Artificial Intelligence: Methods and Applications

Lecture 3: Review of FOPL

Henrik Björklund Ruvan Weerasinghe

Umeå University

What I’d be doing

• Topics in Knowledge Representation

– 12th Nov (Tue) – Revision of FOL

– 15th Nov (Fri) – Reasoning with categories

– 2nd Dec (Tue) – Reasoning with uncertainty

– 6th Dec (Fri) – Probabilistic reasoning

Representing Knowledge

• Explicit vs Tacit

• Declarative vs Procedural

• Facts and Rules

• Need to

– Represent and

– Reason (do inference)

Representing Knowledge

• Natural language

• Database schema

• Propositional logic

– Restrictions on representation (syntax)

– Well defined inferencing (semantics)

• First order (predicate) logic

– Overcomes representational restriction

– Maintains inferencing mechanism

Desirable Properties of a KR System

• Representational adequacy

• Inferential adequacy

• Inferential efficiency

• Acquisitional efficiency

• Hard to optimize all in one formalism

• Main issue: knowledge acquisition bottleneck

Knowledge Engineering

• Identify the questions of interest

• Gather the knowledge (acquisition)

• Define predicates, constants (the ontology)

• Encode the axioms of the domain

• Encode the specific knowledge of the task

• Use the system

• Debug/maintain the system

Example

• Finding out if snakes are dangerous!

– color, head shape, length, movement, marking

• Turns out only some ~10% of the species are actually venomous

– Pythons can still maim or kill (without being venomous)!

• Find out the snake taxonomy

– Related to their dangerousness

Example

• Representing “a snake called slithery”

– snake_slithery / slithery_snake

– snake(slithery) / slithery(snake)

– called_slithery(snake)!

– isa(snake, slithery) / isa(slithery, snake)

– isa or instance?

• Subjects and Predicates transform (usually)

– Slithery is not poisonous

Example

• The general knowledge in the domain?

– Snakes can be venomous

– Snakes can bite

– Snakes can constrict (strangulate)

– Venom, constriction can be fatal

• Other implicit knowledge?

– By default snakes are not harmful?

– If someone dies, they stay dead!

Example

• Specific domain knowledge – Snake habitats

– Snake motion

– Snake color and marking

– Snake length

– Snake food

– When and why do snakes sting, bite, constrict

http://www.sciencekids.co.nz/sciencefacts/animals/snake.html

Exercise

• If it doesn’t rain tomorrow Carl will play tennis

• Stephan is not a graduate but is a good student

• All basketball players are tall

• Some people like durian

• If promising was a virtue Peter would be a saint

• Nobody likes taxes

• Everyone is loyal to someone

Common Mistake 1

• Typically, is the main connective with

• Common mistake: using as the main connective with :

– x At(x,UU) Smart(x)

– means “Everyone is at UU and everyone is smart”

• How do you say “Everyone at UU is smart”?

Common Mistake 2

• Typically, is the main connective with

• Common mistake: using as the main connective with :

– x At(x,UU) Smart(x)

– is true if there is anyone who is not at UU!

• How do you say “Someone at UU is smart”?

Summary of Propositional Logic

PL is declarative

PL allows partial/disjunctive/negated information – (unlike most data structures and databases)

PL is compositional: – meaning of B P is derived from meaning of B and of P

Meaning in PL is context-independent – (unlike natural language, where meaning depends on context)

PL has very limited expressive power – E.g., cannot say “all basketball players are tall“

• except by writing one sentence for each basketball player!

Why First Order (Predicate) Logic?

• Weaknesses of Propositional Logic – Cannot represent/reason at sub-sentence level

e.g. P = It is raining Q = It is not raining But ~P Q

– Cannot generalise (does not allow variables) e.g. P = Socrates is mortal Q = Plato is mortal How to represent ‘All men are mortal’?

Laws of FOPL

• If S is a sentence so is ~S

• If S1 & S2 are sentences so is S1 S2

• If S1 & S2 are sentences so is S1 S2

• If S1 & S2 are sentences so is S1 S2

• If S1 & S2 are sentences so is S1 S2

• If X is a variable and S a sentence then (a) X S is a sentence (b) X S is a sentence

Laws of FOPL (contd.)

(for all) is the universal quantifier

– means true for all values of the variable

(there exists) is the existential quantifier

– means true for some values of the variable

• e.g. Y X mother(X, Y) X Y father(X, Y) mother(X, Y) parent(X, Y)

Relationships of Quantifiers

• ~X p(X) = X ~p(X)

• ~X p(X) = X ~p(X)

X (p(X) q(X)) = X p(X) Y q(Y)

X (p(X) q(X)) = X p(X) Y q(Y)

NB: X p(X) = Y p(Y) and X p(X) = Y p(Y)

Ex.: Why is X (p(X) q(X)) X p(X) Y q(Y) ?

and X (p(X) q(X)) X p(X) Y q(Y) ?

Resolution: An Algorithm for Reasoning with PL

• STEP 1: Convert PL statements to standard form – i.e. into a conjuction of disjuncts

• STEP 2: Prove by refutation – i.e. by proving that the negation of the goal

leads to a contradiction

Example of Resolution for PL

Axiom Clause

P P

(P Q) R ~P ~Q R

(S T) Q ~S Q ~T Q

T T

Example of Resolution for PL

Resolution: Prove R. So we assume ~R ~P ~Q R (2) ~R (G) ~P ~Q P (1) ~T Q (4) ~Q ~T T (5) [] (contradiction) So, we have proved R.

Example for FOPL

• “All Romans who know Marcus either hate Caesar or think that anyone who hates anyone is crazy”

Step I (a)

• Convert to clausal form – e.g.

– “All Romans who know Marcus either hate Caesar or think that anyone who hates anyone is crazy”

X(roman(X) knows(X, marcus)) hate(X, caesar) (Y Z hate(Y, Z) think_crazy(X, Y))

Step I (b)

• Convert to conjuctive normal form – Eliminate by rule

– Standardise variables by renaming

– Move universal quantifiers to the left

– Eliminate existential quantifiers

– Convert to conjunction of disjuncts

– Rename variables taking each conjunct as clause

Step I (b) Example

• Eliminate by rule – i.e. a b to be replaced by ~a b

– further using other rules to convert all to e.g. ~(a b) = ~a ~b we get:

X(~roman(X) ~knows(X, marcus)) (hate(X, caesar) (Y Z ~hate(Y, Z) think_crazy(X, Y)))

Step I (b) Example (contd)

• Standardise variables by renaming – e.g. Xp(X) Xq(X) = Xp(X) Yq(Y)

• Move universal quantifiers to left

XYZ(~roman(X) ~knows(X, marcus)) (hate(X,caesar) (~hate(Y,Z) think_crazy(X, Y)))

Step I (b) Example (contd)

• Eliminate any existential quantifiers – X student(X) replaced by student(S1) where

S1 is a (skolem) constant

– Y Z hate(Y, Z) would however be replaced by a skolem function S2 in Y hate(S2(Y),Y)

• Convert to conjunction of disjuncts – substitute (a b) c = (a c) (b c) etc.

~roman(X) ~knows(X, marcus) hate(X, caesar) ~hate(Y, Z) think_carzy(X, Y)

Step 2

• Resolution works by combining two clauses in opposite form (one negated) to produce a resolvent clause by cancelling out the predicate concerned.

• In cases where such predicates contain arguments that are non-identical, the matching substitution (unification) needs to be applied.

Step 2 Example

winter summer man(john) woman(john) ~winter cold ~man(john) male(john) man(john) woman(john) ~man(X) male(X)

summer cold woman(john) male(john)

woman(john) male(john)

Unification

The terms f(X, a(b,c)) and f(d, a(Z, c)) unify.

Z c

a d

f

b c

a X

f

The terms are made equal if d is substituted for X, and b is substituted for Z. We also say X is instantiated to d and Z is instantiated to b, or X/d, Z/b.

Unification

The terms f(X, a(b,c)) and f(Z, a(Z, c)) unify.

Z c

a Z

f

b c

a X

f

Note that Z co-refers within the term. Here, X/b, Z/b.

Unification (not!)

The terms f(c, a(b,c)) and f(Z, a(Z, c)) do not unify.

Z c

a Z

f

b c

a c

f

No matter how hard you try, these two terms cannot be made identical by substituting terms for variables.

Unification Exercise

Do terms g(Z, f(A, 17, B), A+B, 17) and

g(C, f(D, D, E), C, E) unify?

A B

+ f

g

Z 17

A B 17

C f

g

C E

D E D

Example - English [1] Marcus is a man

[2] Marcus is a Pompeian

[3] All Pompeians are Romans

[4] Caesar is a ruler

[5] All Romans are either loyal to Caesar or hate Caesar

[6] Everyone is loyal to someone

[7] People only try to assassinate rulers they are not loyal to

[8] Marcus tries to assassinate Caesar

Query: Does Marcus hate Caesar?

Example - FOPL [1] man(Marcus)

[2] Pompeian(Marcus)

[3] X Pompeian(X) Roman(X)

[4] ruler(Caesar)

[5] X Roman(X) loyalto(X, Caesar) hate(X, Caesar)

[6] XY loyalto(X, Y)

[7] X Y man(X) ruler(Y) try_assassinate(X, Y) ~loyalto(X, Y)

[8] try_assassinate(Marcus, Caesar)

Example - CNF [1] man(Marcus)

[2] Pompeian(Marcus)

[3] ~Pompeian(X1) Roman(X1)

[4] ruler(Caesar)

[5] ~Roman(X2) loyalto(X2, Caesar) hate(X2, Caesar)

[6] loyalto(X3, f1(X3))

[7] ~man(X4) ~ruler(Y1) ~ try_assassinate(X4, Y1) ~loyalto(X4, Y1)

[8] try_assassinate(Marcus, Caesar)

Example - Resolution • [G] hate(Marcus, Caesar)?

• Assume negation: ~hate(Marcus, Caesar) + [5]

• ~Roman(Marcus) loyalto(Marcus, Caesar) + [3]

• ~Pompeian(Marcus) loyalto(Marcus, Caesar) +[2]

• loyalto(Marcus, Caesar) + [7]

• ~man(Marcus) ~ruler(Caesar) ~try_assassinate(Marcus, Caesar) + [1]

• ~ruler(Caesar) ~try_assassinate(Marcus, Caesar) + [4]

• ~try_assassinate(Marcus, Caesar) + [8]

• [] #contradiction!

• Therefore, negation is false, and so G is true.

Homework

• Marcus was a man • Marcus was a Pompeian • Marcus was born in A.D. 40 • All men are mortal • All Pompeians died when the volcano erupted in A.D. 79 • No mortal lives more than 150 years • It is now 2013 • Alive means not dead

• If someone dies, then he stays dead

Can you find two ways to prove that Marcus is dead?