AI module 2 presentation under VTU Syllabus

PROBLEM SOLVING  PROBLEM-SOLVING AGENTS
• Intelligent agents are supposed to maximize their performance measure.
• Achieving this is sometimes simplified if the agent can adopt a goal and aim at satisfying it.
• Goals help organize behaviour by limiting the objectives that the agent is trying to achieve and hence the
actions it needs to consider.
• Goal formulation, based on the current situation and the agent’s performance measure, is the first step in
problem solving.
• Problem formulation is the process of deciding what actions and states to consider, given a goal.
• The process of looking for a sequence of actions that reaches the goal is called search. A search algorithm
takes a problem as input and returns a solution in the form of an action sequence. Once a solution is found,
the actions it recommends can be carried out. This is called the execution phase.

PROBLEM SOLVING  PROBLEM-SOLVING AGENTS
• Thus, we have a simple “formulate,
search, execute” design for the agent, as
shown in Figure 1.
• After formulating a goal and a problem
to solve, the agent calls a search
procedure to solve it.
• It then uses the solution to guide its
actions, doing whatever the solution
recommends as the next thing to do—
typically, the first action of the sequence
—and then removing that step from the
sequence.
• Once the solution has been executed, the
agent will formulate a new goal.
Figure 1 A simple problem-solving agent.
It first formulates a goal and a problem, searches for a
sequence of actions that would solve the problem, and then
executes the actions one at a time. When this is complete, it
formulates another goal and starts over.

PROBLEM SOLVING  PROBLEM-SOLVING AGENTS  Well-defined Problems and
Solutions
A problem can be defined formally by five components:
1. The initial state that the agent starts in. For example, the initial state for our agent in Romania might be described as
In(Arad).
2. A description of the possible actions available to the agent. Given a particular state s, ACTIONS(s) returns the set of
actions that can be executed in s. We say that each of these actions is applicable in s. For example, from the state In(Arad),
the applicable actions are {Go(Sibiu), Go(Timisoara), Go(Zerind)}.
3. A description of what each action does; the formal name for this is the transition model, specified by a function
RESULT(s, a) that returns the state that results from doing action a in state s. We also use the term successor to refer to any
state reachable from a given state by a single action. For example, we have
RESULT(In(Arad), Go(Zerind)) = In(Zerind)
Together, the initial state, actions, and transition model implicitly define the state space of the problem—the set of all
states reachable from the initial state by any sequence of actions. The state space forms a directed network or graph in
which the nodes are states and the links between nodes are actions. (The map of Romania shown in Figure 2 can be
interpreted as a state-space graph if we view each road as standing for two driving actions, one in each direction) A path
in the state space is a sequence of states connected by a sequence of actions.

PROBLEM SOLVING  PROBLEM-SOLVING AGENTS  Well-defined Problems and
Solutions
4. The goal test, which determines whether a
given state is a goal state. Sometimes there
is an explicit set of possible goal states, and
the test simply checks whether the given
state is one of them. The agent’s goal in
Romania is the singleton set
{In(Bucharest)}.
5. A path cost function that assigns a numeric
cost to each path. The problem-solving
agent chooses a cost function that reflects its
own performance measure. For the agent
trying to get to Bucharest, time is of the
essence, so the cost of a path might be its
length in kilometers. The step cost of taking
action a in state s to reach state s` is denoted
by c(s, a, s`). The step costs for Romania are
shown in Figure 2 as route distances. We
assume that step costs are nonnegative.
Figure 1.2 A simplified road map of part of
Romania.

PROBLEM SOLVING  EXAMPLE PROBLEMS
• The problem-solving approach has been applied to a vast array of task
environments.
• We list some of the best known here, distinguishing between toy and real-world
problems.
• A toy problem is intended to illustrate or exercise various problem-solving
methods. It can be given a concise, exact description and hence is usable by
different researchers to compare the performance of algorithms.
• A real-world problem is one whose solutions people actually care about.

PROBLEM SOLVING  EXAMPLE PROBLEMS  Toy Problem
The first example we examine is the vacuum world. (See Figure 3) This can be formulated as a problem as follows:
• States: The state is determined by both the agent location and the dirt locations. The agent is in one of two locations,
each of which might or might not contain dirt. Thus, there are 2 × = 8 possible world states. A larger environment
with n locations has n * 2n states.
• Initial state: Any state can be designated
as the initial state.
• Actions: In this simple environment, each
state has just three actions: Left, Right,
and Suck. Larger environments might
also include Up and Down.
• Transition model: The actions have their
expected effects, except that moving Left
in the leftmost square, moving Right in
the rightmost square, and Sucking in a
clean square have no effect. The complete
state space is shown in Figure 3.
• Goal test: This checks whether all the
squares are clean.
• Path cost: Each step costs 1, so the path
cost is the number of steps in the path.
Figure 3 The state space for the vacuum world. Links
denote actions: L = Left, R = Right, S = Suck.

The 8-puzzle, an instance of which is shown in Figure 4, consists of a 3×3 board with eight numbered tiles and a
blank space. A tile adjacent to the blank space can slide into the space. The object is to reach a specified goal state,
such as the one shown on the right of the figure. The standard formulation is as follows:
• States: A state description specifies the location of each of the eight tiles and the blank in one of the nine
squares.
• Initial state: Any state can be designated as the initial state. Note that any given goal can be reached from
exactly half of the possible initial states.
• Actions: The simplest formulation defines the
actions as movements of the blank space Left, Right,
Up, or Down.
• Transition model: Given a state and action, this
returns the resulting state; for example, if we apply
Left to the start state, the resulting state has the 5 and
the blank switched.
• Goal test: This checks whether the state matches the
goal state.
Figure 4 A typical instance of the 8-puzzle.
• Path cost: Each step costs 1, so the path cost is the number of steps in the path.

The goal of the 8-queens problem is to place eight queens on a chessboard such that no queen attacks any other.
(A queen attacks any piece in the same row, column or diagonal.) Figure 5 shows an attempted solution that fails:
the queen in the rightmost column is attacked by the queen at the top left.
Figure 5 Almost a solution to the 8-queens
problem.
There are two main kinds of formulation:
 An incremental formulation involves operators that augment
the state description, starting with an empty state; for the 8-
queens problem, this means that each action adds a queen to
the state.
 A complete-state formulation starts with all 8 queens on the
board and moves them around. In either case, the path cost is
of no interest because only the final state counts. The first
incremental formulation one might try is the following:
• States: Any arrangement of 0 to 8 queens on the board is
a state.
• Initial state: No queens on the board.
• Actions: Add a queen to any empty square.
• Transition model: Returns the board with a queen added to the specified square.
• Goal test: 8 queens are on the board, none attacked.

PROBLEM SOLVING  EXAMPLE PROBLEMS  Real-world Problems
• Route-finding problem is defined in terms of specified locations and transitions along links between them.
Route-finding algorithms are used in a variety of applications. Some, such as Web sites and in-car systems that
provide driving directions, are relatively straightforward extensions of the Romania example.
• Touring problems are closely related to route-finding problems, but with an important difference. As with
route finding, the actions correspond to trips between adjacent cities. The state space, however, is quite
different. Each state must include not just the current location but also the set of cities the agent has visited.
• The traveling salesperson problem (TSP) is a touring problem in which each city must be visited exactly
once. The aim is to find the shortest tour. The problem is known to be NP-hard, but an enormous amount of
effort has been expended to improve the capabilities of TSP algorithms.
• A VLSI layout problem requires positioning millions of components and connections on a chip to minimize
area, minimize circuit delays, minimize stray capacitances, and maximize manufacturing yield. The layout
problem comes after the logical design phase and is usually split into two parts: cell layout and channel
routing. In cell layout, the primitive components of the circuit are grouped into cells, each of which performs
some recognized function. Channel routing finds a specific route for each wire through the gaps between the
cells. These search problems are extremely complex, but definitely worth solving.
• Robot navigation is a generalization of the route-finding problem described earlier. Rather than following a
discrete set of routes, a robot can move in a continuous space with an infinite set of possible actions and states.
When the robot has arms and legs or wheels that must also be controlled, the search space becomes many-
dimensional. Advanced techniques are required just to make the search space finite.

PROBLEM SOLVING  SEARCHING FOR SOLUTIONS
• Having formulated some problems, we now need to solve them. A solution is an action sequence, so search algorithms work
by considering various possible action sequences. The possible action sequences starting at the initial state form a search
tree with the initial state at the root; the branches are actions and the nodes correspond to states in the state space of the
problem.
Figure 6 Partial search trees for finding a route from Arad to
Bucharest.
• Figure 6 shows the first few steps in growing the search
tree for finding a route from Arad to Bucharest. The root
node of the tree corresponds to the initial state, In(Arad).
• The first step is to test whether this is a goal state, if not,
then we need to consider taking various actions. We do
this by expanding the current state; that is, applying each
legal action to the current state, thereby generating a
new set of states.
• In this case, we add three branches from the parent node
In(Arad) leading to three new child nodes: In(Sibiu),
In(Timisoara), and In(Zerind).
• Now we must choose which of these three possibilities to
consider further.
• Each of these six nodes is a leaf node, that is, a node
with no children in the tree.
• The set of all leaf nodes available for expansion at any
given point is called the frontier/open list.

• The process of expanding nodes on the frontier continues until either a solution is found or there are no more states to
expand. The general TREE-SEARCH algorithm is shown informally in the Figure 7.
• Search algorithms all share this basic structure; they vary primarily according to how they choose which state to expand
next—the so-called search strategy.
Figure 7 An informal description of the general tree-search and
graph-search algorithms. The parts of GRAPH-SEARCH marked in
bold italic are the additions needed to handle repeated states.
• Can be noticed one peculiar thing about the search tree
shown in Figure 6, it includes the path from Arad to Sibiu
and back to Arad again! We say that In(Arad) is a repeated
state in the search tree, generated in this case by a loopy
path.
• Loopy paths are a special case of the more general concept
of redundant paths, which exist whenever there is more
than one way to get from one state to another.
o Consider the paths Arad–Sibiu (140 km long) and
Arad–Zerind–Oradea–Sibiu (297 km long). Obviously,
the second path is redundant—it’s just a worse way to
get to the same state.
• The way to avoid exploring redundant paths is to remember
where one has been. To do this, we augment the TREE-
SEARCH algorithm with a data structure called the explored
set (also known as the closed list), which remembers every expanded node. Newly generated nodes that match previously
generated nodes—ones in the explored set or the frontier—can be discarded instead of being added to the frontier.

• The search tree constructed by the GRAPH-
SEARCH algorithm contains at most one copy of
each state, so we can think of it as growing a tree
directly on the state-space graph, as shown in
Figure 8.
• The algorithm has another nice property: the
frontier separates the state-space graph into the
explored region and the unexplored region, so that
every path from the initial state to an unexplored
state has to pass through a state in the frontier.
• This property is illustrated in Figure 9. As every
step moves a state from the frontier into the
explored region while moving some states from
the unexplored region into the frontier, we see that
the algorithm is systematically examining the
states in the state space, one by one, until it finds a
solution.
8
9

PROBLEM SOLVING  SEARCHING FOR SOLUTIONS  Infrastructure for Search Algorithms
• Search algorithms require a data structure to keep track of the search tree that is being constructed. For each
node n of the tree, we have a structure that contains four components:
o n.STATE: the state in the state space to which the node corresponds;
o n.PARENT: the node in the search tree that generated this node;
o n.ACTION: the action that was applied to the parent to generate the node;
o n.PATH-COST: the cost, traditionally denoted by g(n), of the path from the initial state to the node, as
indicated by the parent pointers.
• Given the components for a parent node, it is easy to see how to compute the necessary components for a child
node. The function CHILD-NODE takes a parent node and an action and returns the resulting child node:
• The node data structure is depicted in Figure
10.
Figure 10 Nodes are the data structures from which the
search tree is constructed. Each has a parent, a state, and
various bookkeeping fields. Arrows point from child to parent.

PROBLEM SOLVING  SEARCHING FOR SOLUTIONS  Infrastructure for Search Algorithms
• Now that we have nodes, we need somewhere to put them.
• The frontier needs to be stored in such a way that the search algorithm can easily choose the next node to
expand according to its preferred strategy.
• The appropriate data structure for this is a queue.
• The operations on a queue are as follows:
o EMPTY?(queue) returns true only if there are no more elements in the queue.
o POP(queue) removes the first element of the queue and returns it.
o INSERT(element, queue) inserts an element and returns the resulting queue.
• Three common variants are the first-in, first-out FIFO QUEUE or FIFO queue, which pops the oldest element
of the queue; the last-in, first-out or LIFO queue (also known as a stack), which pops the newest element of
the queue; and the priority queue, which pops the element of the queue with the highest priority according to
some ordering function.

PROBLEM SOLVING  SEARCHING FOR SOLUTIONS  Measuring Problem-solving Performance
We can evaluate an algorithm’s performance in four ways:
1. Completeness: Is the algorithm guaranteed to find a solution when there is one?
2. Optimality: Does the strategy find the optimal solution?
3. Time complexity: How long does it take to find a solution?
4. Space complexity: How much memory is needed to perform the search?
In AI, the graph is often represented implicitly by the initial state, actions, and transition model
and is frequently infinite. For these reasons, complexity is expressed in terms of three
quantities:
5. b, the branching factor or maximum number of successors of any node;
6. d, the depth of the shallowest goal node (i.e., the number of steps along the path from
the root); and
7. m, the maximum length of any path in the state space.

PROBLEM SOLVING  UNINFORMED SEARCH STRATEGIES
This section covers several search strategies that come under the heading of uninformed search (also called
blind search).
 The term means that the strategies have no additional information about states beyond that provided in the
problem definition. All they can do is generate successors and distinguish a goal state from a non-goal
state.
Breadth-first Search:
• Breadth-first search is a simple strategy in which the root node is expanded first, then all the successors of
the root node are expanded next, then their successors, and so on.
• In general, all the nodes are expanded at a given depth in the search tree before any nodes at the next level are
expanded.
• Breadth-first search is an instance of the general graph-search algorithm inwhich the shallowest unexpanded
node is chosen for expansion.
• This is achieved very simply by using a FIFO queue for the frontier. Thus, new nodes (which are always
deeper than their parents) go to the back of the queue, and old nodes, which are shallower than the new nodes,
get expanded first.
• Pseudocode is given in Figure 11 and Figure 12 shows the progress of the search on a simple binary tree.
• The time complexity of BFS is and space complexity is also where b is a branching factor and d is a depth.

Figure 11 Breadth-first search on a graph.
Figure 12 Breadth-first search on
a simple binary tree. At each
stage, the node to be expanded
next is indicated by a marker.

Depth-first search:
• Depth-first search always expands the deepest node in the
current frontier of the search tree.
• The progress of the search is illustrated in Figure 13.
• The search proceeds immediately to the deepest level of the
search tree, where the nodes have no successors.
• As those nodes are expanded, they are dropped from the
frontier, so then the search “backs up” to the next deepest
node that still has unexplored successors.
• The depth-first search algorithm is an instance of the graph-
search algorithm; whereas breadth-first-search uses a FIFO
queue, depth-first search uses a LIFO queue.
o A LIFO queue means that the most recently generated
node is chosen for expansion.
o Thismust be the deepest unexpanded node because it is
one deeper than its parent.
• The time complexity of DFS is and space complexity is
where b is a branching factor and m is a maximum depth.
Figure 13 Depth-first search on a binary tree. The unexplored
region is shown in light gray. Explored nodes with no descendants
in the frontier are removed from memory. Nodes at depth 3 have
no successors and M is the only goal node.

AI module 2 presentation under VTU Syllabus

Depth First search
Depth-first search is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root
node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each
branch before backtracking.

Algorithm
Step 1: SET STATUS = 1 (ready state) for each node in G
Step 2: Push the starting node A on the stack and set its
STATUS = 2 (waiting state)
Step 3: Repeat Steps 4 and 5 until STACK is empty
Step 4: Pop the top node N. Process it and set its STATUS =
3 (processed state)
Step 5: Push on the stack all the neighbors of N that are in
the ready state (whose STATUS = 1) and set their STATUS =
2 (waiting state)
[END OF LOOP]

The step by step process to the DFS traversal is given as
follows -
First, create a stack with the total number of vertices in the
graph.
Now, choose any vertex as the starting point of traversal, and
push that vertex into the stack.
After that, push a non-visited vertex (adjacent to the vertex on
the top of the stack) to the top of the stack.
Now, repeat steps 3 and 4 until no vertices are left to visit from
the vertex on the stack's top.
If no vertex is left, go back and pop a vertex from the stack.
Repeat steps 2, 3, and 4 until the stack is empty.

Example of DFS algorithm
example given below, there is a directed graph having 7 vertices.

Now, all the graph nodes have been traversed, and the stack is empty.

AI module 2 presentation under VTU Syllabus

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