Transcript Knowledge Representation and Reasoning

Master of Science in Artificial Intelligence, 2012-2014
Knowledge
Representation
and Reasoning
University "Politehnica" of Bucharest
Department of Computer Science
Fall 2012
Adina Magda Florea
http://turing.cs.pub.ro/krr_11
curs.cs.pub.ro
Lecture 1
Lecture outline
 Course goals
 Grading
 Textbooks and readings
 Syllabus
 Why KR?
 KR&R Challenges
 What is KR&R?
 Formal logic: why and how
 Links for the young researcher
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Course goals
 Provide an overview of existing representational
frameworks developed within AI, their key
concepts and inference methods.
 Acquiring skills in representing knowledge
 Understanding the principles behind different
knowledge representation techniques
 Being able to read and understand research
literature in the area of KR&R
 Being able to complete a project in this research
area
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Grading
 Course grades
Mid-term exam
Final exam
Projects
Laboratory
20%
30%
30%
20%
 Requirements: min 7 lab attendances, min 50% of term
activity (mid-term ex, projects, lab)
 Academic Honesty Policy
It will be considered an honor code violation to give or
use someone else's code or written answers, either for
the assignments or exam tests. If such a case occurs,
we will take action accordingly.
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Textbooks and Readings
 Textbooks
• Artificial Intelligence: A Modern Approach (2010) by
Stuart Russell and Peter Norvig
• Knowledge Representation and Reasoning by Ronald
•
Brachman and Hector Levesque, Morgan Kaufman,
2004
Computational Intelligence: a Logical Approach by
David Poole, Alain Mackworth, and Randy Goebel,
Oxford University Press, 1998
 Readings
• Reading materials will be assigned to you.
• You are expected to do the readings before the class
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Syllabus
1. General knowledge representation issues
Readings:
http://plato.stanford.edu/entries/logic-ai/
2. Logical agents – Logical knowledge representation and
reasoning
• First order predicate logic revisited, ATP
Readings:
AIMA Chapter 7 http://aima.cs.berkeley.edu/newchap07.pdf
• Nonmonotonic logics and reasoning
Readings:
Non-monotonic Logic, Stanford Encyclopedia of Philosophy
http://plato.stanford.edu/entries/logic-nonmonotonic/
Nonmonotonic Reasoning, G. Brewka, I. Niemela, M. Truszczynski
http://www.informatik.uni-leipzig.de/~brewka/papers/NMchapter.pdf
Nonmonotonic Reasoning With WebBased Social Networks
http://www.mindswap.org/~katz/papers/socialnet-defaults.pdf
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Syllabus
• Modal logic, logics of knowledge and beliefs
Readings: Modal logic on Wikipedia
http://en.wikipedia.org/wiki/Modal_logic
+ to be announced
• Semantic networks and description logics,
reasoning services
Readings: to be announced
• Knowledge representation for the Semantic
Web
Readings:
Ontology knowledge representation - from description logic to
OWL
Description Logics as Ontology Languages for the
Semantic Web
http://lat.inf.tu-dresden.de/research/papers/2005/BaSaJS60.pdf
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Syllabus
Midterm exam (written examination) – 1h
3. Rule based agents
• Rete: Efficient unification
Readings:
The RETE algorithm
http://www.cis.temple.edu/~ingargio/cis587/readings/rete.html
• The Soar model, universal subgoaling and chunking –
Readings:
A gentle introduction to Soar, an architecture for human cognition
http://ai.eecs.umich.edu/soar/sitemaker/docs/misc/GentleIntroduction-2006.pdf
• Modern rule based systems
Readings: to be announced
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Syllabus
4. Probabilistic agents
• Probabilistic knowledge representation and reasoning
Readings: to be announced
5. Temporal reasoning
•
Readings: to be announced
6. Reasoning with actions
•
•
Planning
Readings: to be announced
7. Intelligence without representation and reasoning
vs. Strong AI
•
Calls Debate
Final exam
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Links for the young researcher

AI-MAS Links of interest
http://aimas.cs.pub.ro/links

Academic publishing
http://en.wikipedia.org/wiki/Academic_publishing

Writing a Scientific Paper
http://www.oup.com/us/samplechapters/0841234620/?view=usa

ISI Web of Knowledge
http://isiwebofknowledge.com/

Master Journal List
http://science.thomsonreuters.com/mjl/

Conference Proceedings Citation Index
http://wokinfo.com/products_tools/multidisciplinary/webofscience/cpci/

TED – Ideas worth spreading
http://www.ted.com/
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Lecture 1
Readings for Lecture 1:
http://plato.stanford.edu/entries/logic-ai/
Readings for Lecture 2
AIMA Chapter 7
http://aima.cs.berkeley.edu/newchap07.pdf
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1. Why KR?
 We understand by "knowledge" all kinds of
facts about the world.
 Knowledge is necessary for intelligent
behavior (human beings, robots).
 In this course we consider representation of
knowledge and how we can use it in making
intelligent artifacts.
 What is knowledge?
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2. KR&R Challenges
 Challenges of KR&R:
• representation of commonsense knowledge
• the ability of a knowledge-based system to
tradeoff computational efficiency for accuracy
of inferences
• its ability to represent and manipulate
uncertain knowledge and information.
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3. What is KR?
Randall Davis, Howard Shrobe, Peter Szolovits, MIT
 A knowledge representation is most
fundamentally a surrogate, a substitute
for the thing itself, used to enable an entity
to determine consequences by reasoning
about the world.
 It is a set of ontological commitments,
i.e., an answer to the question: In what
terms should I think about the world?
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What is KR?
 It is a fragmentary theory of intelligent
reasoning, expressed in terms of three
components:
• the representation's fundamental
conception of intelligent reasoning;
• the set of inferences the representation
sanctions;
• the set of inferences it recommends.
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What is KR?
 It is a medium for pragmatically efficient
computation, i.e., the computational
environment in which reasoning is
accomplished.
• One contribution to this pragmatic efficiency is
supplied by the guidance a representation provides
for organizing information so as to facilitate making
the recommended inferences.
 It is a medium of human expression, i.e., a
language in which we say things about the
world.
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What is KR?
 If A represents B, then A stands for B and
is usually more easily accessible than B.
 Symbolic representations
 Non-symbolic representations
 Symbolic representations – set of
propositions or statements that are
believed by some agent.
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4. What is Reasoning?
 Reasoning is the use of symbolic
representations of some statements in
order to derive new ones.
 Statements are abstract objects; their
representations are concrete objects and
can be easily manipulated.
 Reasoning should scale well: we need
efficient reasoning algorithms
 http://plato.stanford.edu/entries/logical-consequence/
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5. Models of KRR
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Symbolic logic and ATP
Probabilistic
Temporal
Rules
Structured
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6. Formal logic
 Formal logic is the field of study of entailment
relations, formal languages, truth conditions,
semantics, and inference.
 All propositions/statements are represented as
formulae which have a semantics according to
the logic in question.
 Logical system = Formal language +
semantics
 Formal logics gives us a framework to discuss
different kinds of reasoning.
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6.1 Logical consequence (entailment)
 Proof centered approach to logical
consequence: the validity of a reasoning
process (argument) amounts to there
being a proof of the conclusions from the
premises.
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Logical consequence (entailment)
 Model centered approach to logical
consequence
 Models are abstract mathematical structures that
provide possible interpretations for each of the
objects in a formal language.
 Given a model for a language - define what it is
for a sentence in that language to be true
(according to that model) or not.
 In any model in which the premises are true the
conclusion is true too. (Tarski's definition of logical
consequence from 1936.)
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6.2 Model centered approach
 Interpretation of a formula
 Model of a formula
 Entailment or logical consequence
 A formula F is a logical consequence of a set of
formulas P1,…Pn iff F is true in all interpretations in
which P1,…Pn are true.
 P1,… Pn |= L F
 T Formula F is a logical consequence of a set of
formulas P1,…Pn iff P1,…Pn F is valid.
 T Formula F is a logical consequence of a set of
formulas P1,…Pn iff P1…  Pn  ~F is inconsistent.
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6.3 Proof centered approach
 Theorem, deduction
 Formal system
 Inference rule
S =< A , F , A ,  >
R 
R
R  F n  F , y =  y1,..., y n   x, x, yi  F , i = 1, n
 Premise set
 = {y1 , ... , yn }
E1 = E0 U{x| y  E0n , y  x}
n1
 Consequence of 
E0 =   A
E2 = E1 U{x| y  E1n , y  x}
n1
Ei ( i  0)
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Proof centered approach
 If E0 =   A then x  Ei is deductible from 
 |S x
 Theorems - the elements of Ei if E0 = A ( = )
x  Ei
 Demonstration
| R x
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Proof approach important notions
 Th() – set of provable theorems in 
• Monotonicity
• Idempotence - multiple applications of the
operation do not change the result
 Th() – a fixed point operator which
computes the closure of a set of formulas
 according to the rules of inference
 Th() – the least fixed point of this closure
process
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6.4 Properties of logical systems
Important properties of logical systems:
 Consistency - no theorem of the system contradicts
another.
 Soundness - the system's rules of proof will never
allow a false inference from a true premise. If a system
is sound and its axioms are true then its theorems are also guaranteed to
be true.
 Completeness - there are no true sentences in the
system that cannot, at least in principle, be proved
in the system.
 Some logical systems do not have all three properties. Kurt Godel's
incompleteness theorems show that no standard formal system of
arithmetic can be consistent and complete.
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Properties of logical systems
 A logical system L is complete iff
 |= L  implies  | 
(i.e., all valid formulas are provable)
 A logical system L is sound iff
 |  implies  |= L 
(i.e., no invalid formula is provable)
 FOPL
 Second order logics
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7. Logic based representations
2 possible aims
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
•
•
to make the system function according to the logic
to specify and validate the design
Conceptualization of the world / problem
Syntax - wffs
Semantics - significance, model
Model - the domain interpretation for which a formula is true
Model - linear or structured
M |=S  - " is true or satisfied in component S of the structure M"
Model theory
 Generate new wffs that are necessarily true, given that the old wffs
are true - entailment
KB |=L 
Proof theory
 Derive new wffs based on axioms and inference rules
KB |-i 
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PrL, FOPL
Linear model
Extend PrL, PL
Sentential logic
of beliefs
Uses beliefs atoms BA()
Index PL with agents
Situation calculus
Adds states, actions
Symbol level
Knowledge level
Modal logic
Modal operators
Structured models
Description Logics
Subsumption relationships
Logics of knowledge
and belief
Modal operators B and K
Temporal logic
Modal operators for time
Linear time
Branching time
CTL logic
Branching time
and action
Dynamic logic
Modal operators
for actions
BDI logic
Adds agents, B, D, I
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knowledge
propositional first-order
Paul is a man
a
man(Paul)
Bill is a man
b
man(Bill)
men are mortal
(x) (man(x) 
mortal(x))
First order logic
c
knowledge
first-order
second-order
smaller is
transitive
( x) (( y) (( z)
((<(x,y)  <(y,z) 
<(x,z)))))
( x) (( y) (( z)
((part-of(x,y) 
part-of(y,z) 
part-of(x,z)))))
transitive(<)
part-of is
transitive
R is transitive iff
not expressible
whenever R(x,y) and
R(y,z) hold, R(x,z)
holds too
(see however pseudosecond order)
Higher order logic
transitive(part-of)
( R) ((transitive(R) 
( x) (( y) (( z)
((R(x,y)  R(y,z) 
R(x,z)))))))
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8. Automated Reasoning
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A logical puzzle
Someone who lives in Dreadbury Mansion killed Aunt
Agatha.
Agatha, the butler, and Charles live in Dreadbury
Mansion, and are the only people who live therein.
A killer always hates his victim, and is never richer than
his victim.
Charles hates no one that Aunt Agatha hates.
Agatha hates everyone except the butler.
The butler hates everyone not richer than Aunt Agatha.
The butler hates everyone Aunt Agatha hates.
No one hates everyone.
Agatha is not the butler.
Who killed Aunt Agatha?
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35
36
37
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 Slides 35-38 are from the slides
First-Order Theorem Proving
Peter Baumgartner
NICTA, Logic and Computation Program, Canberra
[email protected]
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