Title: Logics for Data and Knowledge Representation
1Logics for Data and KnowledgeRepresentation
Originally by Alessandro Agostini and Fausto
Giunchiglia Modified by Fausto Giunchiglia, Rui
Zhang and Vincenzo Maltese
2Outline
- Introduction
- Syntax
- Semantics
- Satisfiability and Validity
- Kinds of frames
- Correspondence with FOL
2
3Introduction
- We want to model situations like this one
- 1. Fausto is always happy circumstances
- 2. Fausto is happy under certain
- In PL/ClassL we could have HappyFausto
- In modal logic we have
- 1. ? HappyFausto
- 2. ? HappyFausto
- As we will see, this is captured through the
notion of possible words and of accessibility
relation
4Syntax
- We extend PL with two logical modal operators
- ? (box) and ? (diamond)
-
- ?P Box P or necessarily P or P is
necessary true - ?P Diamond P or possibly P or P is
possible - Note that we define ?P ???P, i.e. ? is a
primitive symbol - The grammar is extended as follows
- ltAtomic Formulagt A B ... P Q ...
? ? - ltwffgt ltAtomic Formulagt ltwffgt ltwffgt?
ltwffgt ltwffgt? ltwffgt - ltwffgt ? ltwffgt ltwffgt ? ltwffgt ?
ltwffgt ? ltwffgt
4
5Different interpretations
Philosophy ?P P is necessary ?P P is possible
Epistemic ?aP Agent a believes P or Agent a knows P
Temporal logics ?P P is always true ?P P is sometimes true
Dynamic logics or logics of programs ?aP P holds after the program a is executed
Description logics ?HASCHILDMALE ? ?HASCHILD.MALE ?HASCHILDMALE ? ?HASCHILD.MALE
5
6Semantics Kripke Model
- A Kripke Model is a triple M ltW, R, Igt where
- W is a non empty set of worlds
- R ? W x W is a binary relation called the
accessibility relation - I is an interpretation function I L ? pow(W)
such that to each proposition P we associate a
set of possible worlds I(P) in which P holds - Each w ? W is said to be a world, point, state,
event, situation, class according to the
problem we model - For "world" we mean a PL model. Focusing on this
definition, we can see a Kripke Model as a set of
different PL models related by an "evolutionary"
relation R in such a way we are able to
represent formally - for example - the evolution
of a model in time. - In a Kripke model, ltW, Rgt is called frame and is
a relational structure.
6
7Semantics Kripke Model
- Consider the following situation
- M ltW, R, Igt
- W 1, 2, 3, 4
- R lt1, 2gt, lt1, 3gt, lt1, 4gt, lt3, 2gt, lt4, 2gt
- I(BeingHappy) 2 I(BeingSad) 1
I(BeingNormal) 3, 4
BeingHappy
1
2
3
BeingSad
BeingNormal
4
BeingNormal
7
8Truth relation (true in a world)
- Given a Kripke Model M ltW, R, Igt, a proposition
P ? LML and a possible world w ? W, we say that
w satisfies P in M or that P is satisfied by w
in M or P is true in M via w, in symbols - M, w ? P in the following cases
- 1. P atomic w ? I(P)
- 2. P ?Q M, w ? Q
- 3. P Q ? T M, w ? Q and M, w ? T
- 4. P Q ? T M, w ? Q or M, w ? T
- 5. P Q ? T M, w ? Q or M, w ? T
- 6. P ?Q for every w?W such that wRw then
M, w ? Q - 7. P ?Q for some w?W such that wRw then M,
w ? Q - NOTE wRw can be read as w is accessible from
w via R
8
9Semantics Kripke Model
- Consider the following situation
- M ltW, R, Igt
- W 1, 2, 3, 4
- R lt1, 2gt, lt1, 3gt, lt1, 4gt, lt3, 2gt, lt4, 2gt
- I(BeingHappy) 2 I(BeingSad) 1
I(BeingNeutral) 3, 4 -
- M, 2 ? BeingHappy M, 2 ? ?BeingSad
- M, 4 ? ?BeingHappy M, 1 ? ?BeingHappy M, 1
? ??BeingSad
BeingHappy
1
2
3
BeingSad
BeingNormal
4
BeingNormal
9
10Satisfiability and Validity
- Satisfiability
- A proposition P ? LML is satisfiable in a Kripke
model M ltW, R, Igt if M, w ? P for all worlds w
? W. - We can then write M ? P
- Validity
- A proposition P ? LML is valid if P is
satisfiable for all models M (and by varying the
frame ltW, Rgt). -
- We can write ? P
10
11Satisfiability
- Consider the following situation
- M ltW, R, Igt
- W 1, 2, 3, 4
- R lt1, 2gt, lt2, 2gt, lt3, 2gt, lt4, 2gt
- I(BeingHappy) 2 I(BeingSad) 1
I(BeingNormal) 3, 4 -
- M, w ? ?BeingHappy for all w ? W, therefore
?BeingHappy is satisfiable in M.
BeingHappy
1
2
3
BeingSad
BeingNormal
4
BeingNormal
11
12Validity
- Prove that P ?A ? ?A is valid
- In all models M ltW, R, Igt,
- (1) ?A means that for every w?W such that wRw
then M, w ? A - (2) ?A means that for some w?W such that wRw
then M, w ? A - It is clear that if (1) then (2) in the example
- (as we will see this is valid in serial frames)
A
1
2
3
A
12
13Kinds of frames
- Given the frame F ltW, Rgt, the relation R is
said to be - Serial iff for every w ? W, there exists w ? W
s.t. wRw - Reflexive iff for every w ? W, wRw
- Symmetric iff for every w, w ? W, if wRw then
wRw - Transitive iff for every w, w, w ? W, if wRw
and wRw then wRw - Euclidian iff for every w, w, w ? W, if wRw
and wRw then wRw - We call a frame ltW, Rgt serial, reflexive,
symmetric or transitive according to the
properties of the relation R -
13
14Kinds of frames
- Serial for every w ? W, there exists w ? W s.t.
wRw - Reflexive for every w ? W, wRw
- Symmetric for every w, w ? W, if wRw then wRw
-
-
1
2
3
1
2
1
2
3
14
15Kinds of frames
- Transitive for every w, w, w ? W, if wRw and
wRw then wRw - Euclidian for every w, w, w ? W, if wRw and
wRw then wRw -
-
1
2
3
1
2
3
15
16Valid schemas
- A schema is a formula where I can change the
variables - THEOREM. The following schemas are valid in the
class of indicated frames - D ?A ? ?A valid for serial frames
- T ?A ? A valid for reflexive frames
- B A ? ??A valid for symmetric frames
- 4 ?A ? ??A valid for transitive frames
- 5 ?A ? ??A valid for Euclidian frames
-
- NOTE if we apply T, B and 4 we have an
equivalence relation - THEOREM. The following schema is valid
- K ?(A ? B) ? (?A ? ?B) Distributivity of ?
w.r.t. ?
16
17Proof for T ? A ? A valid for reflexive frames
- Assuming M, w ? ?A, we want to prove that M, w ?
A. -
- From the assumption M, w ? ?A, we have that for
every w?W such that wRw we have that M, w ? A
(1). -
- Since R is reflexive we also have wRw, we then
imply that M, w ? A (by substituting w to w in
(1))
?A, A
1
2
17
18Proof for B A ? ??A valid for symmetric frames
- Assume M, w ? A. To prove that M, w ? ??A we
need to show that for every w ? W such that wRw
then M, w ? ?A. - M, w ? ?A is that for some w?W such that
wRw then M, w ? A. Therefore we need to
prove that for every w?W such that wRw and for
some w?W such that wRw then M, w ? A -
- Since R is symmetric, from wRw it follows that
wRw. For w?W such that w w, we have that
wRw and M, w ? A. - Hence M, w ? A.
A, ??A
?A
1
2
3
18
19Reasoning services EVAL
- Model Checking (EVAL)
- Given a (finite) model M ltW, R, Igt and a
proposition P ? LML we want to check whether M, w
? P for all w ? W - M, w ? P for all w ?
19
20Reasoning services SAT
- Satisfiability (SAT)
- Given a proposition P ? LML we want to check
whether there exists a (finite) model M ltW, R,
Igt such that M, w ? P for all w ? W - Find M such that M, w ? P for all w
20
21Reasoning services UNSAT
- Unsatisfiability (unSAT)
- Given a (finite) model M ltW, R, Igt and a
proposition P ? LML we want to check that does
not exist any world w such that M, w ? P - Verify that ?? w such that M, w ? P
21
22Reasoning services VAL
- Validity (VAL)
- Given a a proposition P ? LML we want to check
that M, w ? P for all (finite) models M ltW, R,
Igt and w ? W - Verify that M, w ? P for all M and w
22
23Correspondence between ? and ? (? and ?)
- We can define a translation function T LML ? LFO
as follows -
- 1. T(P) P(x) for all propositions P in LML
- 2. T(?P) ?T(P) for all propositions P
- 3. T(P Q) T(P) T(Q) for all propositions
P, Q and ??,?,? - 4. T(?P) ?x T(P) for all propositions P
- 5. T(?P) ?x T(P) for all propositions P
- THEOREM
- For all propositions P in LML, P is modally
valid iff T(P) is valid in FOL.
23