ARTIFICIAL INTELLIGENCE

UNIT – II (Problem Solving by Search-II and Propositional Logic)

Chapter-I: Adversarial Search: Games, Optimal Decisions in Games, Alpha–Beta Pruning,

Imperfect Real-Time Decisions.

ADVERSARIAL SEARCH
 AI Adversarial search: Adversarial search is a game-playing technique where the agents are
surrounded by a competitive environment. A conflicting goal is given to the agents (multiagent).
These agents compete with one another and try to defeat one another to win the game.
 Such conflicting goals give rise to the adversarial search. Here, game-playing means discussing
those games where human intelligence and logic factor is used, excluding other factors such as luck
factor. Tic-tac-toe, chess, checkers, etc., are such type of games where no luck factor works, only
mind works.
 Mathematically, this search is based on the concept of ‘Game Theory.’ According to game theory, a
game is played between two players. To complete the game, one has to win the game and the other
loses automatically.

 Adversarial search is a search, where we examine the problem which arises when we try to plan
ahead of the world and other agents are planning against us.
 In previous topics, we have studied the search strategies which are only associated with a single
agent that aims to find the solution which often expressed in the form of a sequence of actions.
 But there might be some situations where more than one agent is searching for the solution in the
same search space, and this situation usually occurs in game playing.
 The environment with more than one agent is termed as multi-agent environment, in which each
agent is an opponent of other agent and playing against each other. Each agent needs to consider the
action of other agent and effect of that action on their performance.
Mr. Mohammed Afzal, Asst. Professor in AIML
Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
 So, Searches in which two or more players with conflicting goals are trying to explore the same
search space for the solution, are called adversarial searches, often known as Games.
 Games are modelled as a Search problem and heuristic evaluation function, and these are the two
main factors which help to model and solve games in AI.

Types of Games in AI:

Deterministic Chance Moves

Perfect information Chess, Checkers, go, Othello Backgammon, monopoly
Imperfect information Battleships, blind, tic-tac-toe Bridge, poker, scrabble, nuclear war

o Perfect information: A game with the perfect information is that in which agents can look into the
complete board. Agents have all the information about the game, and they can see each other moves
also. Examples are Chess, Checkers, Go, etc.
o Imperfect information: If in a game, agents do not have all information about the game and not
aware with what's going on, such type of games are called the game with imperfect information,
such as tic-tac-toe, Battleship, blind, Bridge, etc.
o Deterministic games: Deterministic games are those games which follow a strict pattern and set of
rules for the games, and there is no randomness associated with them. Examples are chess, Checkers,
Go, tic-tac-toe, etc.
o Non-deterministic games: Non-deterministic are those games which have various unpredictable
events and has a factor of chance or luck. This factor of chance or luck is introduced by either dice
or cards. These are random, and each action response is not fixed. Such games are also called as
stochastic games.
Example: Backgammon, Monopoly, Poker, etc.

Zero-Sum Game
o Zero-sum games are adversarial search which involves pure competition.
o In Zero-sum game each agent's gain or loss of utility is exactly balanced by the losses or gains of
utility of another agent.
o One player of the game try to maximize one single value, while other player tries to minimize it.
o Each move by one player in the game is called as ply.
o Chess and tic-tac-toe are examples of a Zero-sum game.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Zero-sum game: Embedded thinking

The Zero-sum game involved embedded thinking in which one agent or player is trying to figure out:

o What to do.
o How to decide the move.
o Needs to think about his opponent as well.
o The opponent also thinks what to do.

Each of the players is trying to find out the response of his opponent to their actions. This requires
embedded thinking or backward reasoning to solve the game problems in AI.

Formalization of the problem (Elements of Game Playing search):

To play a game, we use a game tree to know all the possible choices and to pick the best one out. A game
can be defined as a type of search in AI which can be formalized of the following elements:
 S0: It is the initial state from where a game begins.
 PLAYER (s): It defines which player is having the current turn to make a move in the state.
 ACTIONS (s): It defines the set of legal moves to be used in a state.
 RESULT (s, a): It is a transition model which defines the result of a move.
 TERMINAL-TEST (s): It defines that the game has ended and returns true.
 UTILITY (s,p): It defines the final value with which the game has ended. This function is also
known as Objective function or Payoff function. The price which the winner will get i.e.
o (-1): If the PLAYER loses.
o (+1): If the PLAYER wins.
o (0): If there is a draw between the PLAYERS.

For example, in chess, tic-tac-toe, we have two or three possible outcomes. Either to win, to lose, or to draw
the match with values +1, -1 or 0.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Game tree:
Let’s understand the working of the elements with the help of a game tree designed for tic-tac-toe. Here,
the node represents the game state and edges represent the moves taken by the players. Game tree involves
initial state, actions function, and result Function.

Example: Tic-Tac-Toe game tree:

The following figure is showing part of the game-tree for tic-tac-toe game. Following are some key points of
the game:
o There are two players MAX and MIN.
o Players have an alternate turn and start with MAX.
o MAX maximizes the result of the game tree.

o MIN minimizes the result.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Example Explanation:
o From the initial state, MAX has 9 possible moves as he starts first. MAX place x and MIN place o,
and both player plays alternatively until we reach a leaf node where one player has three in a row or
all squares are filled.
o Both players will compute each node, minimax, the minimax value which is the best achievable
utility against an optimal adversary.
o Suppose both the players are well aware of the tic-tac-toe and playing the best play. Each player is
doing his best to prevent another one from winning. MIN is acting against Max in the game.
o So, in the game tree, we have a layer of Max, a layer of MIN, and each layer is called as Ply. Max
place x, then MIN puts o to prevent Max from winning, and this game continues until the terminal
node.
o In this either MIN wins, MAX wins, or it's a draw. This game-tree is the whole search space of
possibilities that MIN and MAX are playing tic-tac-toe and taking turns alternately.
Hence adversarial Search for the minimax procedure works as follows:
o It aims to find the optimal strategy for MAX to win the game.
o It follows the approach of Depth-first search.
o In the game tree, optimal leaf node could appear at any depth of the tree.
o Propagate the minimax values up to the tree until the terminal node discovered.
In a given game tree, the optimal strategy can be determined from the minimax value of each node, which
can be written as MINIMAX(n). MAX prefer to move to a state of maximum value and MIN prefer to move
to a state of minimum value then:

Mini-Max Algorithm in Artificial Intelligence

o Mini-max algorithm is a recursive or backtracking algorithm which is used in decision-making and

game theory. It provides an optimal move for the player assuming that opponent is also playing
optimally.
o Mini-Max algorithm uses recursion to search through the game-tree.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
o Min-Max algorithm is mostly used for game playing in AI. Such as Chess, Checkers, tic-tac-toe, go,
and various tow-players game. This Algorithm computes the minimax decision for the current state.
o In this algorithm two players play the game, one is called MAX and other is called MIN.
o Both the players fight it as the opponent player gets the minimum benefit while they get the
maximum benefit.
o Both Players of the game are opponent of each other, where MAX will select the maximized value
and MIN will select the minimized value.
o The minimax algorithm performs a depth-first search algorithm for the exploration of the complete
game tree.
o The minimax algorithm proceeds all the way down to the terminal node of the tree, then backtrack
the tree as the recursion.

Pseudo-code for MinMax Algorithm:

function minimax(node, depth, maximizingPlayer) is

if depth ==0 or node is a terminal node then
return static evaluation of node

if MaximizingPlayer then // for Maximizer Player

maxEva= -infinity
for each child of node do
eva= minimax(child, depth-1, false)
maxEva= max(maxEva,eva) //gives Maximum of the values
return maxEva

else // for Minimizer player

minEva= +infinity
for each child of node do
eva= minimax(child, depth-1, true)
minEva= min(minEva, eva) //gives minimum of the values
return minEva

Initial call: Minimax(node, 3, true)

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Working of Min-Max Algorithm:
o The working of the minimax algorithm can be easily described using an example. Below we have
taken an example of game-tree which is representing the two-player game.
o In this example, there are two players one is called Maximizer and other is called Minimizer.
o Maximizer will try to get the Maximum possible score, and Minimizer will try to get the minimum
possible score.
o This algorithm applies DFS, so in this game-tree, we have to go all the way through the leaves to
reach the terminal nodes.
o At the terminal node, the terminal values are given so we will compare those value and backtrack the
tree until the initial state occurs.
o Following are the main steps involved in solving the two-player game tree:

Step-1: In the first step, the algorithm generates the entire game-tree and apply the utility function to get the
utility values for the terminal states. In the below tree diagram, let's take A is the initial state of the tree.
Suppose maximizer takes first turn which has worst-case initial value =- infinity, and minimizer will take next
turn which has worst-case initial value = +infinity.

Step 2: Now, first we find the utilities value for the Maximizer, its initial value is -∞, so we will compare
each value in terminal state with initial value of Maximizer and determines the higher nodes values. It will
find the maximum among all.
o For node D max(-1,- -∞) => max(-1,4)= 4
o For Node E max(2, -∞) => max(2, 6)= 6

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
o For Node F max(-3, -∞) => max(-3,-5) = -3
o For node G max(0, -∞) = max(0, 7) = 7

Step 3: In the next step, it's a turn for minimizer, so it will compare all nodes value with +∞, and will find
the 3rd layer node values.

o For node B= min(4,6) = 4

o For node C= min (-3, 7) = -3

Step 4: Now it's a turn for Maximizer, and it will again choose the maximum of all nodes value and find the
maximum value for the root node. In this game tree, there are only 4 layers, hence we reach immediately to
the root node, but in real games, there will be more than 4 layers.
o For node A max(4, -3)= 4

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
That was the complete workflow of the minimax two player game.

Properties of Mini-Max algorithm:

o Complete- Min-Max algorithm is Complete. It will definitely find a solution (if exist), in the finite
search tree.
o Optimal- Min-Max algorithm is optimal if both opponents are playing optimally.
o Time complexity- As it performs DFS for the game-tree, so the time complexity of Min-Max
algorithm is O(bm), where b is branching factor of the game-tree, and m is the maximum depth of the
tree.
o Space Complexity- Space complexity of Mini-max algorithm is also similar to DFS which is O(bm).

Limitation of the minimax Algorithm:

 The main drawback of the minimax algorithm is that it gets really slow for complex games such as
Chess, go, etc.
 This type of games has a huge branching factor, and the player has lots of choices to decide.
 This limitation of the minimax algorithm can be improved from alpha-beta pruning which we have
discussed in the next topic.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
ALPHA-BETA PRUNING

o Alpha-beta pruning is a modified version of the minimax algorithm. It is an optimization technique

for the minimax algorithm.
o As we have seen in the minimax search algorithm that the number of game states it has to examine
are exponential in depth of the tree. Since we cannot eliminate the exponent, but we can cut it to
half. Hence there is a technique by which without checking each node of the game tree we can
compute the correct minimax decision, and this technique is called pruning. This involves two
threshold parameter Alpha and beta for future expansion, so it is called alpha-beta pruning. It is
also called as Alpha-Beta Algorithm.
o Alpha-beta pruning can be applied at any depth of a tree, and sometimes it not only prune the tree
leaves but also entire sub-tree.
o The two-parameter can be defined as:
1. Alpha: The best (highest-value) choice we have found so far at any point along the path of
Maximizer. The initial value of alpha is -∞.
2. Beta: The best (lowest-value) choice we have found so far at any point along the path of
Minimizer. The initial value of beta is +∞.
o The Alpha-beta pruning to a standard minimax algorithm returns the same move as the standard
algorithm does, but it removes all the nodes which are not really affecting the final decision but making
algorithm slow. Hence by pruning these nodes, it makes the algorithm fast.
Condition for Alpha-beta pruning:
The main condition which required for alpha-beta pruning is:
α>=β

Key points about alpha-beta pruning:

o The Max player will only update the value of alpha.
o The Min player will only update the value of beta.
o While backtracking the tree, the node values will be passed to upper nodes instead of values of alpha
and beta.
o We will only pass the alpha, beta values to the child nodes.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Pseudo-code for Alpha-beta Pruning:

function minimax(node, depth, alpha, beta, maximizingPlayer) is

if depth ==0 or node is a terminal node then
return static evaluation of node

if MaximizingPlayer then // for Maximizer Player

maxEva= -infinity
for each child of node do
eva= minimax(child, depth-1, alpha, beta, False)
maxEva= max(maxEva, eva)
alpha= max(alpha, maxEva)
if beta<=alpha
break
return maxEva

else // for Minimizer player

minEva= +infinity
for each child of node do
eva= minimax(child, depth-1, alpha, beta, true)
minEva= min(minEva, eva)
beta= min(beta, eva)
if beta<=alpha
break
return minEva

Working of Alpha-Beta Pruning:

Let's take an example of two-player search tree to understand the working of Alpha-beta pruning.
Step 1: At the first step the, Max player will start first move from node A where α= -∞ and β= +∞, these
value of alpha and beta passed down to node B where again α= -∞ and β= +∞, and Node B passes the same
value to its child D.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Step 2: At Node D, the value of α will be calculated as its turn for Max. The value of α is compared with
firstly 2 and then 3, and the max (2, 3) = 3 will be the value of α at node D and node value will also 3.
Step 3: Now algorithm backtrack to node B, where the value of β will change as this is a turn of Min, Now
β= +∞, will compare with the available subsequent nodes value, i.e. min (∞, 3) = 3, hence at node B now α=
-∞, and β= 3.

In the next step, algorithm traverse the next successor of Node B which is node E, and the values of α= -∞,
and β= 3 will also be passed.
Step 4: At node E, Max will take its turn, and the value of alpha will change. The current value of alpha will
be compared with 5, so max (-∞, 5) = 5, hence at node E α= 5 and β= 3, where α>=β, so the right successor
of E will be pruned, and algorithm will not traverse it, and the value at node E will be 5.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Step 5: At next step, algorithm again backtrack the tree, from node B to node A. At node A, the value of
alpha will be changed the maximum available value is 3 as max (-∞, 3)= 3, and β= +∞, these two values
now passes to right successor of A which is Node C. At node C, α=3 and β= +∞, and the same values will
be passed on to node F.
Step 6: At node F, again the value of α will be compared with left child which is 0, and max(3,0)= 3, and
then compared with right child which is 1, and max(3,1)= 3 still α remains 3, but the node value of F will
become 1.

Step 7: Node F returns the node value 1 to node C, at C α= 3 and β= +∞, here the value of beta will be
changed, it will compare with 1 so min (∞, 1) = 1. Now at C, α=3 and β= 1, and again it satisfies the
condition α>=β, so the next child of C which is G will be pruned, and the algorithm will not compute the
entire sub-tree G.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
Step 8: C now returns the value of 1 to A here the best value for A is max (3, 1) = 3. Following is the final
game tree which is the showing the nodes which are computed and nodes which has never computed. Hence
the optimal value for the maximizer is 3 for this example.

Move Ordering in Alpha-Beta pruning:

The effectiveness of alpha-beta pruning is highly dependent on the order in which each node is examined.
Move order is an important aspect of alpha-beta pruning.
It can be of two types:
o Worst ordering: In some cases, alpha-beta pruning algorithm does not prune any of the leaves of
the tree, and works exactly as minimax algorithm. In this case, it also consumes more time because
of alpha-beta factors, such a move of pruning is called worst ordering. In this case, the best move
occurs on the right side of the tree. The time complexity for such an order is O(bm).
o Ideal ordering: The ideal ordering for alpha-beta pruning occurs when lots of pruning happens in
the tree, and best moves occur at the left side of the tree. We apply DFS hence it first search left of
Mr. Mohammed Afzal, Asst. Professor in AIML
Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
the tree and go deep twice as minimax algorithm in the same amount of time. Complexity in ideal
ordering is O(bm/2).

Rules to find good ordering:

Following are some rules to find good ordering in alpha-beta pruning:
o Occur the best move from the shallowest node.
o Order the nodes in the tree such that the best nodes are checked first.
o Use domain knowledge while finding the best move. Ex: for Chess, try order: captures first, then
threats, then forward moves, backward moves.
o We can bookkeep the states, as there is a possibility that states may repeat.

IMPERFECT REAL-TIME DECISIONS

 The minimax algorithm generates the entire game search space, whereas the alpha–beta algorithm
allows us to prune large parts of it.
 However, alpha–beta still has to search all the way to terminal states for at least a portion of the
search space. This depth is usually not practical, because moves must be made in a reasonable
amount of time—typically a few minutes at most.
 The suggestion here is to alter minimax or alpha–beta in two ways: replace the utility function by a
heuristic evaluation function EVAL, which estimates the position’s utility, and replace the terminal
test by a cutoff test that decides when to apply EVAL.
 That gives us the following for heuristic minimax for state s and maximum depth d:

Evaluation functions

 An evaluation function returns an estimate of the expected utility of the game from a given position.
 To design a good evaluation function:
 First, the evaluation function should order the terminal states in the same way as the true utility
function: states that are wins must evaluate better than draws, which in turn must be better than
losses. Otherwise, an agent using the evaluation function might err even if it can see ahead all the
way to the end of the game.
Mr. Mohammed Afzal, Asst. Professor in AIML
Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com
 Second, the computation must not take too long! (The whole point is to search faster.) Third, for
nonterminal states, the evaluation function should be strongly correlated with the actual chances of
winning.

Cutting off search

 The most straightforward approach to controlling the amount of search is to set a fixed depth limit so
that CUTOFF-TEST(state, depth) returns true for all depth greater than some fixed depth d. (It must
also return true for all terminal states, just as TERMINAL-TEST did.)
 The depth d is chosen so that a move is selected within the allocated time. A more robust approach is
to apply iterative deepening.
 When time runs out, the program returns the move selected by the deepest completed search. As a
bonus, iterative deepening also helps with move ordering.

Forward pruning
 It is also possible to do forward pruning, meaning that some moves at a given node are pruned
immediately without further consideration.
 Clearly, most humans playing chess consider only a few moves from each position (at least
consciously).
 One approach to forward pruning is beam search: on each ply, consider only a “beam” of the n best
moves (according to the evaluation function) rather than considering all possible moves.
 Unfortunately, this approach is rather dangerous because there is no guarantee that the best move
will not be pruned away.

Mr. Mohammed Afzal, Asst. Professor in AIML

Mob: +91-8179700193, Email: mdafzal.mbnr@gmail.com