Red Slides Dec06

Artificial IntelligenceCourse No. 320331, Fall 2011

Dr. Kaustubh PathakAssistant Professor, Computer [email protected]

Jacobs University Bremen

December 6, 2011

Kaustubh Pathak (Jacobs University Bremen) Artificial Intelligence December 6, 2011 1 / 396

Introduction

Goal-based Problem-solving Agents using Searching

Non-classical Search Algorithms

Logical Agents

Probability Calculus

Bayesian Networks

Learning

Conclusion

Home-assignments


Introduction

Contents

IntroductionCourse LogisticsWhat is Artificial Intelligence (AI)?Foundations of AIHistory of AIState of the ArtPython IntroductionAgents and their Task-environments


Introduction Course Logistics

Basics

I Textbook: Artificial Intelligence, a Modern Approach by StuartRussell and Peter Norvig, III Edition, Pearson, 2010.

I I/II editions are also usable. Reserve copies in the library.

I Additional material from other sources will supplement the textbookmaterial.



Grading

I Break-down: Home-assignments (60%), Mid-term exam (15%), Finalexam (25%). Exams compulsory.

I In all, 10 home-assignments: 8 (5%) and 2 (10%).

I Mid-term exam date 25 Oct. 2011 (Tue.) after the reading days.

I Coding in Python (preferred) or C++.

I Teaching Assistant: Mr. Mohammad [email protected]

I Home-assignments email to both the instructor and the TA with [AI]in the subject-line.



Syllabus

I Introduction to AI; Intelligent agents: Chapters 1,2.I A brief introduction to Python.

I Solving problems by searching: Chapters 3,4.I BF, DF, A? search; GA, Simulated Annealing.

I Uncertain knowledge & reasoning: Chapters 13, 14, 15.I Bayesian Networks.

I Learning: Chapters 18, 20.I By examples; probabilistic models.


Introduction What is Artificial Intelligence (AI)?

Defining AIHuman-centered vs. Rationalist Approaches

Thinking Humanly [Theautomation of] activities that

we associate with human think-

ing, activities such as decision-

making, problem-solving, learn-

ing... (Bellman, 1978)

Thinking Rationally Thestudy of computations that make

it possible to perceive, reason, and

act. (Winston, 1992)

Acting Humanly The artof creating machines that per-

form functions that require intelli-

gence when performed by people.

(Kurzweil, 1990)

Acting Rationally Computa-tional Intelligence is the study of

the design of intelligent agents.

(Poole et al., 1998)



Acting Humanly

The Turing Test (1950)

The test is passed if a human interrogator,after posing some written questions,cannot determine whether the responsescome from a human or from a computer.

Total Turing Test

There is a video signal for the interrogatorto test the subjects perceptual abilities, aswell as a hatch to pass physical objectsthrough.

Figure 1: Alan Turing(1912-1954)



Capabilities required for passing the Turing test

The Turing Test

I Natural language processing

I Knowledge representation

I Automated reasoning

I Machine learning

The Total Turing Test

I Computer vision

I Robotics



Thinking HumanlyCognitive Science

Trying to discover how human mindswork.

I Introspection

I Psychological experiments onhumans

I Brain imaging: FunctionalMagnetic Resonance Imaging(fMRI), Positron EmissionTomography (PET), EEG, etc.

Figure 2: fMRI image (source: V.Tsytsarev http://www.umsl.edu/~tsytsarev)

Youtube video (1:00-4:20)

Reading mind by fMRI



Thinking Rationally

I Aristotles Syllogisms.

I The logicist tradition in AI. Logical programming.I Problems:

I Cannot handle uncertaintyI Does not scale-up due to high computational requirements.



Acting RationallyThe Rational Agent Approach

Definition 1 (Agent)

An agent is something that acts. A rational agent is one that acts so as toachieve the best outcome, or when there is uncertainty, the best expectedoutcome.

I This approach is more amenable to scientific development than theones based on human behavior or thought.

I Rationality is well defined mathematically in a way, it is justoptimization under constraints. When, due to computationaldemands in a complicated environment, the agent cannot maintainperfect rationality, it resorts to limited rationality.


Introduction Foundations of AI

Disciplines Contributing to AI. I

Philosophy

I Rationalism: Using power of reasoning to understand the world.I How does the mind arise from the physical brain?

I Dualism: Part of mind is separate from matter/nature. ProponentRene Descartes, among others.

I Materialism: Brains operation constitutes the mind. Claims that freewill is just the way perception of available choices appears to thechoosing entity.

MathematicsLogic, computational tractability, probability theory.

EconomicsUtility theory, decision theory (probability theory + utility theory), gametheory.



NeuroscienceThe exact way the brain enables thought is still a scientific mystery.However, the mapping between areas of the brain and parts of body theycontrol or receive sensory input from.

Parietal LobeFrontal Lobe

Occipital Lobe

Temporal Lobe

Motor Cortex

Visual Cortex

Dorsal Stream

Ventral Stream

Cerebellum

Spinal Cord

Figure 3: The human cortex with the various lobes shown in different colors. Theinformation from the visual cortex gets channeled into the dorsal (where/how)and the ventral (what) streams.



Psychology

Behaviorism (stimulus/response), Cognitive psychology.

Computer Engineering

Hardware and Software. Computer vision.

Linguistics

Natural language processing.



Control Theory and Cybernetics

I The basic idea of control theory is to use sensory feedback to altersystem inputs so as to minimize the error between desired andobserved output. Basic example: controlling the movement of anindustrial robotic arm.

I Norbert Wiener (18941964)

I Modern control theory and AI have a considerable overlap. Controltheory focuses more on calculus of continuous variables, Matrixalgebra, whereas AI also uses tools of logical inference and planning.


Introduction History of AI

History of AI I

Gestation Period (1943-1955)

McCulloch and Pitts (1943) proposed a model for the neuron. Hebbianlearning (1949) for updating inter-neuron connection strengths developed.Alan Turing published Computing Machinery and Intelligence (1950),proposing the Turing test, machine learning, genetic algorithms, andreinforcement learning.

Birth of AI (1956)

The Dartmouth workshop organized by John McCarthy of Stanford.

Early Enthusiasm (1952-1969)

LISP developed. Several small successes including theorem proving etc.Perceptrons (Rosenblatt, 1962) developed.

Reality hits (1966-1973)



History of AI II

After the Sputnik launch (1957), automatic Russian to English translationattempted. Failed miserably.

1. The spirit is willing, but the flesh is weak. Translated to:

2. The wodka is good but the meat is rotten.

Computational complexity scaling-up could not be handled. Perceptronswere found to have very limited representational power. Most ofgovernment funding stopped.

Knowledge-based Systems (1969-1979)

Use of expert domain specific knowledge for inference. Examples:DENDRAL (1969) for inferring molecular structure from massspectrometer results; MYCIN (Blood infection diagnosis).

AI in Industry (1980-present)



History of AI III

Companies like DEC, DuPont etc. developed expert systems. Governmentfunding restored.

Return of Neural Networks (1986-present)

Back-propagation learning algorithm developed.

AI embraces Control Theory and Statistics (1987-present)

Rigorous mathematical methods began to be resused instead of ad hocmethods. Example: Hidden Markov Models (HMM), Bayesian Networks,etc.

Intelligent agents (1995-present)

Growth of the Internet. AI in web-based applications (-bots).

Huge data-sets (2001-present)



History of AI IV

Learning based on very large data-sets. Example: Filling in holes in aphotograph; Hayes and Efros (2007). Performance went from poor for10,000 samples to excellent for 2 million samples.


Introduction State of the Art

Successful Applications

Intelligent Software Wizards

Figure 4: Microsoft Office Assistant Clippit



Logistics Planning

Dynamic Analysis and Replanning Tool (DART). Used during Gulf war(1990s) for scheduling of transportation. DARPA stated that this singleapplication paid back DARPAs 30 years investment in AI.DART won DARPAs outstanding Performance by a Contractor award, formodification and transportation feasibility analysis for Time-Phased Forceand Deployment Data that was used during Desert Storm.http://www.bbn.com



DARPA Urban Challenge 2007

Figure 5: Video (6:05)



Simultaneous Localization and Mapping (SLAM)




Big Dog: A Control Engineering Marvel




Digression

New Sensing Technologies: Example Kinect

(a) The Microsoft Kinect 3D camera(from Wikipedia)

(b) A point-cloud obtained from it (fromWillow Garage).



Probabilistic Grasp Planning




Object Recognition & Pose Estimation




DARPA Grand Challenge 2005 I

Figure 10: Autonomous driving in the Mojave Desert region of the United Statesalong a 150-mile route. The winning car Stanley from Stanford.



DARPA Grand Challenge 2005 II

Touareg interface

Laser mapper

Wireless E-Stop

Top level control

Laser 2 interface

Laser 3 interface

Laser 4 interface

Laser 1 interface

Laser 5 interface

Camera interface

Radar interface Radar mapper

Vision mapper

UKF Pose estimation

Wheel velocity

GPS position

GPS compass

IMU interface Surface assessment

Health monitor

Road finder

Touch screen UI

Throttle/brake control

Steering control

Path planner

laser map

vehicle state (pose, velocity)

velocity limit

map

vision map

vehicle state

obstacle list

trajectory

road center

RDDF database

driving mode

pause/disable command

Power server interface

clocks

emergency stop

power on/off

Linux processes start/stop heart beats

corridor

SENSOR INTERFACE PERCEPTION PLANNING&CONTROL USER INTERFACE

VEHICLE

INTERFACE

RDDF corridor (smoothed and original)

Process controller

GLOBAL

SERVICES

health status

data

Data logger File system

Communication requests

vehicle state (pose, velocity)

Brake/steering

Communication channels

Inter-process communication (IPC) server Time server

Figure 11: Stanleys architecture. Watch http://youtu.be/TDqzyd7fDRc.


Introduction Python Introduction

Built-in Data-types

Type Example ImmutableNumbers 12, 3.4, 7788990L, 6.1+4j, Decimal XStrings "abcd", abc, "abcs" XBoolean True, False XLists [True, 1.2, "vcf"]Dictionaries {"A" : 25, "V" : 70}Tuples ("ABC", 1, Z) XSets/FrozenSets {90,a}, frozenset({a, 2}) X/XFiles f= open(spam.txt, r)}Single Instances None, NotImplemented. . .



Sequences Istr, list, tuple

Creation and Indexing

a= "1234567"

a[0]

b= [z, x, a, k]

b[-1] == b[len(b)-1], b[-1] is b[len(b)-1]

x= """This is a

multiline string"""

print x

y= me

too

print y

len(y)



Sequences IIstr, list, tuple

Immutability

a[1]= q # Fails

b[1]= s

c= a;

c is a, c==a

a= "xyz"; c is a

Help

dir(b)

help(b.sort)

b.sort()

b # In-place sorting



Sequences IIIstr, list, tuple

Slicing

a[1:2]

a[0:-1]

a[:-1], a[3:]

a[:]

a[0:len(a):2]

a[-1::-1]

Repetition & Concatenation

c=a*2

b*3

a= a + 5mn

d= b + [abs, 1, False]



Sequences IVstr, list, tuple

Nesting

A=[[1,2,3],[4,5,6],[7,8,9]]

A[0]

A[0][2]

A[0:-1][-1]

A[3] # Error

List Comprehension

q= [x.isdigit() for x in a]

print q

p=[(r[1]**2) for r in A if r[1]< 8] # Power

print p



Sequences Vstr, list, tuple

Dictionaries

D= {0:Rhine, 1:"Indus", 3:"Hudson"}

D[0]

D[6] # Error

D[6]="Volga"

dir(D)



Numbers I

I Math Operations

1 a= 10; b= 3; c= 10.5; d=1.2345

2 a/b

3 a//b, c//b # Floor division: b*(a//b) + (a%b) == a

4 d**c # Power

5 type(10**40) # Unlimited integers

6 import math

7 import random

8 dir(math)

9 math.pi # repr(x)

10 print math.pi # str(x)

11 s= "e is %08.3f and list is %s" % (math.e, [a,1,1.5])

12 random.random() # [0,1)

13 random.choice([apple,orange,banana,kiwi])



Numbers II

I Booleans

1 s1= True

2 s2= 3 < 5


Dynamic Typing I

I Variables are names and have no types. They can refer to objects ofany type. Type is associated with objects.

1 a= "abcf"

2 b= "abcf"

3 a==b, a is b

4 a= 2.5

I Objects are garbage-collected automatically.

I Shared references



Dynamic Typing II1 a= [4,1,5,10]

2 b=a

3 b is a

4 a.sort()

5 b is a

6 a.append(w)

7 a

8 b is a

9 a= a + [w]

10 a

11 b is a

12 b

13 x= 42

14 y= 42

15 x is y, x==y

16 x= [1,2,3]; y=[1,2,3]



Dynamic Typing III

17 x is y, x==y

18 x=123; y= 123

19 x is y, x==y # Wassup?

20 # Assignments create references

21 L= [1,2,3]

22 M= [x, L, c]

23 M

24 L[1]= 0

25 M

26 # To copy

27 L= [1,2,3]

28 M= [x, L[:], c]

29 M

30 L[1]= 0

31 M



Control Statements I

I Mind the indentation! One extra carriage return to finish ininteractive mode.

1 import sys

2 tmp= sys.stdout

3 sys.stdout = open(log.txt, a)

4 x= random.random();

5 if x < 0.25:

6 [y, z]= [-1, 4]

7 elif 0.25


Loops I

I While

1 i= 0;

2 while i< 5:

3 s= raw_input("Enter an int: ")

4 try:

5 j= int(s)

6 except:

7 print invalid input

8 break;

9 else:

10 print "Its square is %d" % j**2

11 i += 1

12 else:

13 print "exited normally without break"



Loops II

I For

1 X= range(2,10,2) # [2, 4, 6, 8]

2 N= 7

3 for x in X:

4 if x> N:

5 print x, "is >", N

6 break;

7 else:

8 print no number > , N, found

9

10 for line in open(test.txt).xreadlines():

11 print line.upper()



Functions I

I Arguments are passed by assignment

1 def change_q(p, q):

2 for i in p:

3 if i not in q: q.append(i)

4 p= abc

5

6 x= [a,b,c]; # Mutable

7 y= bdg # Immutable

8 print x, y

9 change_q(q=x,p=y)

10 print x, y

I Output

[a, b, c] bdg

[a, b, c, d, g] bdg



Functions II

I Scoping rule: LEGB= Local-function, Enclosing-function(s), Global(module), Built-ins.

1 v= 99

2 def local():

3 def locallocal():

4 v= u

5 print "inside locallocal ", v

6 u= 7; v= 2

7 locallocal()

8 print "outside locallocal ", v

9

10

11 def glob1():

12 global v

13 v += 1



Functions III

14

15 local()

16 print v

17 glob1()

18 print v

I Output

inside locallocal 7

outside locallocal 2

99

100



Packages, Modules IPython Standard Library http://docs.python.org/library/

I Folder structure

root/

pack1/

__init__.py

mod1.py

pack2/

__init__.py

mod2.py

I root should be in one of the following: 1) program home folder, 2)PYTHONPATH 3) standard lib folder, or, 4) in a .pth file on path. Thefull search-path is in sys.path.

I Importing



Packages, Modules IIPython Standard Library http://docs.python.org/library/

import pack1.mod1

import pack1.mod3 as m3

from pack1.pack2.mod2 import A,B,C



Classes I

I Example

class Animal(object): # new style classes

count= 0

def __init__(self, _name):

Animal.count += 1

self.name= _name

def __str__(self):

return I am + self.name

def make_noise(self):

print (self.speak()+" ")*3

class Dog(Animal):

def __init__(self, _name):

Animal.__init__(self, _name)

self.count= 1



Classes II

def speak(self):

return "woof"

I Full examples in python examples.tgz



Useful External Libraries

I The SciPy library is a vast Python library for scientific computations.I http://www.scipy.org/I In Ubuntu install python-scitools in the package-manager.I Library for doing linear-algebra, statistics, FFT, integration,

optimization, plotting, etc.

I Boost is a very mature and professional C++ library. It has Pythonbindings. Refer to:http://www.boost.org/doc/libs/1_47_0/libs/python/doc/

I For creation of Python graph data-structures (leveraging boost) lookat: http://projects.skewed.de/graph-tool/


Introduction Agents and their Task-environments

A general agent

Agent Sensors

Actuators

Enviro

nment

Percepts

Actions

?

Definition 2 (A Rational Agent)

For each possible percept sequence, a rational agent should select anaction that is expected to maximize its performance measure, given theevidence provided by the percept sequence, and whatever built-in (prior)knowledge the agent has.



The PEAS Description of the Task EnvironmentPerformance, Environment, Actuators, Sensors

Agent-Type Performance-measure

Environment Actuators Sensors

Part-pickingrobot

Percentage ofparts in thecorrect bin

Conveyor beltwith parts; bins

Motors tomove the var-ious joints ofthe arm

Camera, motorangle-encoders

Medical diag-nosis system

Healthy pa-tient, reducedcosts

Patient, hospi-tal

Tests, treat-ments, refer-rals

Symptoms,Test-findings



Properties of the Task EnvironmentFully observable: relevant environmentstate fully exposed by the sensors.

Partially observable: e.g. a limitedfield-of-view sensor. Unobservable

Single Agent Multi-agentDeterministic: If the next state is com-pletely determined by the current stateand the action of the agent

Stochastic: Uncertainties quantified byprobabilities.

Episodic: Agents experience is dividedinto atomic episodes, each independentof the last, e.g. assembly-line robot.

Sequential: current action affects futureactions, e.g. chess-playing agent.

Static: Environment unchanging.Semi-dynamic: agents performancemeasure changes with time, env. static.

Dynamic: The environment changeswhile the agent is deliberating.

Discrete: state of the environment isdiscrete, e.g. chess-playing, traffic con-trol.

Continuous: The state smoothlychanges in time, e.g. a mobile robot.

Known: rules of the game/laws ofphysics of the env. are known to theagent.

Unknown: The agent must learn therules of the game.

I Hardest case: Partially observable, multiagent, stochastic, sequential,dynamic, continuous, and unknown.



Example of a Partially Observable Environment

Figure 12: A mobile robot operating GUI.



Agent Types

Four basic types in order of increasing generality:

I Simple reflex agents

I Reflex agents with state

I Goal-based agents

I Utility-based agents

All these can be turned into learning agents



Simple reflex agents

Agent

Environment

Sensors

What the worldis like now

What action Ishould do nowConditionaction rules

Actuators

Algorithm 1: Simple-Reflex-Agent

input : perceptoutput : actionpersistent: rules, a set of

condition-action rules

state Interpret-Input(percept) ;rule Rule-Match(state, rules) ;action rule.action;return action



Model-based reflex agents I

Agent

Environment

Sensors

What action Ishould do now

State

How the world evolves

What my actions do

Conditionaction rules

Actuators




Model-based reflex agents II

Algorithm 2: Model-Based-Reflex-Agent

input : perceptoutput : actionpersistent: state, agents current conception of worlds state

model , how next state depends on the current state and actionrules, a set of condition-action rulesaction, the most recent action, initially none

state Update-State(state, action, percept, model) ;rule Rule-Match(state, rules) ;action rule.action;return action



Goal-based agents

Agent

Environment

Sensors

What it will be like if I do action A


State


What my actions do

Goals

Actuators




Utility-based agents

Agent

Environment

Sensors

What it will be like if I do action A

How happy I will be in such a state


State


What my actions do

Utility

Actuators




Learning agents

Performance standard

Agent

Environment

Sensors

Performance element

changes

knowledgelearning goals

Problem generator

feedback

Learning element

Critic

Actuators



Contents

Goal-based Problem-solving Agents using SearchingThe Graph-Search AlgorithmUninformed (Blind) SearchInformed (Heuristic) Search



Problem Solving Agents

Algorithm 3: Simple-Problem-Solving-Agentinput : perceptoutput : actionpersistent: seq, an action sequence, initially empty

state, agents current conception of worlds stategoal , a goal, initially nullproblem, a problem formulation

state Update-State(state, percept) ;if seq is empty then

goal Formulate-Goal(state) ;problem Formulate-Problem(state, goal) ;seq Search(problem) ;if seq = failure then return a null action

action First(seq) ;seq Rest(seq) ;return action



Searching for Solutions

I State: The system-state of x Xparameterizes all properties of interest. Theinitial-state of the state is x0 and the set ofgoal-states is Xg . The set X of valid statesis called the state-space.

I Actions or Inputs: At each state x, thereare a set of valid actions u(x) U(x) thatcan be taken by the search agent to alter thestate.

I State Transition Function: How a newstate x is created by applying an action u tothe current state x.

x = f(x,u) (2.1)

The transition may have a cost k(x,u) > 0.

u1u2

Initial State x0

Goal State xg

Valid Actions

Figure 13: Nodes arestates, and edges arestate-transitions causedby actions.



Examples I

2

Start State Goal State

51 3

4 6

7 8

5

1

2

3

4

6

7

8

5

(a) An instance

1

23

45

6

7

81

23

45

6

7

8

State Node depth = 6g = 6

state

parent, action

(b) A node of the search-graph. Arrowspoint to parent-nodes.

Figure 14: The 8-puzzle problem



Examples II

Figure 15: An instance of the 8-Queens problem



Examples III

Giurgiu

UrziceniHirsova

Eforie

NeamtOradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DobretaCraiova

Sibiu Fagaras

Pitesti

Vaslui

Iasi

Rimnicu Vilcea

Bucharest

71

75

118

111

70

75120

151

140

99

80

97

101

211

138

146 85

90

98

142

92

87

86

Figure 16: The map of Romania. An instance of the route planning problemgiven a map.



Examples IV

Figure 17: A 2D occupancy grid map created using Laser-Range-Finder (LRF).



Examples V

Figure 18: Result of A path-planning algorithm on a multi-resolution quad-tree.


Goal-based Problem-solving Agents using Searching The Graph-Search Algorithm

Graph-SearchCompare with Textbook Fig. 3.7

Algorithm 4: Graph-Search

input : x0,XgD = , The explored-set/dead-set/passive-set;F .Insert(x0, g(x0) = 0) The frontier/active-set;while F not empty dox, g(x) F .Choose() Remove x from the frontier;if x Xg then return SUCCESS;D D {x};for u U(x) do

1 x f(x,u), g(x) g(x) + k(x,u) ;if (x / D) and (x / F) thenF .Insert(x, g(x) + h(x));

else if (x F) then2 F .Resolve-Duplicate(x);


Goal-based Problem-solving Agents using Searching The Graph-Search Algorithm

Measuring Problem-Solving Performance

I Completeness: Is the algorithm guaranteed to find a solution if thereis one?

I Optimality: Does the strategy find optimal solutions?I Time & Space Complexity: How long does the algorithm take and

how much memory is needed?I Branching factor b: The maximum number of successors (children) of

any node.I Depth d : The shallowest goal-node level.I Max-length m: Maximum length of any path in state-space.


Goal-based Problem-solving Agents using Searching Uninformed (Blind) Search

Breadth-First Search (BFS)

I The frontier F is implemented as a FIFO queue. The oldest elementis chosen by Choose().

I For a finite graph, it is complete, and optimum if all edges have samecost. It finds the shallowest goal node.

I The Graph-Search can return as soon as a goal-state is generatedin line 1.

I Number of nodes generated b + b2 + . . .+ bd = O(bd). This is thespace and time complexity.

I The explored set will have O(bd1) nodes and the frontier will haveO(bd) nodes.

I Mememory becomes more critical than computation time, e.g. forb = 10, d = 12, 1 KB/node, search-time is 13 days, andmemory-requirements 1 petabyte.



BFS Example

A

B C

D E F G



Dijkstra Algorithm or Uniform-Cost Search

I The frontier F is implemented as a priority-queue. Choose() selectsthe element with the highest priority, i.e. the minimum path-lengthg(x).

I F .Resolve-Duplicate(x) function on line 2 updates the path-costg(x) of x in the frontier F , if the new value is lower than the storedvalue. If the cost is decreased, the old parent is replaced by the newone. The priority queue is reordered to reflect the change.

I It is complete and optimum.

I When a node x is chosen from the priority-queue, the minimum lengthpath from x0 to it has been found. Its length is denoted as g(x).

I In other words, the optimum path-lengths of all the explored nodes inthe set D have already been found.



Correctness of Dijkstra Algorithm I

D

F

x

xg

x0

Figure 19: The graph separation by the frontier. The node x in the frontier ischosen for further expansion. The set D is a tree.



Correctness of Dijkstra Algorithm II

Observations

I Unexplored nodes can only be reached through the frontier nodes.

I An existing frontier node xf s cost can only be reduced thorough anode xc which currently has been chosen from the priority-queue as ithas the smallest cost in the frontier. The reduced cost g(xf ) > g(xc),and now parent(x)= xc .

I Note that D remains a tree.

I The frontier expands only through the unexplored children of thechosen frontier node, all of the children will have costs worse thantheir parent.

I The costs of the successively chosen frontier nodes arenon-decreasing.



Correctness of Dijkstra Algorithm IIIRemark 3 (When a frontier node xc is chosen for exansion, itsoptimum path has been found: Proof by induction.)

I Initially, x0 is added to D and expanded. Its child with the minimumpath-cost is then chosen from the frontier to be added to D. Itsfound path-cost is optimum.

I Assume that the paths of all nodes in D from x0 are optimum. Nowwe choose a new node xc to expand from the frontier. Then, we needto show that the path from x0 to xc through parent(xc) D is theoptimum.If this were not the case, then the optimum path to xc will passthrough some other node in the frontier, but their optimumpath-costs cannot be less than g(xc).

Thus sub-paths of an optimum path are also optimum: this is arestatement of the Bellmans Principle of Optimality.



Depth-first Search (DFS)

I The frontier F is implemented as a LIFO stack. The newest elementis chosen by Choose().

I For a finite graph, it is complete, and but not optimum.

I Explored nodes with no descendants in the frontier can be removedfrom the memory! This gives a space-complexity adavantage: O(bm)nodes. This happens automatically if the algorithm is writtenrecursively.

I Assuming that nodes at the same depth as the goal-node have nosuccessors, b = 10, d = 16, 1 KB/node, DFS will require 7 trillion(1012) times less space than BFS! That is why, it is popular in AIresearch.



Algorithm 5: Depth-limited-Search

input : current-state x, depth dif x Xg then

return SUCCESS ;else if d = 0 then

return CUTOFFelse

for u U(x) dox f(x,u);result Depth-limited-Search(x, d 1);if result =SUCCESS then

return SUCCESSelse if result =CUTOFF then

cutoff-occurred trueif cutoff-occurred then return CUTOFF ;else return NOT-FOUND ;



DFS ExampleGoal node M

A

B C

D E F G

H I J K L M N O



Iterative Deepening Search (IDS)

I As O(bd) O(bd1), one can combine the benefits of BFSand DFS.

I All the work from previous iteration is redone, but this isacceptable, as the frontier-size is dominant.

Algorithm 6: Iterative-Deepening-Search

for d= 0 to doresult Depth-limited-Search(x0, d);if result 6= CUTOFF then

return result



Limit = 3

Limit = 2

Limit = 1

Limit = 0 A A

A

B C

A

B C

A

B C

A

B C

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H I J K L M N O

A

B C

D E F G

H J K L M N OI

A

B C

D E F G

H I J K L M N O


Goal-based Problem-solving Agents using Searching Informed (Heuristic) Search

A Search I

est. path-cost from x0 to xg `(x)

= g(x) + est. path-cost from x to xg h(x)

(2.2)

`(x) , g(x) + h(x) (2.3)`(xg ) g(xg ), as h(xg ) = 0. (2.4)

I h(x) is a heuristically estimated cost, e.g., for the map route-findingproblem, h(x) = xg x.

I `(x) is the estimated cost of the cheapest solution through node x.

I A is a Graph-Search where, the frontier F is a priority-queue withhigher priority given to lower values of the evaluation function `(x).

I If no heuristics are taken, i.e. h(x) 0, A reduces to Dijkstrasalgorithm.



A Search II

I Similar to Dijkstras Algorithm, F .Resolve-Duplicate(x) online 2 of Algo. 4 updates the cost `(x) in the frontier F , if the newvalue is lower than the stored value. If this occurs, the old parent ofx is replaced by the new one. The priority queue is reordered toreflect the change.



Example heuristic function

Urziceni

NeamtOradea

Zerind

Timisoara

Mehadia

Sibiu

PitestiRimnicu Vilcea

Vaslui

Bucharest

GiurgiuHirsova

Eforie

Arad

Lugoj

DrobetaCraiova

Fagaras

Iasi

0160242161

77151

366

244226

176

241

25332980

199

380234

374

100193

Figure 20: Values of hSLD straight-line distances to Bucharest.



Conditions for Optimality of A IA will find the optimal path, if the heuristic cost h(x) is admissible andconsistent.

Definition 4 (Admissibility)

h(x) is admissible, if it never over-estimates the cost to reach the goal h(x) is always optimistic.Definition 5 (Consistency)

h(x) is consistent, if for every child (generated by action ui ) xi of a node x

the triangle-inequality holds:

h(xg ) = 0, (2.5)

h(x) k(x,ui , xi ) + h(xi ), i (2.6)

This is a stronger condition than admissibility, i.e. consistency admissibility.



A proof I

Lemma 6 (`(x) is non-decreasing along any optimal path)

x0 xm

xnxp

Figure 21: A dashed line between two nodes denotes that the nodes areconnected by a path, but are not necessarily directly connected.

To prove: Let xp be a node for which the optimum-path with cost g(xp)has been found (see figure): Nodes xm and xn lie on this optimum pathsuch that xm precedes xn, then

`(xm) `(xn) (2.7)



A proof IIProof.

I First note that since xm and xn lie on the optimum path to xp, theirpaths are also optimum and have lengths g(xn) and g(xm)respectively.

I Let us first assume that xm is the parent of xn, then

`(xn) = h(xn) + g(xn) (2.8)= h(xn) + g

(xm) + k(xm, xn) (2.9)(2.6)

h(xm) + g(xm) = `(xm) (2.10)I Now if xm is not the parent of xn but a predecessor, the inequality

can be chained for every child-parent node on the path between them,and we reach the same conclusion.



A proof III

Lemma 7 (At selection for expansion, a nodes optimum path hasbeen found)

To prove: In every iteration of A, the node x selected for expansion bythe frontier F having the minimum value of `(x) is such that, at selection:

g(x) = g(x). (2.11)



Proof of Lemma 7 I

x0

xn

Fk+1

D xs

xp

xm

(x0 : xm)

(x0 : xm)

(xp : xn)

Figure 22: The path shown in blue is the assumed optimal path. (x0 : xm) isthe optimum path found for a node xm. Since xm D at iteration k + 1, by theinduction hypothesis it must have been selected by Fi , i < k + 1, and hence itspath is optimum.



Proof of Lemma 7 II

I Proof is by induction. At N = 1, the frontier F selects its only nodex0 with cost g(x0) = g(x0) = 0.

I Assume that the induction hypothesis holds for N = 1 . . . k , Now weneed to show that it holds for iteration k + 1.

I Assume that Fk+1 selects xn at this iteration. All frontier nodes havetheir parents in D. At the time of selection, parent(xn)= xs . Supposethat the path through xs is not the optimal path for xn, but theoptimal path is , as shown in the figure in blue. The path consistsof edges existing in the search-graph which consists of all possibleparent-child edges of all nodes. This path exists in the graph atiteration k + 1 whether or not it will ever be discovered in futureiterations is irrelevant.



Proof of Lemma 7 IIII Note that the assumed optimal-path has to pass through a node

which is in Fk+1 because the frontier separates the dead-nodes Dand unknown nodes and all expansion occurs through the frontiernodes.

I Let xp be the first of all the nodes in the frontier Fk+1 (ref. figure).Since it is the first, its parent xm D. The entire assumed optimalpath consists of the following sub-paths ( stands forpath-concatenation):

(x0 : xn) = (x0 : xm) (xm : xp) (xp : xn) (2.12)

The sub-path (x0 : xm) of the assumed optimal path might bedifferent from (x0 : xm) already found at a previous iteration due tothe induction hypothesis (ref. figure). However, since all sub-paths ofan optimal path are also optimal, we conclude:

(x0 : xm) (x0 : xm). (2.13)Kaustubh Pathak (Jacobs University Bremen) Artificial Intelligence December 6, 2011 94 / 396


Proof of Lemma 7 IVI Note that the path may wander around the frontier Fk+1 and the

unexplored space before it reaches xn, but it cannot re-enter D, onceit leaves it at xm. Why?

I Node xp was added to the frontier Fk+1 with path-cost

gk+1(xp) = g(xm) + k(xm, xp) (2.14)

As xp is part of the assumed optimal path , its path-cost is alsooptimal. Combining this with (2.13),

g(xp) = gk+1(xp) (2.15)`k+1(xp) = `

(xp) = h(xp) + g(xp). (2.16)

I As xp lies on the optimum-path to xn, from Lemma 6,

`(xp) `(xn) (2.17a) `k+1(xn), the cost of xn at k+1. (2.17b)



Proof of Lemma 7 VI However, xn was selected by Fk+1 for expansion. Therefore, at

iteration k + 1, it must be true that,

`k+1(xn) `k+1(xp) (2.16)= `(xp). (2.18)

Combining (2.16), (2.17), and (2.18), we get:

`(xp) `(xn) `k+1(xn) `(xp), (2.19) `(xn) = `k+1(xn), (2.20)

g(xn) + h(xn) = gk+1(xn) + h(xn), (2.21)

g(xn) = gk+1(xn), (2.22)

or, the path-cost of g(xn) at k + 1 is indeed optimum. This is incontradiction to what we assumed, with our alternate optimal-path assumption. Therefore, we conclude that the optimal path g(xn) isfound at N = k + 1 when xn is selected by Fk+1.



Proof of Lemma 7 VI

I By extension, when the goal point is selected by the frontier, itsoptimum path with cost g(xg ) has been found.



Figure 23: Region searched before finding a solution: Dijkstra path search



Figure 24: Region searched before finding a solution: A path search. Thenumber of nodes expanded is the minimum possible.



Lemma 8 (Monotonicity of Expansion)

Let the selected node by Fj be xj , and that selected by Fj+1 be xj+1.Then it must be true that

`(xj) `(xj+1). (2.23)

Why? This shows that at iteration N = j , if Fj selects xj then:I All nodes x with `(x) < `(xj) have already been expanded, and

some nodes with `(x) = `(xj) have also been expanded.I In particular, when the first goal is found, then, All nodes x with`(x) < g(xg ) have already been expanded, and some nodes with`(x) = g(xg ) have also been expanded.



Properties of A

I The paths are optimal w.r.t. the cost function, but do you notice anyundesirable properties of the planned paths?

I Why are there less colors in Fig. 7 than in Fig. 6?



Selection of the Heuristic Cost

I The heuristic cost-to-go functions h can be generated from relaxedproblems where no search is required. E.g. straight-line distance formap route planning.

I Possibilities for the 8-puzzle problem (Fig. 14): 1) h is the number ofmisplaced tiles, 2) h is the Manhattan distance of tiles from their goalpositions. Which is better?

I From Lemma 8, at the iteration when the goal is found, themaximum possible number of nodes expanded uptil then form a set

Sh = {x | h(x) g(xg ) g(x)} (2.24)

Thus, if we have two heuristics h1, h2 s.t. x

h2(x) h1(x) |Sh2 | |Sh1 | h2 is more efficient. (2.25)I Can always take h(x) = max{h1(x), . . . , hm(x)}



Contents

Non-classical Search AlgorithmsHill-ClimbingSampling from a PMFSimulated AnnealingGenetic Algorithms



Local Search

currentstate

objective function

state space

global maximum

local maximumflat local maximum

shoulder

Figure 25: A 1-D state-space landscape. The aim is to find the global maximum.

Local search algorithms are used whenI The search-path itself is not important, but only the final optimal

state, e.g. 8-Queen problem, job-shop scheduling, IC design, TSP, etc.I Memory efficiency is needed. Typically only one node is retained.I The aim is to find the best state according to an objective function to

be optimized. We may be seeking the global maximum or theminimum. How can we reformulate the former to the latter?


Non-classical Search Algorithms Hill-Climbing

Algorithm 7: Hill-Climbing

input : x0, objective (value) function v(x)to maximize

output: x, the state where a localmaximum is achieved

x x0 ;while True do

y the highest-valued child of x ;if v(y) v(x) then return x ;x y

To avoid getting stuck in plateaux:

I Allow side-ways movements (upto alimit)

I Random-restart: perform search frommany randomly chosen x0 till anoptimal solution is found.

Figure 26: A ridge. The localmaxima are not directlyconnected to each other.


Non-classical Search Algorithms Sampling from a PMF

Sampling from a Probability Mass Function (pmf)

Definition 9 (PMF)

Given a discrete random-variable A with an exhaustive and ordered (canbe user-defined, if no natural order exists) list of its possible values[a1, a2, . . . , an], its pmf P(A) is a table, with probabilitiesP(A = ai ), i = 1 . . . n. Obviously,

ni=1 P(A = ai ) = 1.

Problem 10Sampling a PMF A uniform random number generator in the unit-intervalhas the probability distribution function pu[0,1](x) as shown below in thefigure. Python random.random() returns a sample x [0.0, 1.0). Howcan you use it to sample a given discrete distribution (PMF)?

1

0 1

pu[0,1](x)

xa b

P (x [a, b] ; 0 a b < 1) = b a



Definition 11 (Cumulative PMF FA for a PMF P(A))

FA(aj) , P(A aj) =n

i=1

P(A = ai )u(aj ai ), where, (3.1)

u(x) ,{

0 x < 0

1 x 0 (3.2)

FA(aj) ij

P(A = ai ) (3.3)

u(x) is called the discrete unit-step function or the Heaviside stepfunction. What is FA(an)?



I Form a vector of half-closed intervals

s ,

[0,FA(a1))

[FA(a1),FA(a2))...

[FA(an2),FA(an1))[FA(an1), 1)

, Define a0 s.t. FA(a0) , 0. (3.4)

I Let ru be a sample from a uniform random-number generator in theunit-interval [0, 1). Then,

P(ru s[i ]) = P(FA(ai1) ru < FA(ai ))= P(FA(ai1) ru FA(ai ))= FA(ai ) FA(ai1)(3.3)= P(ai ) (3.5)


Non-classical Search Algorithms Simulated Annealing

Algorithm 8: Simulated-Annealing

input : x0, objective (cost) function c(x)to minimize

output: x, a locally optimum statex x0 ;for k k0 to do

T Schedule(k) ;if 0 < T < then return x ;y a randomly selected child of x ;E c(y) c(x) ;if E < 0 then

x yelse

x y with probability P(E ,T ) ;P(E ,T ) = 1

1+eE/T eE/T ;

(Boltzmann distribution)

An example of aschedule is

Tk = T0ln(k0)

ln(k)(3.6)

Applications:

VLSI layouts,Factory-scheduling.


Non-classical Search Algorithms Simulated Annealing

Another Schedule

We first generate some random rearrangements, and use themto determine the range of values of E that will be encounteredfrom move to move. Choosing a starting value of T which isconsiderably larger than the largest E normally encountered,we proceed downward in multiplicative steps each amounting toa 10 % decrease in T . We hold each new value of T constantfor, say, 100N reconfigurations, or for 10N successfulreconfigurations, whichever comes first. When efforts to reduceE further become sufficiently discouraging, we stop.

Numerical Recipes in C: The Art of Scientific Computing.


Non-classical Search Algorithms Genetic Algorithms

Algorithm 9: Genetic Algorithm

input : P = {x}, a population of individuals,a Fitness() function to maximize

output: x, an individualrepeatPn ;for i 1 to Size(P) do

x Random-Selection(P, Fitness()) ;y Random-Selection(P, Fitness()) ;c Reproduce(x, y) ;if small probability then Mutate(c) ;Add c to Pn

P Pnuntil x P, Fitness(x) > Threshold, or enough timeelapsed ;return best individual in P



Algorithm 10: Reproduce(x, y)

N Length(x) ;R random-number from 1 to N (cross-over point);c Substring(x, 1, R) + Substring(y, R + 1, N) ;return c

(a)Initial Population

(b)Fitness Function

(c)Selection

(d)Crossover

(e)Mutation

24

23

20

11

29%

31%

26%

14%

32752411

24748552

32752411

24415124

32748552

24752411

32752124

24415411

32252124

24752411

32748152

24415417

24748552

32752411

24415124

32543213

Figure 27: The 8-Queens problem. The ith number in the string is the position ofthe queen in the ith column. The fitness function is the number of non-attackingpairs (maximum fitness 28).



Properties of Genetic Algorithms (GA)

I Crucial issue: encoding.

I Schema e.g. 236 , an instance of this schema is 23689745. Ifaverage fitness of instances of schema are above the mean, then thenumber of instances of the schema in the population will grow overtime. It is important that the schema makes some sense within thesemantics/physics of the problem.

I GAs have been used in job-shop scheduling, circuit-layout, etc.

I The identification of the exact conditions under which GAs performwell requires further research.


Logical Agents

Contents

Logical AgentsPropositional LogicEntailment and InferenceInference by Model-CheckingInference by Theorem ProvingInference by ResolutionInference with Horn ClausesAgents based on Propositional LogicTime out from Logic


Logical Agents

Knowledge-Base

A Knowledge-Base is a set of sentences expressed in a knowledgerepresentation language. New sentences can be added to the KB and itcan be queried about whether a given sentence can be inferred from whatis known.A KB-Agent is an example of a Reflex-Agent explained previously.

Algorithm 11: Knowledge-Base (KB) Agent

input : KB, a knowledge-base,t, time, initially 0.

Tell(KB, Make-Percept-Sentence(percept, t)) ;action Ask(KB, Make-Action-Query(t)) ;Tell(KB, Make-Action-Sentence(action, t)) ;t t + 1 ;return action



Propositional Logic Semantics ITruth-Value

The truth-tables of (negation), (and/conjunction), (or/disjunction)are well-known.

P Q P Q P QT T T TT F F FF T T FF F T T

P Q P Q Premise implies Conclusion (4.2a)P Q Q P Contraposition (4.2b)P Q (P Q ) (Q P ) Biconditional (4.2c)



Propositional Logic Semantics IITruth-Value

R (P Q ) (R P ) Q Associativity (4.2d)R (P Q ) (R P ) Q Associativity (4.2e)(P Q ) P Q De Morgan (4.2f)(P Q ) P Q De Morgan (4.2g)

R (P Q ) (R P ) (R Q ) Distributivity (4.2h)R (P Q ) (R P ) (R Q ) Distributivity (4.2i)



Wumpus World

PIT

1 2 3 4

1

2

3

4

START

Stench

Stench

Breeze

Gold

PIT

PIT

Breeze

Breeze

Breeze

Breeze

Breeze

Stench

Figure 28: Actions=[Move-Forward, Turn-Left, Turn-Right, Grab, Shoot, Climb],Percept=[Stench, Breeze, Glitter, Bump, Scream]



Wumpus World KB

Px ,y is true if there is a pit in [x , y ]Wx ,y is true if there is a Wumpus in [x , y ]Bx ,y is true if the agent perceives a breeze in [x , y ]Sx ,y is true if the agent perceives a stench in [x , y ]

R1 : P1,1 (4.3)R2 : B1,1 (P1,2 P2,1), R3 : B2,1 (P1,1 P2,2 P3,1) (4.4)

(4.5)

We also have percepts.

R4 : B1,1, R5 : B2,1 (4.6)

Query to the KB: = P1,2 or = P2,2.



Model and Sentences

Definition 12 (Model)

I A model is a possible-world, where we fix the truth-value of everyrelevant sentence.

I If a sentence is true in model m, we say that m satisfies , or m isa model of . For example, m = {P = F ,Q = F} is a model of = (P Q ).

I The notation M() means the set of all models of the sentence ,that is all possible assignments of the relevant truth-values, for which evaluates to true.

I A sentence is valid (is a tautology) is it is true for all models, e.g.P P .

I A sentence is satisfiable if it is true in some model.


Logical Agents Entailment and Inference

Entailment

Definition 13 (Entailment)

The sentence entails the sentence , i.e.

, if and only if (iff) M() M() (4.7)

Theorem 14 (Deduction Theorem)

For any sentences and , iff the sentence ( ) is valid, i.e.true in all models.

Theorem 15 (Proof by Contradiction)

For any sentences and , iff the sentence ( ) isunsatisfiable. Assume to be false, and show that it leads tocontradictions with known axioms of .


Logical Agents Entailment and Inference

Logical Inference

I Inference is the use of entailment to draw conclusions. For example,KB , shows that the sentence can be concluded from what theagent knows.

I KB `i denotes that the inference algorithm i derives from KB .An algorithm which derives only entailed sentences is called sound.An algorithm is complete if it can derive any sentence that is entailed.

I Simplest inference is by model-checking: list all models in which KBis true, and show that for all those models, is true.


Logical Agents Inference by Model-Checking

Model-Checking Based InferenceAlgorithm 12: TT-Entails(KB, )

input: KB, a knowledge-base; , a query sentencesymbols list of proposition symbols in KB and ;return TT-Check(KB, , symbols, model={})Algorithm 13: TT-Check(KB, , symbols, model)

input: KB, a knowledge-base; , a sentence; symbols; modelif Empty(symbols) then

if Pl-true(KB, model) thenreturn Pl-true(, model)

elsereturn true // KB is false

elseP First(symbols); tail Rest(symbols) ;return TT-Check(KB, , tail, model {P = T}) andTT-Check(KB, , tail, model {P = F})



Conjunctive Normal Form (CNF)

Definition 16 (Clause)

A clause is a disjunction of literals. Example: (P Q R)Definition 17 (CNF)

Every sentence of propositional logic is equivalent to a conjunction ofclauses: This form is called the Conjunctive Normal Form (CNF). TheCNF can be arrived at by making use of the equivalence-relationships likeDe Morgan laws, distributivity, etc. Example:

A B (A B) (B A)



Davis-Putnam-Logemann-Loveland (DPLL) AlgorithmA complete backtracking algorithm

TT-Entails can be made more efficient. First entailment is now to beinferred by solving the SAT problem: iff the sentence ( ) isunsatisfiable. DPLL has 3 improvements over TT-Entails:I Early Termination: A clause is true, if any literal is true. A sentence

(in CNF) is false if any clause is false, which occurs when all itsliterals are false. Sometimes a partial model suffices. Example:(A B) (C A) is true if A = True .

I Pure Symbol: A symbol which occurs with the same sign in allclauses. Example: (A B), (C B), (C A). If a sentence has amodel, then it has a model with pure symbols assigned to make theirliteral true: this never makes the clause false.

I Unit Clause is a single literal clause or a clause where all literals butone have already been assigned false by the model. To make aunit-clause true, an appropriate truth-value for the sole literal can befound. Example: (C B) with B already assigned true. Assigninga unit-clause may create another: unit propagation.



Algorithm 14: DPLL-Satisfiable(s)

input: s, a sentence in propositional logicclauses the set of clauses in the CNF of s ;symbols a list of propositional symbols in s ;return DPLL(clauses, symbols, { })Algorithm 15: DPLL(clauses, symbols, model)

if every clause in clauses is true in model then return true ;if some clause in clauses is false in model then return false ;P, v Find-Pure-Symbol(symbols, clauses, model) ;if P 6= then return DPLL(clauses, symbols P, model { P=v }) ;P, v Find-Unit-Clause(clauses, model) ;if P 6= then return DPLL(clauses, symbols P, model { P=v }) ;P First(symbols); rest Rest(symbols) ;return DPLL(clauses, rest, model { P= true }) or DPLL(clauses,rest, model { P= false })



WalkSATA local search algorithm for satisfiability

Algorithm 16: WalkSAT

input : C , a set of propositional clauses; p, probability of random-walk,typically 0.5; N maximum flips allowed

output: A satisfying model or failuremodel a random assignment of T/F to symbols in C . ;for i = 1 to N do

if model satisfies C then return model ;clause a randomly clause in C that is false in model ;if sample true with probability p then

flip the value in model of a randomly selected symbol from clause ;else

flip whichever symbol in clause maximizes the number of satisfiedclauses


Logical Agents Inference by Theorem Proving

Propositional Theorem Proving I

Use inference rules and equivalence-relationships to produce a chain ofconclusions which leads to the desired goal sentence.

Inference Rules

,

Modus Ponens (( ) ) (4.8)

And-Elimination ( ) (4.9)

Monotonicity of Logical Systems

The set of entailed sentences can only increase as information is added tothe KB. If KB then KB Search for Proving


Logical Agents Inference by Theorem Proving

Propositional Theorem Proving II

I Initial State: The initial KB.

I Action: All inference rules applied to all the sentences that match thetop half of the inference rule.

I Result: The application of an action results in adding the sentence inthe bottom half of the inference rule to the KB.

I Goal: The sentence were trying to prove.

The search can be performed by e.g. IDS. This search is sound, but is itcomplete?


Logical Agents Inference by Resolution

Resolution

Applies to clauses: disjunctions of literals. The inference-rule calledresolution leads to a complete inference algorithm when combined with acomplete search algorithm.

s1 . . . sk , sis1 . . . si1 si+1 . . . sk unit resolution rule (4.10)

(4.11)

If si and mj are complementary literals, the full resolution rule states:

s1 . . . sk , m1 . . . mns1 . . . si1 si+1 . . . sk m1 . . . mj1 mj+1 . . . mn

(4.12)

To show soundness, consider the two possible truth values of si . Theresulting clause after resolution should contain only one copy of eachliteral: removal of multiple copies is called factoring.



Completeness of the Resolution Algorithm I

Remark 18 (Caution)

When resolving (A B C ) and (A B), the resolvent is eitherB B C or A A C , both of which are equivalent to True . Theresolvent is not C . You can validate

(A B C ) (A B) 2 C

by a truth-table.

Definition 19 (Resolution Closure RC (S))

The set of all clauses derivable by repeated application of resolution to aset of clauses S .

Definition 20 (Resolution Applied to Contradicting Clauses)

An empty-clause results from applying resolution to contradicting clauses,e.g. to A and A.



Completeness of the Resolution Algorithm II

Theorem 21 (Ground Resolution Theorem)

If a set of clauses is unsatisfiable, then the resolution closure of thoseclauses contains the empty clause.



Explicit Model Construction when Empty-Clause / RC (S)Algorithm 17: SatisfyingModelConstruction

input : S , a set of propositional clauses;RC (S), the set of clauses which is the resolution-closure of S and

does not contain the null-clauseoutput: A satisfying model[P1, . . . ,Pn] The list of all symbols in S ;for i = 1 to n do

Find a clause RC (S) s.t. = (Pi False . . . False ) aftersubstituting P1 . . .Pi1 assigned in previous iterations ;if such a clause found then

Pi False ;else

Pi True ;return The constructed model [P1, . . . ,Pn].



A Worked-out Example

I Let the sentence S consist of the following clausesC1 : X Y Z , C2 : Z R S , C3 : S T .

I Its closure RC (S) contains, in addition to S , (using G to denoteresolution) C4 = C1 G C2 : X Y R S ,C5 = C2 G C3 : Z R T , andC6 = C1 G C5 : X Y R T .

I We now trace the Algo. 17 for [P1, . . . ,Pn] = [X ,Y ,R,T ,S ,Z ]. Inthe for-loop over i , the following selections will be made:

I i=1: X = True .I i=2: Y = True .I i=3: R = True .I i=4: T = False , due to C6 under previous assignments.I i=5: S = True .I i=6: Z = False , due to C1 under previous assignments.

It can be verified, that under this model, all clauses of RC (S) areTrue .



Proof of the Ground Resolution Theorem I

I Proof by contraposition: we prove that if the closure RC (S) does notcontain the empty-clause, then S is satisfiable. One such modelsatisfying S , which is in CNF, can be recursively constructed byassigning truth-values to all symbols [P1, . . . ,Pn] in S using theAlgo. 17 SatisfyingModelConstruction.

I Algo. SatisfyingModelConstruction produces a valid modelfor S , i.e. all of its clauses are true in this model. To prove this,assume the opposite: at some iteration i = k of the for-loop inAlgo. 17, the assignment to Pk causes a clause C of RC (S) tobecome false for the first time. For this to occur,C = (false . . . false Pk) or C = (false . . . false Pk). Ifonly one of these two is present in RC (S), then the assignment rulechooses the appropriate value for Pk to make C true.



Proof of the Ground Resolution Theorem II

I The problem occurs if both are in RC (S). But in this case, theirresolvent (false . . . false ) also has to be in RC (S), which meansthat the resolvent is already false by the assignment P1, . . . ,Pk1.This contradicts our assumption that the first falsified clause appearsat stage k. Thus, the construction never falsifies a clause in RC (S)-it thus produces a valid model for RC (S) and in particular, for S .

I Therefore, this proves that if the closure RC (S) does not contain theempty-clause, then S is satisfiable. It also proves its contraposition,namely, the Ground Resolution Theorem.



A Resolution Algorithm for Inferring Entailment

Algorithm 18: PL-Resolution(KB, )

input: KB, a knowledge-base; , a sentence; symbols; modelclauses the set of clauses in the CNF of (KB ) ;N {} ;while true do

for clause-pair Ci ,Cj clauses doresolvents Resolve(Ci ,Cj) ;if resolvents contains the empty-clause then return true ;N N resolvents

if N clauses then return false ;clauses clauses N



Wumpus World

P2,1 B1,1 B1,1 P1,2 P2,1 P1,2 B1,1 B1,1 P1,2

P2,1 P1,2P1,2 P2,1 P2,1 B1,1 P2,1 B1,1 P1,2 P2,1 P1,2B1,1 P1,2 B1,1

^

^

^

^^ ^ ^

^ ^ ^

^

^

Figure 29: CNF of B1,1 (P1,2 P2,1) along with the observation B1,1. Itentails P1,2


Logical Agents Inference with Horn Clauses

Definition 22 (Definite Clause)

A clause with only one positive literal. Every definite clause can be writtenas an implication. Example: (A B C ) (B C A).Definition 23 (Horn Clause)

I A clause with at most one positive literal. Includes definite clauses.

I Horn clauses are closed under resolution. Why?

I Horn clauses with no positive literals are called goal clauses.

I Inference with Horn clauses can be done with forward or backwardchaining. Deciding entailment with Horn clauses can be done in timethat is linear in size of the KB!


Logical Agents Inference with Horn Clauses

Definition 22 (Definite Clause)

A clause with only one positive literal. Every definite clause can be writtenas

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