Artificial Intelligence: A Modern Approach

AIMA Python file:

"""Markov Decision Processes (Chapter 17)

First we define an MDP, and the special case of a GridMDP, in which
states are laid out in a 2-dimensional grid.  We also represent a policy
as a dictionary of {state:action} pairs, and a Utility function as a
dictionary of {state:number} pairs.  We then define the value_iteration
and policy_iteration algorithms."""

from utils import *

class MDP:
    """A Markov Decision Process, defined by an initial state, transition model,
    and reward function. We also keep track of a gamma value, for use by
    algorithms. The transition model is represented somewhat differently from
    the text.  Instead of T(s, a, s') being  probability number for each
    state/action/state triplet, we instead have T(s, a) return a list of (p, s')
    pairs.  We also keep track of the possible states, terminal states, and
    actions for each state. [page 615]"""

    def __init__(self, init, actlist, terminals, gamma=.9):
        update(self, init=init, actlist=actlist, terminals=terminals,
               gamma=gamma, states=set(), reward={})

    def R(self, state):
        "Return a numeric reward for this state."
        return self.reward[state]

    def T(state, action):
        """Transition model.  From a state and an action, return a list
        of (result-state, probability) pairs."""

    def actions(self, state):
        """Set of actions that can be performed in this state.  By default, a
        fixed list of actions, except for terminal states. Override this
        method if you need to specialize by state."""
        if state in self.terminals:
            return [None]
            return self.actlist

class GridMDP(MDP):
    """A two-dimensional grid MDP, as in [Figure 17.1].  All you have to do is
    specify the grid as a list of lists of rewards; use None for an obstacle
    (unreachable state).  Also, you should specify the terminal states.
    An action is an (x, y) unit vector; e.g. (1, 0) means move east."""
    def __init__(self, grid, terminals, init=(0, 0), gamma=.9):
        grid.reverse() ## because we want row 0 on bottom, not on top
        MDP.__init__(self, init, actlist=orientations,
                     terminals=terminals, gamma=gamma)
        update(self, grid=grid, rows=len(grid), cols=len(grid[0]))
        for x in range(self.cols):
            for y in range(self.rows):
                self.reward[x, y] = grid[y][x]
                if grid[y][x] is not None:
                    self.states.add((x, y))

    def T(self, state, action):
        if action == None:
            return [(0.0, state)]
            return [(0.8, self.go(state, action)),
                    (0.1, self.go(state, turn_right(action))),
                    (0.1, self.go(state, turn_left(action)))]

    def go(self, state, direction):
        "Return the state that results from going in this direction."
        state1 = vector_add(state, direction)
        return if_(state1 in self.states, state1, state)

    def to_grid(self, mapping):
        """Convert a mapping from (x, y) to v into a [[..., v, ...]] grid."""
        return list(reversed([[mapping.get((x,y), None)
                               for x in range(self.cols)]
                              for y in range(self.rows)]))

    def to_arrows(self, policy):
        chars = {(1, 0):'>', (0, 1):'^', (-1, 0):'<', (0, -1):'v', None: '.'}
        return self.to_grid(dict([(s, chars[a]) for (s, a) in policy.items()]))

Fig[17,1] = GridMDP([[-0.04, -0.04, -0.04, +1], [-0.04, None, -0.04, -1], [-0.04, -0.04, -0.04, -0.04]], terminals=[(3, 2), (3, 1)])
value_iteration(mdp, epsilon=0.001): "Solving an MDP by value iteration. [Fig. 17.4]" U1 = dict([(s, 0) for s in mdp.states]) R, T, gamma = mdp.R, mdp.T, mdp.gamma while True: U = U1.copy() delta = 0 for s in mdp.states: U1[s] = R(s) + gamma * max([sum([p * U[s1] for (p, s1) in T(s, a)]) for a in mdp.actions(s)]) delta = max(delta, abs(U1[s] - U[s])) if delta < epsilon * (1 - gamma) / gamma: return U def best_policy(mdp, U): """Given an MDP and a utility function U, determine the best policy, as a mapping from state to action. (Equation 17.4)""" pi = {} for s in mdp.states: pi[s] = argmax(mdp.actions(s), lambda a:expected_utility(a, s, U, mdp)) return pi def expected_utility(a, s, U, mdp): "The expected utility of doing a in state s, according to the MDP and U." return sum([p * U[s1] for (p, s1) in mdp.T(s, a)])
policy_iteration(mdp): "Solve an MDP by policy iteration [Fig. 17.7]" U = dict([(s, 0) for s in mdp.states]) pi = dict([(s, random.choice(mdp.actions(s))) for s in mdp.states]) while True: U = policy_evaluation(pi, U, mdp) unchanged = True for s in mdp.states: a = argmax(mdp.actions(s), lambda a: expected_utility(a,s,U,mdp)) if a != pi[s]: pi[s] = a unchanged = False if unchanged: return pi def policy_evaluation(pi, U, mdp, k=20): """Return an updated utility mapping U from each state in the MDP to its utility, using an approximation (modified policy iteration).""" R, T, gamma = mdp.R, mdp.T, mdp.gamma for i in range(k): for s in mdp.states: U[s] = R(s) + gamma * sum([p * U[s] for (p, s1) in T(s, pi[s])]) return U

AI: A Modern Approach by Stuart Russell and Peter NorvigModified: Jul 18, 2005