfies the Markov property: it contains sufficient information to predict the future.","explanation":"This pattern is ubiquitous in deep RL: when single observations don't satisfy Markov, we augment with history (frame stacking) or use recurrent architectures that maintain hidden memory."},"type":"example"},{"data":{"icon":"AlertCircle","title":"Markov Violations in Practice","content":"Many real-world problems are not naturally Markov: a robot might need to remember where it was to infer where it is (localization), a dialogue agent needs conversational history, a trading agent needs past price trends. Recognizing when your problem violates Markov—and designing state representations that restore it—is a critical RL engineering skill.","variant":"warning"},"type":"callout"}],"titleType":"number","titleNumber":5},{"id":"complete-mdp-specification","title":"Complete MDP Specification: Worked Example","content":[{"data":{"content":"Let's put everything together with a complete MDP specification for a classic problem: the **Frozen Lake** environment.\n\n**Problem Description:**\n\nAn agent navigates a 4×4 frozen lake. The surface has:\n- Safe frozen tiles (F): Agent can stand here\n- Holes (H): Agent falls in and episode ends (failure)\n- Start tile (S): Where the agent begins\n- Goal tile (G): Reaching here ends the episode (success)\n\nThe catch: the ice is slippery! When the agent chooses an action, it only moves in that direction with probability 1/3—it might slip sideways."},"type":"text"},{"data":{"code":"$75","fileName":"frozen_lake_mdp.py","languages":[{"label":"Python","value":"python"}],"showLineNumbers":true},"type":"code"},{"data":{"icon":"CheckCircle","title":"What Makes This an MDP","content":"Notice how every component is precisely specified: states are numbered 0-15, actions are the 4 directions, transitions are stochastic (1/3 probability each outcome), rewards are sparse (1 at goal, 0 elsewhere), and γ=0.99. Given this specification, we can apply any MDP algorithm—value iteration, policy iteration, Q-learning—without ambiguity.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":6},{"id":"finite-vs-infinite","title":"Finite vs. Infinite Horizon MDPs","content":[{"data":{"content":"An MDP's **horizon** refers to the length of the decision-making process:\n\n**Finite Horizon MDPs:**\n- Episode terminates after exactly $T$ steps\n- Objective: Maximize $\\sum_{t=0}^{T-1} R_t$ (often no discounting needed)\n- Optimal policy may be *time-dependent*: $\\pi^*_t(s)$ varies with $t$\n- Example: Game with fixed number of moves, planning a trip with departure date\n\n**Infinite Horizon MDPs:**\n- Episode can run indefinitely (or until terminal state)\n- Objective: Maximize $\\sum_{t=0}^{\\infty} \\gamma^t R_t$ (discounting required for convergence)\n- Optimal policy is *stationary*: $\\pi^*(s)$ independent of time\n- Example: Continuous control, ongoing resource management\n\n**Why does discounting matter for infinite horizons?**\n\nWithout discounting ($\\gamma = 1$) in infinite horizon settings, the sum of rewards could diverge to infinity, making comparison meaningless. The discount factor ensures:\n\n$$\\sum_{t=0}^{\\infty} \\gamma^t R_t \\leq \\sum_{t=0}^{\\infty} \\gamma^t R_{max} = \\frac{R_{max}}{1 - \\gamma} < \\infty$$"},"type":"text"},{"data":{"rows":[["Episode length","Fixed $T$ steps","Unbounded or until terminal"],["Discounting","Optional (often $\\gamma=1$)","Required ($\\gamma < 1$)"],["Optimal policy","Time-dependent $\\pi^*_t(s)$","Stationary $\\pi^*(s)$"],["Value function","$$V^\\pi_t(s)$ depends on time","$$V^\\pi(s)$ time-independent"],["Computation","Backward induction from $T$","Fixed point iteration"],["Applications","Games, planning with deadline","Control, ongoing optimization"]],"caption":"Comparing Horizon Types","headers":["Property","Finite Horizon","Infinite Horizon"]},"type":"table"}],"titleType":"number","titleNumber":7},{"id":"episodic-vs-continuing","title":"Episodic vs. Continuing Tasks","content":[{"data":{"content":"A separate but related distinction concerns whether the task naturally breaks into **episodes**:\n\n**Episodic Tasks:**\n- Experience naturally divides into independent episodes\n- Each episode starts fresh, ends in a terminal state\n- Examples: Games (each game is an episode), robot trials, dialogue sessions\n- Agent can learn from many episodes, each a complete trajectory\n\n**Continuing Tasks:**\n- No natural end—interaction continues indefinitely\n- Must consider the single, never-ending experience stream\n- Examples: Process control, lifelong robotics, server load balancing\n- Often use *average reward* formulation instead of discounted sum\n\n**Unified Notation (Sutton & Barto):**\n\nTo handle both cases elegantly, we can treat episode termination as a transition to an **absorbing state** $s_{\\text{terminal}}$ with $R=0$ and self-loops. This makes episodic tasks a special case of continuing tasks, simplifying algorithm design."},"type":"text"},{"data":{"icon":"Lightbulb","title":"Practical Implications","content":"In practice, most RL implementations use episodic formulations even for continuing tasks by artificially truncating episodes after T steps, then bootstrapping from the value estimate at truncation. This gives the learning benefits of episodic resets while approximating the continuing objective.","variant":"tip"},"type":"callout"}],"titleType":"number","titleNumber":8},{"id":"summary","title":"Summary: The MDP Foundation","content":[{"data":{"content":"We have established the formal framework that will support every algorithm and concept in reinforcement learning. Let's consolidate the key insights:"},"type":"text"},{"data":{"type":"ordered","items":["**MDPs are defined by the 5-tuple** $(\\mathcal{S}, \\mathcal{A}, P, R, \\gamma)$ — states, actions, transitions, rewards, discount","**The Markov property is fundamental** — current state is sufficient to predict the future; enables recursive decomposition","**State spaces can be discrete or continuous** — the choice affects algorithm selection and computational complexity","**Action spaces may depend on state** — not all actions are legal in all states (e.g., chess moves)","**Horizon matters** — finite horizon allows time-dependent policies; infinite horizon requires discounting and yields stationary policies","**Episodic vs. continuing affects formulation** — but both can be handled with absorbing state notation"],"title":{"icon":"CheckCircle","text":"Key Takeaways","type":"icon","color":"primary","variant":"outline-circle"},"variant":"accent"},"type":"list"},{"data":{"content":"**What's Next:**\n\nWith the MDP structure established, we turn next to **state transitions**—examining how the transition probability function $P(s'|s,a)$ governs environment dynamics, and how agents can reason about uncertainty in outcomes."},"type":"text"},{"data":{"icon":"CheckCircle","title":"Page Complete","content":"You now understand the formal mathematical framework of Markov Decision Processes—the foundation upon which all reinforcement learning rests. This 5-tuple formalism transforms intuitive notions of sequential decision-making into precise mathematical objects amenable to rigorous analysis and algorithm design.","variant":"success"},"type":"callout"}],"titleType":"number","titleNumber":9}],"isLocked":false},{"id":"seed_105b4da2a7df57cb514aaecd9d49f025","title":"State Transitions","order":1,"content":null,"isLocked":true},{"id":"seed_45cdf65b967fd294b81f007835337ee3","title":"Reward Signals","order":2,"content":null,"isLocked":true},{"id":"seed_04d8628cfd9f5cfa7043d96e60d0572f","title":"Discount Factor","order":3,"content":null,"isLocked":true},{"id":"seed_f5dc224b84809ea9eadbede9875a855e","title":"Bellman Equations","order":4,"content":null,"isLocked":true}]}},"dataUpdateCount":1,"dataUpdatedAt":1792287736245,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getModule"],{"input":{"slug":"markov-decision-processes"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getModule\"],{\"input\":{\"slug\":\"markov-decision-processes\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287736268,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287736246,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["learningContent","getProgress"],{"input":{"moduleId":"seed_acab1927294332a5e5a8454558b39c4b"},"type":"query"}],"queryHash":"[[\"learningContent\",\"getProgress\"],{\"input\":{\"moduleId\":\"seed_acab1927294332a5e5a8454558b39c4b\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287736268,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287736246,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","get"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_f5dc224b84809ea9eadbede9875a855e"},"type":"query"}],"queryHash":"[[\"feedback\",\"get\"],{\"input\":{\"contentId\":\"seed_f5dc224b84809ea9eadbede9875a855e\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287736268,"state":{"data":{"json":null},"dataUpdateCount":1,"dataUpdatedAt":1792287736246,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["feedback","getUserLike"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_f5dc224b84809ea9eadbede9875a855e"},"type":"query"}],"queryHash":"[[\"feedback\",\"getUserLike\"],{\"input\":{\"contentId\":\"seed_f5dc224b84809ea9eadbede9875a855e\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287736268,"state":{"data":{"json":[]},"dataUpdateCount":1,"dataUpdatedAt":1792287736246,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"success","fetchStatus":"idle"},"queryKey":[["highlights","getHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_f5dc224b84809ea9eadbede9875a855e"},"type":"query"}],"queryHash":"[[\"highlights\",\"getHighlights\"],{\"input\":{\"contentId\":\"seed_f5dc224b84809ea9eadbede9875a855e\",\"contentType\":\"LEARNING_PAGE\"},\"type\":\"query\"}]"},{"dehydratedAt":1792287736268,"state":{"data":"$undefined","dataUpdateCount":0,"dataUpdatedAt":0,"error":null,"errorUpdateCount":0,"errorUpdatedAt":0,"fetchFailureCount":0,"fetchFailureReason":null,"fetchMeta":null,"isInvalidated":false,"status":"pending","fetchStatus":"fetching"},"queryKey":[["highlights","getPopularHighlights"],{"input":{"contentType":"LEARNING_PAGE","contentId":"seed_f5dc224b84809ea9eadbede9875a855e","minCount":2},"type":"query"}],"queryHash":"[[\"highlights\",\"getPopularHighlights\"],{\"input\":{\"contentId\":\"seed_f5dc224b84809ea9eadbede9875a855e\",\"contentType\":\"LEARNING_PAGE\",\"minCount\":2},\"type\":\"query\"}]","promise":"$@76"}]},"children":"$L77"}]