Transcript Representaзгo do Conhecimento

Auto-Epistemic Logic
• Proposed by Moore (1985)
• Contemplates reflection on self knowledge
(auto-epistemic)
• Allows for representing knowledge not just
about the external world, but also about the
knowledge I have of it
Syntax of AEL
• 1st Order Logic, plus the operator L (applied
to formulas)
• L j means “I know j”
• Examples:
MScOnSW → L MScSW
(or  L MScOnSW →  MScOnSW)
young (X)  L studies (X) → studies (X)
Meaning of AEL
• What do I know?
– What I can derive (in all models)
• And what do I not know?
– What I cannot derive
• But what can be derived depends on what I
know
– Add knowledge, then test
Semantics of AEL
• T* is an expansion of theory T iff
T* = Th(T{Lj : T* |= j}  {Lj : T* |≠ j})
• Assuming the inference rule j/Lj :
T* = CnAEL(T  {Lj : T* |≠ j})
• An AEL theory is always two-valued in L, that
is, for every expansion:
 j | Lj  T*  Lj  T*
Knowledge vs. Belief
• Belief is a weaker concept
– For every formula, I know it or know it not
– There may be formulas I do not believe in,
neither their contrary
• The Auto-Epistemic Logic of knowledge
and belief (AELB), introduces also operator
B j – I believe in j
AELB Example
• I rent a film if I believe I’m neither going to
baseball nor football games
Bbaseball  Bfootball → rent_filme
• I don’t buy tickets if I don’t know I’m going to
baseball nor know I’m going to football
 L baseball   L football →  buy_tickets
• I’m going to football or baseball
baseball  football
• I should not conclude that I rent a film, but do
conclude I should not buy tickets
Axioms about beliefs
• Consistency Axiom
B
• Normality Axiom
B(F → G) → (B F → B G)
• Necessitation rule
F
BF
Minimal models
• In what do I believe?
– In that which belongs to all preferred models
• Which are the preferred models?
– Those that, for one same set of beliefs, have a minimal
number of true things
• A model M is minimal iff there does not exist a
smaller model N, coincident with M on Bj e Lj
atoms
• When j is true in all minimal models of T, we
write T |=min j
AELB expansions
• T* is a static expansion of T iff
T* = CnAELB(T  {Lj : T* |≠ j}
 {Bj : T* |=min j})
where CnAELB denotes closure using the
axioms of AELB plus necessitation for L
The special case of AEB
• Because of its properties, the case of theories
without the knowledge operator is especially
interesting
• Then, the definition of expansion becomes:
T* = YT(T*)
where YT(T*) = CnAEB(T  {Bj : T* |=min j})
and CnAEB denotes closure using the axioms
of AEB
Least expansion
• Theorem: Operator Y is monotonic, i.e.
T  T1  T2 → YT(T1)  YT(T2)
• Hence, there always exists a minimal
expansion of T, obtainable by transfinite
induction:
– T0 = CnAEB(T)
– Ti+1 = YT(Ti)
– Tb = Ua < b Ta (for limit ordinals b)
Consequences
• Every AEB theory has at least one
expansion
• If a theory is affirmative (i.e. all clauses
have at least a positive literal) then it has at
least a consistent expansion
• There is a procedure to compute the
semantics
LP for
Knowledge Representation
• Due to its declarative nature, LP has
become a prime candidate for Knowledge
Representation and Reasoning
• This has been more noticeable since its
relations to other NMR formalisms were
established
• For this usage of LP, a precise declarative
semantics was in order
Language
• A Normal Logic Programs P is a set of rules:
H  A1, …, An, not B1, … not Bm (n,m  0)
•
•
•
•
where H, Ai and Bj are atoms
Literal not Bj are called default literals
When no rule in P has default literal, P is called
definite
The Herbrand base HP is the set of all instantiated
atoms from program P.
We will consider programs as possibly infinite sets of
instantiated rules.
Declarative Programming
• A logic program can be an executable
specification of a problem
member(X,[X|Y]).
member(X,[Y|L])  member(X,L).
• Easier to program, compact code
• Adequate for building prototypes
• Given efficient implementations, why not use it to
“program” directly?
LP and Deductive Databases
• In a database, tables are viewed as sets of facts:
flight
from
to
flight (lisbon, adam).
Lisbon Adam
 flight (lisbon, london)
Lisbon London
M
M
M
• Other relations are represented with rules:
connection( A, B)  flight ( A, B).
connection( A, B)  flight ( A, C ), connection(C , B).
chooseAnother ( A, B)  not connection( A, B).
LP and Deductive DBs (cont)
• LP allows to store, besides relations, rules for
deducing other relations
• Note that default negation cannot be classical
negation in:
connection( A, B)  flight ( A, B).
connection( A, B)  flight ( A, C ), connection(C , B).
chooseAnother ( A, B)  not connection( A, B).
• A form of Closed World Assumption (CWA) is needed
for inferring non-availability of connections
Default Rules
• The representation of default rules, such as
“All birds fly”
can be done via the non-monotonic operator not
flies( A)  bird ( A), not abnormal ( A) .
bird ( P)  penguin( P).
abnormal ( P)  penguin( P).
bird (a).
penguin( p).
The need for a semantics
• In all the previous examples, classical logic is not
an appropriate semantics
– In the 1st, it does not derive not member(3,[1,2])
– In the 2nd, it never concludes choosing another
company
– In the 3rd, all abnormalities must be expressed
• The precise definition of a declarative semantics
for LPs is recognized as an important issue for its
use in KRR.
2-valued Interpretations
• A 2-valued interpretation I of P is a subset
of HP
– A is true in I (ie. I(A) = 1) iff A  I
– Otherwise, A is false in I (ie. I(A) = 0)
• Interpretations can be viewed as representing possible
states of knowledge.
• If knowledge is incomplete, there might be in some
states atoms that are neither true nor false
3-valued Interpretations
• A 3-valued interpretation I of P is a set
I = T U not F
where T and F are disjoint subsets of HP
– A is true in I iff A  T
– A is false in I iff A  F
– Otherwise, A is undefined (I(A) = 1/2)
• 2-valued interpretations are a special case, where:
HP = T U F
Models
• Models can be defined via an evaluation function Î:
– For an atom A, Î(A) = I(A)
– For a formula F, Î(not F) = 1 - Î(F)
– For formulas F and G:
• Î((F,G)) = min(Î(F), Î(G))
• Î(F  G)= 1 if Î(F)  Î(G), and = 0 otherwise
• I is a model of P iff, for all rule H  B of P:
Î(H  B) = 1
Minimal Models Semantics
• The idea of this semantics is to minimize positive
information. What is implied as true by the program is
true; everything else is false.
ableMathematician( X )  physicist ( X )
physicist (einstein)
president ( cavaco )
• {pr(c),pr(e),ph(s),ph(e),aM(c),aM(e)} is a model
• Lack of information that cavaco is a physicist, should
indicate that he isn’t
• The minimal model is: {pr(c),ph(e),aM(e)}
Minimal Models Semantics
D [Truth ordering] For interpretations I and J, I  J iff for
all atom A, I(A)  I(J), i.e.
TI  TJ and FI  FJ
T Every definite logic program has a least (truth ordering)
model.
D [minimal models semantics] An atom A is true in
(definite) P iff A belongs to its least model. Otherwise, A
is false in P.
TP operator
• The minimal models of a definite P can be
computed (bottom-up) via operator TP
D [TP] Let I be an interpretation of definite P.
TP(I) = {H: (H  Body)  P and Body  I}
T If P is definite, TP is monotone and continuous. Its
minimal fixpoint can be built by:
 I0 = {}
and
In = TP(In-1)
T The least model of definite P is TPw({})
On Minimal Models
• SLD can be used as a proof procedure for the
minimal models semantics:
– If the is a SLD-derivation for A, then A is true
– Otherwise, A is false
• The semantics does not apply to normal
programs:
– p  not q has two minimal models:
{p} and {q}
There is no least model !