1. Artificial Intelligence 6. Representing Knowledge in Predicate Logic
Course V231
Department of Computing
Imperial College, London
Jeremy Gow

2. The Case of the Silk Gloves
“It was elementary my dear Watson. The killer always left a silk glove at the scene of the murder. That was his calling card. Our investigations showed that only three people have purchased such gloves in the past year. Of these, Professor Doolally and Reverend Fisheye have iron-clad alibis, so the murderer must have been Sergeant Heavyset. When he tried to murder us with that umbrella, we knew we had our man.”

3. Not So Elementary
“The killer always left a silk glove at the scene of the murder.”
(This is inductive reasoning - guessing at a hypothesis)
“That was his calling card.”
(This is abductive reasoning - choosing an explanation)
“…only three people have purchased such gloves in the past year.”
(This is model generation - constructing examples)
“Professor Doolally and Reverend Fisheye have iron-clad alibis.”
(This is constraint-based reasoning - ruling out possibilities)
“…so the murderer must have been Sergeant Heavyset.”
(This is deductive reasoning - inferring new information from ‘known’ facts)

4. Bring on the Lawyers The murderer always confesses in some way
So, Holmes never has to defend his reasoning
This is a good job, as it’s mostly unsound
A good lawyer might point out that all the victims were - coincidentally - members of the silk glove appreciation society. Where’s your case now, Mr. Holmes?

5. Automated Reasoning Very important area of AI research
“Reasoning” usually means deductive reasoning
New facts are deduced logically from old ones
Inductive reasoning (later in course)
Guessing facts from old ones and from evidence
Two main aspects of deductive reasoning
Logical representations (thousands of them)
Rules of deduction (how to deduce new things)
Lecture 6 and Lecture 7

6. Applications of Automated Reasoning
Automated theorem proving
Automated mathematics
Axioms(A) are given, theorem statement(T) is given
Reasoning agent searches from A to T (or from T to A)
Using rules of deduction to move around the search space
Automated verification
Hardware and Software verification
That they perform as specified
Remember the Intel chip fiasco?
Intel now have lots of people working on automated verification

7. First Order Predicate Logic
Very important way of representing knowledge
FOPL
Lecture 7
Lecture 6
Prolog
Deduction
Expert
Systems
Theorem
Proving

8. Syntax and Semantics Keep thinking of predicate logic as a language
We need to communicate (share knowledge) with AI agent
Using logic as the knowledge representation scheme
Need to both inform and understand
Syntax
Symbols used and how sentences are put together
Predicates, connectives, constants, functions, variables, quantifiers
Semantics
How we interpret sentences
How we translate between the two languages
How we tell the truth of a sentence

9. Predicates Predicates are statements that Example: father(bob,bill)
Certain things are related in specific ways
Predicate name identifies the relationship
Arguments are the things being related
Arity is the number of things being related
Example: father(bob,bill)
Father is the predicate name
Relationship is (nearly) obvious: bob is bill’s father
Bob and bill are the arguments (arity here is 2)
Predicates can relate
Constants, functions and variables

10. Connectives: And, Or “And” represented by: (i) & (ii) ∧ (iii) ,
If predicates p1 and p2 are true then (p1 ∧ p2) is true
“Or” represented by: (i) ∨ (ii) | (iii) ;
If either predicate p1 or predicate p2 is true (or both)
Then (p1 ∨ p2) is true
Also note the use of brackets
Indicate where to stop and work out truth values

11. Connectives: Not, Implies, Equivalence
“Not” represented by: (i) ┓(ii) ∼ (iii) \+
Changes predicate truth (true to false, or false to true)
“Implies” represented by: → or ←
If one statement is true, then another is also true
“is equivalent to” represented by: ↔
You’ll also hear:
“if and only if” & “necessary and sufficient condition”

12. Examples of Connectives in Use
“Simon lectures AI and bioinformatics”
lectures_ai(simon) ∧ lectures_bioinformatics(simon)
Better:
lectures(simon,ai) ∧ lectures(simon,bioinfo)
“If Simon isn’t lecturing AI, Bob must be”
┓lectures(simon,ai) → lectures(bob,ai)
“George and Tony will win or Saddam will lose”
(will_win(george) ∧ will_win(tony)) ∨ ┓will_win(saddam)

13. Constants Stand for actual things
Such as england or barbara_woodhouse
Also use constants for specific words like blue
But blue has different shades…
In this case you should have:
Made blue a predicate name, then you could have had:
shade_of_blue(aqua_marine), shade_of_blue(navy), etc.
Choosing predicates and constants is important
Convention: use lower case letters for constants

14. Functions Special predicates Important
Where we think of them having inputs and an output
If arity n, then the first n-1 arguments are inputs
And the final argument is thought of the output
Important
Only a function if a set of inputs has a unique output
Use the equals sign to make I/O clear
And to make it clear this is a function

15. Example of a Function “The cost of an omelette at the Red Lion is £5”
Normally:
cost_of(omelette,red_lion,five_pounds)
But cost_of is a function
Input: the name of a meal and the name of a pub
Output: the cost of the meal
So, we can write this as:
cost_of(omelette,red_lion) = five_pounds

16. Using Functions for Abbreviation
We want to minimise misunderstanding
So, if we can, we should make sentences succinct
If we want to, we can abbreviate:
If we are talking about the output of a function
We can replace it with the predicate part
i.e., replace the RHS of equality with the LHS
Example:
“Omelettes at the Red Lion cost less than
pancakes at House of Pancakes”
less_than(cost_of(omelette,red_lion),cost_of(pancake,pancake_house))

17. Variables Want to be more expressive now cost_of(meal,red_lion) = 3
“There’s a meal at the Red Lion which costs £3”
cost_of(meal,red_lion) = 3
But meal is a constant, like omelette or pancake
Doesn’t express what we wanted it to
We’re really talking about some general meal
Not a meal in particular
Call this meal X (a variable)
cost_of(X,red_lion) = 3
Not quite right yet

18. Variables Continued Need to further express our beliefs about X
We say that:
“There is such a meal X”
“There exists such a meal X”
We need a symbol for “there exists”
This is represented as: ∃
∃ X (cost_of(X,red_lion) = 3)
Still not quite right (see later)

19. Quantifiers The exists sign is known as a quantifier
A sentence is known as “existentially quantified”
Using quantifiers is known as quantification
One other quantifier: ∀ (“forall sign”)
Example: “all students enjoy AI lectures”
∀ X (student(X) → enjoys(X,ai_lectures))
This is “universally quantified”
Forall says that each predicate involving the quantified variable is true for every possible choice of that variable
Terminology:
Constants are ‘ground variable’, and we ‘instantiate’ a variable
“All meals cost £3” becomes “Spaghetti costs £3”

20. Be careful with quantifiers!!
“There is a meal at the Red Lion which costs three pounds”
∃ X (meal(X) ∧ cost_of(red_lion, X) = 3)
What about:
“All the meals at the Red Lion cost three pounds”
Replace exists by forall:
∀ X (meal(X) ∧ cost_of(red_lion, X) = 3)
∀ X (meal(X) → cost_of(red_lion, X) = 3)
∀ X (meal(X) & serves(red_lion, X) → cost_of(red_lion, X) =3)

21. More Translation Pitfalls to Avoid
Getting ∧ and ∨ mixed up
Example from lecture 4
“Every Monday and Wednesday I go to John’s house for dinner”
∀ X (day(X,mon) ∨ day(X,weds)
→ go(me,house(john)) ∧ eat(me,dinner))

22. Even More Pitfalls Getting confused with quantifiers
Translate: “All things in the bag are red”
Which of these translations is correct?
1. ∃ X (in_bag(X) → red(X))
2. ∀ X (red(X) → in_bag(X))
3. ∀ X (∀ Y (bag(X) ∧ in_bag(Y,X) → red(Y)))

23. Translating Logic to English
Take explicit sentence, make it more succinct
Example:
∃ X (meal(X) ∧ cost_of(red_lion, X) = 3)
1. “There is something called X, where X is a meal and X costs three pounds at the Red Lion”
2. “There is a meal, X, which costs three pounds at the Red Lion”
3. “There is a meal which costs £3 at the Red Lion”

24. The Prolog Programming Language
Declarative rather than procedural
Ask a friend to buy you some groceries…
For declarative languages (Kowalski)
Algorithm = Logic + Control
That is, two most important aspects are:
The underlying logical representation chosen
The search techniques underpinning the language
We look at the logic and search in Prolog

25. From FOPL to Logic Programs
Restrict representation to only implications
Restrict implications to have the format
∀ Xi (P1 ∧ P2 ∧ … ∧ Pn → H)
These are called Horn clauses
P1 ∧ P2 ∧ … ∧ Pn is called the body
H is called the head
A collection of Horn clauses is called
A Logic Program

26. From Logic Programs to Prolog
1. Drop the universal quantification
This is assumed in Prolog
2. Write the other way around
So that P1 ∧ P2 ∧ … ∧ Pn → H
Becomes: H ← P1 ∧ P2 ∧ … ∧ Pn
3. Write ← as :-
Gives us: H :- P1 ∧ P2 ∧ … ∧ Pn
4. Write ∧ as commas and put a full stop on the end:
Gives us: H :- P1 , P2 , … , Pn.
It’s now in Prolog format

27. Example Translation Written as a Prolog Logic Program:
“In every lecture, if the lecturer is good, or the subject matter interesting, then the students are bright and attentive”
∀ X (has_good_lecturer(X) ⌵ subject_is_interesting_in(X)) → students_bright_in(X) ∧ students_attentive_in(X))
Written as a Prolog Logic Program:
students_bright_in(X) :- has_good_lecturer(X)
students_bright_in(X) :- subject_is_interesting(X)
students_attentive_in(X) :- has_good_lecturer(X)
students_attentive_in(X) :- subject_is_interesting(X)

28. Search in Prolog Prolog programs consist of Given a query:
A database of many Horn clauses
Given a query:
?- query_predicate(X).
Prolog searches the database
From top to bottom
To match the head and arity of the query predicate
To a Horn clause already in the database

29. Search in Prolog Continued
Once it has found a match for the head
It checks to see whether it can also match
The variables in the body
Does this using unification (see later lecture)
Variables can be instantiated to constants
Variables can be matched
If it gets a match, we say it has proved the query
Prolog has a closed world assumption
If it cannot prove a query, then the query is false

30. Search Example
president(X) :- first_name(X, georgedubya), second_name(X, bush).
prime_minister(X) :- first_name(X, maggie), second_name(X, thatcher).
prime_minister(X) :- first_name(X, tony), second_name(X, blair).
first_name(tonyblair, tony).
first_name(georgebush, georgedubya).
second_name(tonyblair, blair).
second_name(georgebush, bush).
?- prime_minister(P).
P = tonyblair
?- \+ president(tonyblair)
Yes

31. Prolog Details Check out the notes (and Russell & Norvig)
How arithmetic is carried out
How performance is measured
Logical inferences per second (LIPS)
How performance is greatly improved
By compiling the code
Either to something like C
Or an intermediate language like the WAM
LIPS are now in millions

32. OR-Parallelism of Prolog
When trying to find a match to query
?- prime_minister(X)
One processor takes this clause
prime_minister(X) :- first_name(X, tony), second_name(X, blair).
And another processor takes this:
prime_minister(X) :- first_name(X, maggie), second_name(X, thatcher).

33. AND-Parallelism of Prolog
When trying to find a match to query
?- prime_minister(X)
And looking in the database at:
prime_minister(X) :- first_name(X, tony), second_name(X, blair).
One processor tries to match this:
first_name(X, tony)
And another processor tries to match this:
second_name(X, thatcher)
More tricky than OR-parallelism (variables have to match)

34. Expert Systems Speak to experts in the field Major success in AI
Determine a set of important rules for the task they do
Encode them in a knowledge base, e.g., a Prolog program
Use the program to try to do the expert’s task
Major success in AI
In particular medical/agricultural diagnosis systems
"A leading expert on lymph-node pathology describes a fiendishly difficult case to the expert system, and examines the system's diagnosis. He scoffs at the system's response. Only slightly worried, the creators of the system suggest he ask the computer for an explanation of the diagnosis. The machine points out the major factors influencing its decision and explains the subtle interaction of several of the symptoms in this case. The experts admits his error, eventually.“
From Russell and Norvig

35. Example of a Prolog Expert System
Card game from last lecture
Overkill to use minimax to solve it
As we can express what each player should do
Using rules about which card to choose in which situation
Examples (rules for player one, first move)
If there are three or four even numbered cards
Then player one should choose the biggest even number
If there are three or four odd numbered cards
Then player one should choose the biggest odd number
See notes for Prolog implementation of this