Knowledge representation review

Download Knowledge representation review

Post on 13-Feb-2017

212 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • Artificial Intelligence: Methods and Applications

    Lecture 3: Review of FOPL

    Henrik Bjrklund Ruvan Weerasinghe

    Ume University

    What Id 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

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

  • Exercise

    If it doesnt 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,

Recommended

View more >