Artificial IntelligenceArtificial Intelligence
Chapter 3Chapter 3
Structures and Strategies For State Structures and Strategies For State Space SearchSpace Search
Contents
• Introduction• Structures for State Space search
•Graph Theory•The Finite State Machine•State representation of problems
• Strategies for Space State Search•Data driven – Goal driven search•Depth Search – Breadth search•Depth First search with Iterative Deepening
• Using the State Space to represent Reasoning with the propositional and predicate calculus
•And /Or Graphs•Examples
Introduction The importance of the problem space
The choice of a problem space makes a big difference
in fact, finding a good abstraction is half of the problem
Intelligence is needed to figure out what problem space to use
the human problem solver is conducting a search in the space of problem spaces
Introduction
The theory of state space search is a primary tool for representing a problem as a state space graph , using graph theory to analyze the structure and complexity of the problem and the search procedure we employ to solve it
The city of Königsberg Swiss Leonhard Euler 18th cen Problem: if there is a walk around the city that
crosses each bridge exactly once?
RepresentationsEuler’s invented Graph of the Königsberg
bridge system- Euler Path
rb : river bank
I : island
b : bridge
Graph theoryNodes rb - iLinkes arcs - bEULER noted WALK WAS IMPOSSIBLE UNLESS A GRAPH HAS EXACTLY ZERO OR TWO NODES OF ODD DEGREESDegree no of arcs joining a node
Representing Königsberg bridge system- using predicate calculus
Predicate calculus: connect(X, Y, Z)connect(i1, i2, b1) connect(i2, i1, b1)
connect(rb1, i1, b2) connect(i1, rb1, b2)
connect(rb1, i1, b3) connect(i1, rb1, b3)
connect(rb1, i2, b4) connect(i2, rb1, b4)
connect(rb2, i1, b5) connect(i1, rb2, b5)
connect(rb2, i1, b6) connect(i1, rb2, b6)
connect(rb2, i2, b7) connect(i2, rb2, b7)
Connect :tow lands connected by a particular bridge
Graph Theory
A graph consists of a set of finite nodes N1,N2… and a set of arcs connecting the nodes
Arcs are orded pairs of nodes ie arc(N3,N4)… . If a directed arc connects Nj and Nk,
Nj is called the parent, and Nk is called the child. If Nj is also connected to Ni, Nk and Nj are siblings.
A rooted tree has a unique node which has no parents called the root. Each node in a rooted tree has a unique parent.
The edge in the rooted tree are directed away from the root
The path in a rooted tree is of length n-1 for a sequence of nodes (N1,N2,…,Nn.)
Graph Theory (cont’d)
On the path of the rooted tree a node is called an ancestor of all node after it and a descendant of all nodes before it
path that contains any node more than once is said to contain a cycle or loop.
A tree is a graph in which there is a unique path between every pair of nodes.
Two nodes are said to be connected if a path exists that includes them both.
Tip or Leaf node is the node that has no children
Strucures for State Space Search
A labeled directed graph
Family relationships in a rooted tree
a is ancestor of g, h and i
(e f g h I j) Leaves or tips
a is Root
b is parent of( e f)
(g, h , i) children of c
(b , c , d ) are siblings
)b,e (are connected
Finite State Machine Finite-state machines can solve a large number of problems, among which
electronic design automation, communication protocol design, parsing and other engineering applications. In biology and artificial intelligence research, state machines or hierarchies of state machines are sometimes used to describe neurological systems and in linguistics — to describe the grammars of natural languages.
There are several action types: Entry action
which is performed when entering the state Exit action
which is performed when exiting the state Input action
which is performed depending on present state and input conditions Transition action
which is performed when performing a certain transition
http://en.wikipedia.org/wiki/Finite-state_machine
Finite State Machine (FSM)
(a) The finite state graph for a flip flop and
(b) its transition matrix.
Flip Flop FSM
The Finite State Machine
The Finite State Machine is finite directed connected graph (S, I , F): S : Set of states – nodes I : Set of input values - path F : State transaction function
FSM is an abstract model of computation. FSM Used to recognize component of a
formal language
(a)The finite state graph
(b)The transition matrix for string recognition example
(c)This called Moore machine – FS accepting M
String RecognitionExample for FSM for recognizing all strings contain the exact sequence “abc”
Starting
Accepting state
Finite State Accepting Machine
There are tow FSM techniques Deterministic FSM: transition function for any input value to a state gives a unique
next state Probabilistic FSM: the transition function defines a distribution of output states for
each input to a state
The State Space representation of Problems The problem space consists of: a state space which is a set of states representing the possible
configurations of the world- with initial state a set of operators (legal moves)which can change one state into
another - arcs The problem space can be viewed as a graph where the states
are the nodes and the arcs represent the operators. Pathes are searched until either the goal description is satisfied
or abandoned It is important to choose the best path according too the needs of
the problem
State Space and Search
State Space of Tic – Tac – Toe
•Start state : empty board
•Goal state:board with 3x’s in row column diagonal
•States are all possible configuration of X’s O’s
•It’s a directed graph rather than a tree - goal can be reached by different states
•No cycle
•Complexity = 9!
The 8-puzzle problem as state space search
states: possible board positions operators: one for sliding each square in
each of four directions,or, better, one for moving the blank square in each of four directions
initial state: some given board position goal state: some given board position Note: the “solution” is not interesting here, we
need the path.
• generated by “move blank” operations
-- up
-- left
-- down
-- left
• Cycles may occur:so it’s a graph-multiple parents
State Space of the 8-Puzzle
The travelling salesperson problem
Find the shortest path for the salesperson to travel, visiting each city and returning to the starting city
Search for the travelling salesperson problem. Each arc is marked with the total weight of all paths from the start node (A) to its endpoint.
The goal is the lowest – cost path - Goal 2 reach path not a state
The travelling salesperson problem
Complexity = (n-1)!
Other techniques for solving TSP –
Branch & bound with complexity=(1.26)n
Greedy nearest neighbour( E, D, B, C, A), at a cost of 550, is not the shortest path. The comparatively high cost of arc (C, A) defeated the heuristic.
Strategies for State Space Search
Data-driven search – forward chaining Begin with the given facts and a set of legal rules for
changing states Apply rules to facts to produce new facts Continue until it generate a path that satisfies the goal
condition Goal-driven search – backward chaining
Begin with the goal and a set of facts and legal rules Search rules that generate this goal Determine conditions must be true to use these rules (sub
goals) Continue until it works back to the facts of the problem
Goal Driven search
Goal driven is suggested if : A Goal is given or easily be formulated
(diagnostic systems) (theorem to be proved) Data are not given but acquired by solver Large no. of rules match the facts of the problem
State space in which goal-directed search effectively prunes extraneous search paths.
Goal-driven Search
Data Driven search
Data Driven is suggested if All or most of the data are given (PROSPECTOR
- Dipmeter) Large no of goals but few ways to use facts Difficult to form a goal
Dipmeter" is a measuring instrument to measure resonant frequency of radio frequency circuits. It measures the amount of absorption of a high frequency inductively coupled magnetic field by nearby object. http://en.wikipedia.org/wiki/Dipmeter
State space in which data-directed search prunes irrelevant data and their consequents and determines one of a number of possible goals.
Data-driven Search
GDS & DDS
Both goal-driven and data-driven search the same state space graph
Order and no. of states are different Preferred strategy is determined by:
Complexity of the rules Shape of state space Availability of problem data
Searching Strategies
•Heuristic search search process takes place by traversing search space with applied rules (information).
•Techniques: Greedy Best First Search, A* Algorithm
•There is no guarantee that solution is found.
•Blind search traversing backtracking the search space until the goal nodes is found (might be doing exhaustive search).
•Techniques : Breadth First,Depth first, Interactive Deepening search. Uniform Cost •guarantee that solution is found
Backtracking Search Search – find a path from start until reaches a
goal (quit) or dead end (backtrack) Backtrack – when the path is dead, try others
Backtrack to the most recent node
on the path having unexamined siblings Continue until find goal or
all children been searched
(backtrack fails back)
Backtrack algorithm
SL: start list-if the goal is found have the solution path
NSL : nodes waiting evaluation DE: dead end CS :state to evaluate : goal state
Stack : From left
Stack : From left
Backtracking search of a hypothetical state space space.
The Breadth-first search
Breadth-first search When a state is examined, all of its children are examined after
any of its siblings (all nodes in a given level before any node in the next level)
Explore the search space in a level-by-level fashion queue structured- states are added to the right and removed
from the left FIFO
1 234
Order of search : A , B , C , D , E , F , G
The breadth-first search algorithm
A trace of breadth-first search
The graph at iteration 6 of breadth-first search. States on open and closed are highlighted
Breadth-first search of the 8-puzzle, showing order in which states were removed from open
Depth-first search
Depth-first search When a state is examined, all of its children and
their descendants are examined before any of its siblings
Go deeper into the search space where possible Stack structure descendant states are added and
removed from the left of (open) list- LIFO
Depth-first search34
5
…….
The depth-first search algorithm
A trace of depth-first search
The graph at iteration 6 of depth-first search. States on open and closed are highlighted
Depth-first search of 8-puzzle with a depth bound of 5
Evaluation Criteria (Russel – Norvig)
completeness Is the problem solver guaranteed to find a solution?
time complexity how long does it take to find the solution?
space complexity memory required for the search
optimality When the solution is found is it guranteed to be optimal?
main factors for complexity considerations: branching factor b, depth d of the shallowest goal node,
maximum path length m
Comparison between breadth- and depth-first search (Luger)
Choosing between depends on the problem properties (shortest path-branching factor-available time length – no of needed sol’s-
Breadth-first Always find the shortest path to a goal- High branching factor (high no of
children for a state-in open list-
(need memory), complexity Bn child on n level
Depth-first More efficient (gets quickly into a deep search space) suitable for graphs
with many branches Complexity B*N (open contains just the children of a single state) May get lost missing shorter path or stuck with infinite long path does not
lead to goal
Depth – first Search with Iterative Deepening
Use the the depth search with depth bound 1-if it fails another search with depth bound 2 Solution lies within a certain depth or time constraints Level by level search = shortest path & space usage =in n
level = B*n Guaranteed to find a shortest solution (BFS) & space
usage = B*n Complexity O(b*n) Time complexity = O(B^n) = no of nodes grows
exponentaily
All these strategies- Blind - get exponentail time complexity for worst case
Iterative Deepening DFS (ID-DFS)
“Blind search”
BFS and DFS are blind in the sense that they have no knowledge about the problem at all other than the problem space
Such techniques are also called brute-force search, uninformed search, or weak methods
Using the State Space to Represent Reasoning with propositional and
predicate Calculus
State Space Description of a Logical SystemAND/OR GraphsFurther Examples and Applications
MACSYMA (integration)Where is Fred?The Financial AdvisorEnglish Grammar
State Space Description of a Logical System
The propositional and Predicate calculus can be used as the formal specification language for making nodes distinguishable as well as for mapping the nodes of a graph onto the state space.
Inference rules can be used to create and describe the arcs between states.
Problems in the predicate calculus, such as determining whether a particular expression is a logical consequence of a given set of assertions, may be solved using search.
propositional calculus Propositions that are logical consequences of the
given set of assertions correspond to the nodes that
may be reached along a directed path from a state
representing a true proposition. [s,r,p] corresponds to the sequence of inferences:
s and sr yields r.r and rp yields p.
Determining whether a given proposition is a logical
consequence of a set of propositions becomes a
problem of finding a path from a boxed node to the
goal node.
propositional calculus A set of assertions :
q p; r p; v q; s r; t r;
s u; s; t; State space graph of a set of implications
the arcs correspond to logical implications () propositions given true (s and t) correspond to
the given data of the problem and represented as boxed nodes.
Boxed nodes
Logical Operators
General NameFormal NameSymbols
NotNegation
AndConjunction
OrDisjunction
If… Then/ImpliesConditional
If and only ifBiconditional
The Truth Table
PQNOT(P)
ANDP ^ Q
ORP v Q
ImpliesP Q
BiconditionalP Q
TTFTTTT
TFFFTFF
FTTFTTF
FFTFFTT
Predicate Calculus Logic
Basic idea : operator (variables_1, variables_2,…) Example: “She likes chocolate” likes (she,
chocolate). Universal quantifier (X) to show all object is
true [Eg: All students (X (student (X))] Existential quantifier (X) to show existence /
partial object is true [ Eg: Some people ( X (people (X))]
And/Or graphs If the premises of an implication are connected
by an operator, they are called AND nodes, and
the arcs from this node are joined by a curved
link.
And/Or graph is actually a specialization of a type
of graph known as a hypergraph, which connects
nodes by sets of arcs.
And/or graph
• And/or graph of expression q r p
Or – separate arcs
• And/or graph of the expression q r → p
• and -- connected
, :operators
HYPERGRAGH
And/or graph of a set of propositional calculus expressions.
a
b
c
a b d
b d f
a c e
f g
a e h
•Is h true? Example for goal directed strategy
And/Or graph Search operator(and nodes) indicates a problem
decomposition in which the problem is broken into subproblems such that all of the subproblems must be solved to solve the original problem.
operator indicates a selection, a point at which a choice may be made between alternative problem-solving strategies.
Examples and Applications for and/or graph searching
Macsyma “Where is fred?” An English Language Parser and Sentence
Generator The Financial Advisor Revisited
MACSYMA : An Example for goal directed- and/or graph
And/or graph of part of the state space for integrating a function (goal directed)And/or graph of part of the state space for integrating a function (goal directed)
The facts and rules of this example are given as English sentences followed by their predicate calculus equivalents:
“Where is fred?”: Example for Checking English sentences Goal-directed and/or graph
The solution subgraph showing that Fred is at the museum.
An English Language Parser and Sentence Generator
Rewrite rules and transform it into another the pattern on one site of the with the pattern on the other side : For changing expression from language to
another Determine if sentences are well-formed
sentences
Rules for a simple subset of English grammar are:
And/or graph for the grammar. Some of the nodes (np, art, etc) have been written more than once to simplify drawing the graph.
An English Language Parser and Sentence Generator:
A parser needs :
Grammar rules
Terminals : dictionary of words in the language ( a , the , man , dog , likes , bites)
Parse tree for the sentence “The dog bites the man.”
•A Well –Formed expression in the grammar consists of terminals and that can be reduce to the sentence symbol
•Use data –driven parsing (maching right hand side and replace by the pattern on left
•Parsing is important for natural Languages– structuring compilers and interpretors for computer languages
Generating legal sentences by a goal driven search Begin with sentence as top level goal and end when no more
rules to apply Create all acceptable sentence
A sentence ia a np followed by a vp(1) Np is replaced by n (2) giving n vp Man is the first n available (8) giving man vp Np is satisfied and vp is attemped (3) replace vp with v giving
man v Replace v with likes (10 Man likes the first acceptable sentence – there are 80 correct
sentences Repeat until all possible state space has been searched
Parsing and generating can be used for completing sentences (correctnes not symentic)
And/or graph searched by the financial advisor.
Blind Search: Uniform-Cost
S
Size of Queue: 0
Nodes expanded: 0
Queue: Empty
Current action: Waiting…. Current level: n/a
Press space to begin the search
Current action: Expanding Current level: 0
Queue: SSize of Queue: 1
A
B
C
Size of Queue: 3 Queue: A, B, C
1
15
5
Nodes expanded: 1
S
Current level: 1
G
10A
Nodes expanded: 2
Queue: B, G11, C
Current action: Backtracking Current level: 0Current level: 1Current action: Expanding
5
Nodes expanded: 3
Queue: G10, G11, C15
B
Current level: 2
Queue: EmptySize of Queue: 0
FINISHED SEARCH
GGGGG
The goal state is achieved and the path S-B-G is returned. In relation to path cost, UCS has found the optimal route. Press space to end.