Artificial Intelligence
Logic
Course 7
Knowledge Representation
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As the degree of “connectedness” and “understanding”
increase, we progress from data through information and
knowledge to wisdom
Degree of
Connectedness
Wisdom
Knowledge
Information
Understanding
patterns
Understanding
relations
Data
Understanding
principles
Degree of
understanding
Models of Representation
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Symbolic representation
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Expert systems
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Semantics networks
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Conceptual dependency
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Frame representation
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Script representation
Logic and Natural Language
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A language with concrete rules
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Many ways to translate between languages
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A statement can be represented in different logics
And perhaps differently in same logic
Expressiveness of a logic
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No ambiguity in representation (may be other errors!)
Allows unambiguous communication and processing
Very unlike natural languages e.g. English
How much can we say in this language?
Not to be confused with logical reasoning
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Logics are languages, reasoning is a process (may use logic)
Syntax and Semantics
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Syntax
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Semantics
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Rules for constructing legal sentences in the logic
Which symbols we can use (English: letters, punctuation)
How we are allowed to combine symbols
How we interpret (read) sentences in the logic
Assigns a meaning to each sentence
Example: “All lecturers are seven foot tall”
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A valid sentence (syntax)
And we can understand the meaning (semantics)
This sentence happens to be false (there is a counterexample)
Example
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In a restaurant, your Father has ordered
Fish, your Mother ordered Vegetarian, and
you ordered Meat. Out of the kitchen
comes some new person carrying the
three plates. What will happen?
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Questions
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“Who has the meat?”
“Who has the meat?”
Logic
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Propositional Logic
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Predicate Logic
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The study of statements and their connectivity
structure
The study of individuals and their properties
Propositional logic more abstract and
hence less detailed than predicate logic
Proposition Logic
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Also known as sentential logic
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Is a formal system in which knowledge is
represented as propositions
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These propositions can be joined in various ways
using logical operators
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These expressions can then be interpreted as
truth-preserving inference rules that can be used
to derive new knowledge from the old, or test the
existing knowledge.
Propositional Logic
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The logic formalism allows the derivation
of new knowledge from already known
knowledge, using logic-mathematic
deduction
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Applicability
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Automate demonstration of theorems
Artificial intelligence
Propositional Logic
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A proposition is a statement that is either
true or false (but not both)
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In propositional logic, we assume a
collection of atomic propositions are given:
p, q, r, s, t, ….
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Then we form compound propositions by
using logical connectives (logical
operators)
Propositional Logic
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Syntax
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Propositions, e.g. “it is wet”
Connectives: and, or, not, implies, iff
(equivalent)
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Brackets, T (true) and F (false)
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Semantics
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Define how connectives affect truth
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“P and Q” is true if and only if P is true and Q is true
Use truth tables to work out the truth of
statements
Propositional Logic
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A formula is valid if it holds under every
assignment. |= F to denote this. A valid formula
is called a tautology
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A sentence is contingent if and only if some
interpretation satisfies it and some interpretation
falsifies it.
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P: ‘2+2 = 4 ‘
P: ‘It will rain tomorrow ‘
A formula is unsatisfiable if it holds under no
assignment. An unsatisafiable formula is called a
contradiction
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P: ‘1=0’
Inference rules
P
P→ Q
Q
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Modus Ponens
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Chain rule
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AND introduction
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Transposition
P→ Q
Q→ R
P→ R
P
Q
P∧ Q
P→ Q
~Q→ ~ P
Example
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p = “AI covers logic programming.”
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q = “AI only covers fun topics.”
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r = “Logic programming is a fun topic.”
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¬p = “AI does not cover logic programming.”
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pΛq = “AI covers logic programming and AI only covers fun
topics.”
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pVq →r = “If AI covers logic programming and AI only
covers fun topics, then logic programming is a fun topic.”
Predicate Logic
Propositional logic combines atoms
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Predicates allow us to talk about objects. It is a
property or description of subjects in the universe
of discourse
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An atom contains no propositional connectives
Have no structure (today_is_wet, john_likes_apples)
Properties: is_wet(today)
Relations: likes(john, apples)
True or false
In predicate logic each atom is a predicate
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e.g. first order logic, higher-order logic
Predicate Logic
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More expressive logic than propositional)
Constants are objects: john, apples
Predicates are properties and relations:
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Functions transform objects:
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likes(john, apples)
likes(john, fruit_of(apple_tree))
Variables represent any object: likes(X, apples)
Quantifiers qualify values of variables
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True for all objects (Universal):
apples)
Exists at least one object (Existential):
apples)
∀X. likes(X,
∃X. likes(X,
Example
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“Every rose has a thorn”
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For all X
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if (X is a rose)
then there exists Y
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(X has Y) and (Y is a thorn)
Example
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“On Mondays and Wednesdays I go to
John’s house for dinner”
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Note the change from “and” to “or”
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Translating is problematic
Forward Chaining
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A Horn clause is a clause with at most one
positive literal
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Categories
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A rule: 1 positive literal, at least 1 negative literal. A
rule has the form "~P1 V ~P2 V ... V ~Pk V Q". This is
logically equivalent to "[P1^P2^ ... ^Pk] => Q"; thus,
an if-then implication with any number of conditions but
one conclusion.
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Examples:
 "~man(X) V mortal(X)" (All men are mortal);
 "~parent(X,Y) V ~ancestor(Y,Z) V ancestor(X,Z)" (If X is a
parent of Y and Y is an ancestor of Z then X is an ancestor of
Z.)
Forward Chaining
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Categories
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A fact or unit: 1 positive literal, 0 negative literals.
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Examples:
 "man(socrates)", "parent(elizabeth,charles)", "ancestor(X,X)"
(Everyone is an ancestor of themselves (in the trivial sense).)
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A negated goal : 0 positive literals, at least 1 negative literal.
In virtually all implementations of Horn clause logic, the
negated goal is the negation of the statement to be proved;
the knowledge base consists entirely of facts and goals. The
statement to be proven, therefore, called the goal, is therefore
a single unit or the conjuction of units; an existentially
quantified variable in the goal turns into a free variable in the
negated goal.
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Example:
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If the goal to be proven is "exists (X) male(X) ^
ancestor(elizabeth,X)" (show that there exists a male descendent of
Elizabeth) the negated goal will be "~male(X) V
~ancestor(elizabeth,X)".
The null clause: 0 positive and 0 negative literals. Appears
only as the end of a resolution proof.
Forward Chaining
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Starts with the atomic sentences in the
knowledge base and apply Modus Pones in
the forwarded direction, adding new
atomic sentences, until no future inference
can be done
Forward chaining
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Idea: fire any rule whose premises are satisfied in the KB,
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add its conclusion to the KB, until query is found
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Forward chaining example
Backward Chaining
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Starts from the goal, chaining through rules to find knows
facts that support the rule
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Depth-First Search
Idea: work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal
1.
has already been proved true, or
2.
has already failed
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Backward chaining example
Beyond True and False
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Multi-valued logics
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More than two truth values
e.g., true, false & unknown
Fuzzy logic uses probabilities, truth value in [0,1]
Modal logics
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Modal operators define mode for propositions
Epistemic logics (belief)
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e.g. p (necessarily p), p (possibly p), …
Temporal logics (time)
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e.g. p (always p), p (eventually p), …
Next
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Course
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Probabilistic Reasoning
Laboratory
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Verification of A* project and evolutionary
algorithms project