An Introduction to Normal Multimodal Logics:
Interaction Axioms, Prefixed Tableau Calculus,
some [un]Decidability Results, and Applications
Matteo Baldoni
Dipartimento di Informatica - Universita` degli Studi di Torino
C.so Svizzera, 185 - I-10149 Torino (Italy)
e-mail: [email protected]
URL: http://www.di.unito.it/~baldoni
In the presentation ...
• an introduction to Modal and Multimodal Logics
• a tableau calculus for a wide class of normal multimodal logics
(inclusion [Fariñas del Cerro and Penttonen, 1988] and incestual
[Catach, 1988] multimodal logics) modular w.r.t. the axiom
systems;
• some (un)decidability results for the class of inclusion and
incestual multimodal logics;
• an application of inclusion modal logics to logic programming: the
logic programming languages NemoLOG and DyLOG.
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An Introduction to
Normal Multimodal Logics
2
1
(Mono)Modal Logic
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An Introduction to
Normal Multimodal Logics
Knowledge
3
Beliefs
Modal Logics
Actions
Dynamic changes
Time
Modal logics are suitable to deal with reasoning about
distributed knowledge
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An Introduction to
Normal Multimodal Logics
4
2
The Modal Operator “
“
This means that the
meaning of this
formula does not
depend only on the
truth-value of its
subformulae.
no truth-functional
ϕ
This means that ϕ is
not only true but
necessarily true, it is
true independently
from the scenario
(or state, world, etc.)
It qualifies the truth value of ϕ
Genova, 3 maggio 2000
An Introduction to
Normal Multimodal Logics
The Modal Operator “
ϕ is necessarily
true
ϕ is believed
5
“
ϕ
ϕ is true in any
possible scenario
ϕ is known
ϕ is always true
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3
The Modal Operator “
accessibility
relation
w1
ϕ
ϕ
if and only if
ϕ
∀wi : wRwi
M , wi ϕ
wj
wn
¬ϕ
ϕ
M , wn ϕ
M , w1 ϕ
M , wk ϕ
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An Introduction to
Normal Multimodal Logics
The Modal Operator “
ϕ
w1
ϕ
w
ϕ
wj
ϕ
7
“: Kripke semantics
Kripke
interpretation
wk
wn
ϕ
M,w
wk
w
M
“: Kripke semantics
M=
W,R ,V
¬ϕ
[Hughes and Cresswell, 1996; Fitting, 1993]
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An Introduction to
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4
Axiomatization
• all axiom schemas for the propositional calculus;
K : (ϕ ⊃ ψ ) ⊃ ( ϕ ⊃ ψ )
• the axiom schema:
• the modus ponens rule of inference;
• the necessitation rule of inference:
if I can infer ϕ then I can infer
ϕ
• some other properties...
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Properties for the Modal Operator “
“
w1
w2
4: ϕ ⊃
ϕ
w
Transitivity
(positive introspection)
w3
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5
Properties for the Modal Operator “
“
w1
w2
B :ϕ ⊃
ϕ
w
Simmetry
w3
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An Introduction to
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11
Properties for the Modal Operator “
“
w1
w2
T : ϕ ⊃ϕ
w
Reflexivity
w3
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6
Properties for the Modal Operator “
“
w1
D: ϕ ⊃ ϕ
w2
w
Seriality
w3
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Properties for the Modal Operator “
“
w1
w2
5: ϕ ⊃
ϕ
𪠪
w
Euclideanness
w3
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(negative introspection)
An Introduction to
Normal Multimodal Logics
14
7
Multimodal Logic
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An Introduction to
Normal Multimodal Logics
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Multimodal Operators
w1
a
[a]ϕ
b
w
[b]ψ
M
a
b
wn
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ψ
ϕ
a
wk
ψ•
wj
ϕ
ϕ
more than one modal
operator
• they are named by means
of labels
• “a” often identifies the
name of an agent
[Halpern and Moses, 1992]
An Introduction to
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8
The Multimodal Operator “[a]“
ϕ is necessarily
true for the
agent “a”
[a]ϕ
ϕ is believed
by “a”
ϕ is true after
executing
the action “a”
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ϕ
w
an
M=
w1
a1
a1
an
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ϕ
wk
an
wj
ϕ
17
“: Kripke semantics
ϕ
wn
ϕ is known
by “a”
An Introduction to
Normal Multimodal Logics
The Modal Operator “
an
a1
ϕ is true in any
possible scenario
of “a”
W ,R1,L,Rn, V
¬ϕ
[Genesereth and Nilsson, 1989]
An Introduction to
Normal Multimodal Logics
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9
Multimodal systems [Catach, 1991]
• Complex modalities (obtained by composing modal operators of
different types).
• Several modal aspects can be captured at the same time (e.g.,
knowledge and time, knowledge and beliefs, beliefs and actions,
etc.).
• They allow agent situations to be designed:
– different ways of reasoning;
– different ways of interacting between each other.
• Properties of modalities as set of axioms.
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An example: The fox and the raven (1)
… the fox tries to capture the raven’s cheese, in order to do so the
fox charmes the raven ...
[ fox ]
it represents what the fox believes
[ praise]
it represents the action in which the
fox prises the raven
[sing ]
it represents the action in which the
raven sings
[always]
it expresses the facts that are always
true after executing any actions
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10
An example: The fox and the raven (2)
[ fox ]
axiomatized only by K
[ praise]
axiomatized only by K
[sing ]
axiomatized only by K
[always]
axiomatized by
T ( always ) : [ always ]ϕ ⊃ ϕ
4(always ) : [ always ]ϕ ⊃ [always ][always ]ϕ
4 M (always , praise ) : [always ]ϕ ⊃ [ praise ][always ]ϕ
4 M ( always , sing ) : [ always ]ϕ ⊃ [ sing ][always ]ϕ
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An example: The fox and the raven (3)
[ fox ][ praise ]charmed (raven )
the fox believes that if it praises the
raven, then the raven is charmed
[ fox ][ praise ](charmed (raven) ⊃ sing dropped (cheese))
the fox believes that after praising
the raven may sing and so it drops
the cheese
[ fox ][ praise ] sing dropped (cheese)
the fox believes that in any moment
if the raven is charmed then it is
possible that the raven sings and so
drops the cheese
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11
An example: the friends puzzle (1)
Two friends, John and Peter, have an appointment ...
[ j ] [ p]
• modalities to represent what John and Peter know:
T ( p):[ p]ϕ ⊃ ϕ
4( p):[ p]ϕ ⊃ [ p][ p]ϕ
T ( j ):[ j ]ϕ ⊃ ϕ
4( j ):[ j ]ϕ ⊃ [ j ][ j ]ϕ
S 4( p)
[ w ( p )]
• modality to represent what Peter’s wife believes:
S 4( j )
K( j)
• interaction axioms between Peter, Peter’s wife, and John:
- if Peter knows that John knows something, then John knows that Peter
knows that thing:
P ( p , j ):[ p ][ j ]ϕ ⊃ [ j ][ p ]ϕ
- if Peter’s wife believes something, then Peter believes the same thing:
I ( w( p), p):[ w( p)]ϕ ⊃ [ p]ϕ
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An example: the friends puzzle (2)
... does each of the two friends know that the other one knows that he
has an appointment?
[ j ][ ]
i t
t [ ][ j ]
i t
t
• Peter knows the time of the appointment and that John knows the place of
their appointment:
[ p ]time
[ ][ j ] l
• Peter’s wife believes that if Peter knows the time of their appointment, then
John knows that too:
[ ( )]([ ]ti
[ j ]ti
)
• Peter knows that if John knows the place and the time of their appointment,
then John knows that he has an appointment:
[ ][ j ]( l
Genova, 3 maggio 2000
ti
i t
An Introduction to
Normal Multimodal Logics
t)
24
12
Interaction axioms:Inclusion Modal Logics
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Inclusion Modal Logics
• We are interested in the class of inclusion multimodal logics
[Fariñas del Cerro and Penttonen, 1988]
• They are characterized by set of logical axioms of the form
[t1][t2 ]...[tn ]ϕ ⊃[s1][s2 ]...[sm]ϕ (n > 0, m ≥ 0)
• Motivations:
– non-homogeneous
– interaction axioms
– they have interesting computational properties
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13
Inclusion Modal Logics: examples
• reflexivity
T (t ):[t ]ϕ ⊃ ϕ
• transitivity
4(t ):[t ]ϕ ⊃ [t ][t ]ϕ
• inclusion
• mutual trans.
• persistency
• seriality
• simmetry
• euclideanness
I (t , t' ):[t ]ϕ ⊃ [t' ]ϕ
4M (t, t' ):[t ]ϕ ⊃ [t' ][t ]ϕ
P(t, t' ):[t ][t' ]ϕ ⊃ [t' ][t ]ϕ
[t1 ][t2 ]...[tn ]ϕ ⊃ [s1 ][s2 ]...[sm ]ϕ
D(t ):[t]ϕ ⊃ t ϕ
B( t ):ϕ ⊃ [t ] t ϕ
5(t ): t ϕ ⊃ [t ] t ϕ
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W , {ℜt t ∈ MOD}, V
Inclusion Modal
Logics: possibleworlds semantics
• W is a set of “worlds”;
• the ’s are the accessibility
ℜ
relations,
one for each modality;
• V is a valuation function.
ℜt1 oℜt2 o...oℜtn ⊇ℜs1 oℜs2 o...oℜsm
[t1][t2]...[tn]ϕ ⊃[s1][s2]...[sm]ϕ
1
s1 s2 L sm ϕ ⊃ t1 t2 L tn ϕ
t
t
Genova, 3 maggio 2000
An Introduction to
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'
' 1
28
14
Inclusion Modal Logics: examples
• reflexivity
T (t ):[t ]ϕ ⊃ ϕ
ℜt ⊇ I
• transitivity
4(t ):[t ]ϕ ⊃ [t ][t ]ϕ
ℜt ⊇ ℜt o ℜt
ℜt ⊇ ℜt'
I (t , t' ):[t ]ϕ ⊃ [t' ]ϕ
• inclusion
ℜt ⊇ ℜt' o ℜt
4M (t, t' ):[t ]ϕ ⊃ [t' ][t ]ϕ
• mutual trans.
P(t, t' ):[t ][t' ]ϕ ⊃ [t' ][t ]ϕ
• persistency
ℜt o ℜt ' ⊇ ℜt ' o ℜt
D(t ):[t]ϕ ⊃ t ϕ
• seriality
B( t ):ϕ ⊃ [t ] t ϕ
• simmetry
ℜt1 o ℜt2 o...oℜtn ⊇ℜs1 o ℜs2 o...oℜsm
5(t ): t ϕ ⊃ [t ] t ϕ
• euclideanness
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Inclusion Modal Logics
• [Fariñas del Cerro & Penttonen, 88];
• for simulating the behaviour of
grammars;
[t1 ][t2 ]...[tn ]ϕ ⊃ [s1 ][s2 ]...[sm ]ϕ
t1t2 ... tn → s1s2 ... sm
• undecidability result;
Example:
but
• no proof method (a part of axiom
systems);
• no (un)decidability results of
restricted subclasses.
Genova, 3 maggio 2000
An Introduction to
Normal Multimodal Logics
a → aa
a→ε
[a ]ϕ ⊃ [a ][a ]ϕ
[a ]ϕ ⊃ ϕ
30
15
Inclusion Modal Logics
Proof Theory:
A tool for
• a prefixed tableau calculus
to deal in a uniform way with
all logics in the class by
using
directly
the
characterizing axioms as
rewriting rules
(Un)Decidability:
• about
some
subclasses
defined on the analogy with
the grammar productions of
rewriting
systems
(eg.
context-sensitive,
contextfree, right-regular).
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An Introduction to
Normal Multimodal Logics
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Proof theory: A Tableaux Calculus
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16
Axiom system vs other calculi
• Easy and intuitive.
• It is not an appropiate choice for automatization.
• ‘‘Subformula principle’’ (everything one needs in order to prove
or disprove a formula is contained in the formula itself):
– resolution;
– sequent calculi;
– tableau calculi.
• Relatively few works on this topics.
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Proof theory : a prefixed tableau calculus
resolution methods
translation methods
tableaux methods
[Fitting, 1983; ...]
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[Fariñas del Cerro and Enjalbert, 1989;
...]
[Ohlbach, 1991;
Auffray and Enjalbert, 1992;
Gasquet, 1994; ...]
Prefixed tableaux:
[Fitting, 1983; Nerode, 1989;
Catach, 1991; Massacci, 1994;
Goré, 1995; Governatori, 1995;
Cunningham and Pitt, 1996,
Beckert and Goré, 1997;
Fariñas del Cerro et al., 1998; ... ]
An Introduction to
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17
Proof Theory : a tableau calculus
It is an attempt to build an interpretation in which a given formula
is satisfiable; i.e. a refutation method.
• it does not require any normal forms;
• tableau calculi have a strong relationship with the semantics
issue, then they are easier and more natural to develop especially
for non-classical logics for which, generally, the semantics is
known better than the computational properties;
• tableau methods can supply a return answer.
[Fitting, 83; Massacci, 94; Goré, 95; ]
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Proof theory: a prefixed tableau calculus
It is a labeled tree where each node consists of:
a prefixed signed formulae, or of an accessibility relation formula
wρt w'
w: Tϕ
prefix
(constant)
prefix
prefix
formula
label: name of
accessibility relation
sign
They describe a
graph
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18
Proof theory: a prefixed tableau calculus
w:T¬ϕ
T¬
w': Fϕ
w: F¬ϕ
F¬
w':Tϕ
w'
wρt w'
t
w : F (ϕ ⊃ ψ )
F⊃
w : Tϕ
w : Fψ
w : T (ϕ ⊃ ψ )
T ⊃
w : Fϕ w : Tψ
wρt w'
w:T [t ]ϕ
T [t ]
w':Tϕ
w
T [t ]ϕ
wρt w'
w: F [t ]ϕ
F [t ]
w':Tϕ
Tϕ
w'
t
Fϕ
w F[t ]ϕ
wρt w'
They describe a calculus for K(t) !
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Proof theory: a prefixed tableau calculus
w ρ s1 w1 ... w m −1 ρ s m w '
ρ
w ρ w ' ... w ' ρ w
t1
n −1
1
wρs1 w1
wρt1 w'1
Genova, 3 maggio 2000
[t1][t2]...[tn]ϕ ⊃[s1][s2]...[sm]ϕ
tn
w1
wm−1
s1
sm
t1
tn
w'1
w'n−1
An Introduction to
Normal Multimodal Logics
wm−1ρsm w'
w'n−1 ρsn w'
38
19
Graph vs path representation
a
Axiom: [a][b]ϕ ⊃ [c]ϕ
(≡11
. a)
w2
b
(≡11
. c)
(≡1.) i
w1
c
1 i : F([a]p ∧ 〈c〉q ⊃ 〈a〉 p)
(≡11
. a .1b )
2 i : T[a]p
3 i : T〈c〉q
1 1.: F([a]p ∧〈c〉q ⊃〈a〉) p
4 i : F〈a〉 p
2 1.:T[a]p
5 w1 : Tq
6 iρcw1
iρaw2
3 1.:T〈c〉q
w2ρbw1
4 1.: F〈a〉 p
5 11
. c.:Tq
7 w2 : Fp
8 w2 : Tp
×
6 11
. a.1b.:Tq
The subprefix 11
.a
does not occur on
the branch!
?
Genova, 3 maggio 2000
An Introduction to
Normal Multimodal Logics
39
Graph vs path representation
Axioms:
[a]ϕ ⊃ [c]ϕ
a
c
(≡1.) i
(≡11
. a)
w1 ( ≡ 11
. b)
[b]ϕ ⊃ [c]ϕ
1 i : F([a] p ∧ 〈c〉q ⊃ 〈b〉 p)
2 i : T[a] p
3 i : T〈c〉q
4 i : F〈b〉 p
5 w1 : Tq
6 iρcw1 iρaw1 iρbw1
7 w1 : Fp
8 w1 :Tp
×
Genova, 3 maggio 2000
(≡11
. c)
b
1 1.: F([a]p ∧〈c〉q ⊃〈b〉 p)
2
3
4
5
1.:T[a]p
1.:T〈c〉q
1.: F〈b〉 p
11
. c.:Tq
6 11
. a.:Tq
7 11
. b.:Tq
8 11
. a.: Fp
9 11
. b.:Tp
?
An Introduction to
Normal Multimodal Logics
.b
. a , 11
The subprefix 11
11
.
and c must be
identified!
40
20
(Un)Decidability results
Genova, 3 maggio 2000
An Introduction to
Normal Multimodal Logics
Decidability
ϕ
is valid in a given IML?
Yes!
No!
41
Completeness
of
the
tableau
calculus
implies
the
semidecidability of the inclusion modal
logics.
But it is possible to define a decision
procedure which works for the whole
class of propositional inclusion
modal logics?
The class of inclusion modal logics is undecidable
[Fariñas del Cerro and Penttonen, 88]
Thue Logics are undecidable:
word problem - satisfiability
Genova, 3 maggio 2000
[t1 ][t2 ]...[tn ]ϕ ⇔ [s1 ][s2 ]...[sm ]ϕ
An Introduction to
Normal Multimodal Logics
t1t2 ... tn ↔ s1s2 ... sm
42
21
Decidability for Modal Logics
• Finite Model Property (f.m.p.): a modal system L has the f.m.p. if
and only if each non-theorem of L is false in some finite model.
• Filtration method by [Fisher and Ladner, 1979].
• Each of fifteen normal system obtained by D, T, B, 4, 5 is
decidable (has the f.m.p.) [Chellas, 1980].
• A decision procedure based on a tableau system (prefixed tableau
[Fitting, 1983]).
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(Un)Decidability results
Unrestricted grammars
t1t2 ... t n → s1s2 ... sm
U
[t1 ][t2 ]...[t n ]ϕ ⊃ [ s1 ][ s2 ]...[ sm ]ϕ
U
U
Context sensitive grammars
D
t1t 2 ... t n → s1 s2 ... s m ( n ≤ m )
Context-free grammars
t → s1 s2 ... s m
t ∈ V , s i ∈ (V ∪ T ) *
U
Right-regular grammars
t → s1 s 2 ... s m − 1 s m
t ∈ V , si < m ∈ T * , sm ∈ (V ∪ T ) *
Genova, 3 maggio 2000
Thue systems
t1t2 ... t n ↔ s1s2 ... sm
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22
Undecidability results
Unrestricted and Thue grammars:
G = (V , T , P , S )
S ⇒*G s1s2... sm
[S ]p ⊃ [s1][s2 ]...[sm]p
iff
has a tableau proof
T[S ]p
F[s1 ][s2 ]...[sm ] p
S
s1
F[s2 ]...[sm ] p
s2
F [s3 ]...[ sm ] p
S
...
s1
...
s2
F[sm ] p
sm
sm
Fp
Tp
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Undecidability results
Context sensitive, Context-free and deterministic grammars:
G = (V1 ∪ V 2 , T1 ∪ T2 , P , S )
G1 = (V1 , T1 , P1 , S1 ) G2 = (V2 , T2 , P2 , S 2 )
L ( G1 ) ∩ L ( G 2 ) ≠ ∅
iff
P = P1 ∪ P2 ∪ {S → t , S → S t t ∈ T }
∧ t∈T (〈t 〉 q ∧ [ S ]〈t 〉 q) ⊃ [ S1 ] p ⊃ 〈 S 2 〉 p has a tableau
proof
S1
T
T [S1 ]
t1
s1 s2
...
sm
...
t1
t tn
... i ...
...
S2
Genova, 3 maggio 2000
t1
T
F S p
2
ti
s1
(〈t 〉
[ S ]〈 t 〉 )
tn
...
t1
s2
t tn
... i ...
...
An Introduction to
Normal Multimodal Logics
S1
S2
t tn
... i ...
...
...
sm
F
T
46
23
A Decision Procedure?
[a ]ϕ ⊃ [b ][a ]ϕ
[a ]ϕ ⊃ [b ]ϕ
a
a
a
a
i
b
w1
a
a
b
b
w2
Genova, 3 maggio 2000
w3
...
1 i : F ( 〈 b 〉 p ⊃ 〈 a 〉[ b ] p )
2
i : T 〈b〉 p
3
i : F 〈 a 〉[ b ] p
4
w1 : Tp
5
i ρ b w1
6
w1 : F [b ] p
7
w 2 : Fp
8
w1 ρ b w2
9
w2 : T [b ] p
10
w3 : Fp
11
w 2 ρ b w3
...
iρ a w1
w 2 ρ a w1
w 2 ρ a w3
iρ a w2
w1 ρ a w3
An Introduction to
Normal Multimodal Logics
i ρ a w3
47
Incestual Modal Logics
• We are interested in the class of incestual multimodal logics
[Catach, 1988]
• They are characterized by set of logical axioms of the form
〈 a 〉[ b ]ϕ ⊃ [ c ]〈 d 〉ϕ
• Motivations:
– non-homogeneous
– interaction axioms
Modal operators can be labeled by
complex parameters built up from
atomic labels by means of union
“U” and composition “;”
[ ]
[t t' ] [t ][t' ]
[t t' ] [t] [t' ]
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24
Some well-known incestual axioms
•
•
•
•
•
•
•
•
•
•
seriality:
simmetry:
transitivity
euclideanness:
determinism:
density:
mutual ser.:
relative incl.:
persistency:
...
D(t ):[t]ϕ ⊃ t ϕ
〈 a = ε 〉[b = t ]ϕ ⊃ [c = ε ]〈 d = t 〉ϕ
B(t ):ϕ ⊃ [t] t ϕ
4( t ):[t ]ϕ ⊃ [t ][t ]ϕ
5(t ): t ϕ ⊃ [t ] t ϕ
〈 a = ε 〉[b = t ]ϕ ⊃ [c = t ; t ]〈 d = ε 〉ϕ
δ ( t ): t ϕ ⊃ [t ]ϕ
De( t ): t ϕ ⊃ t t ϕ
D(t, t' ):[t ]ϕ ⊃ t' ϕ
[t ]ϕ ⊃ ([t' ]ϕ ⊃ [t"]ϕ )
P( t , t ' ):[t ][t ' ]ϕ ⊃ [t ' ][t ]ϕ
Genova, 3 maggio 2000
〈 a〉[b]ϕ ⊃ [c]〈 d 〉ϕ
An Introduction to
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49
W , {ℜt t ∈ MOD}, V
IMLs: possibleworlds semantics
• W is a set of “worlds”;
• the ℜt ’s are the accessibility
relations, one for each modality;
• V is a valuation function.
ℜb oℜ−d1 ⊇ℜ−a1 oℜc
〈a〉[b]ϕ ⊃ [c]〈d 〉ϕ
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ℜ
ℜ
ℜb
ℜd
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An example: the three wise men puzzle
• At least one of the wise men has a white spot
[any ]( ws( a ) ∨ ws( b ) ∨ ws( c ))
• Whenever one of them has (not) a white spot, the others know this fact.
[any](¬ws( X ) ⊃ [Y ]¬ws( X ))
[any ]( ws( X ) ⊃ [Y ]ws( X ))
• [any] is a weak common
knowledge operator:
• whenever a wise men does (not)
know something, the others know
that he does (not) know that thing:
T(any) [any]ϕ ⊃ ϕ
4(any) [any]ϕ ⊃ [any][any]ϕ
5( X , Y ) ¬[ X ]ϕ ⊃ [Y ]¬[ X ]ϕ
I (any, X ) [any]ϕ ⊃ [ X ]ϕ
4 M ( X , Y ) [ X ]ϕ ⊃ [Y ][ X ]ϕ
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An Introduction to
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Incestual Modal Logics:
tableau calculus (main rules)
w ρt ∪ t ' w '
ρ
w ρt w ' w ρt ' w ' β
∈ℜ t ∪t '
51
wρt ;t ' w'
ρ
wρt w'' α
w'' ρt ' w'
∈ℜ t ;t '
w''
∈ℜ t ∪t '
∈ℜ t
∈ℜ t '
∈ℜ t '
wρa w' wρc w"
ρ
w' ρb w * w" ρd w *
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An Introduction to
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ℜ
ℜb
ℜd
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An example of a tableau proof
1i : F 〈 a 〉 ([b ' ] p ∧ [b" ] p ) ⊃ [ c ] p )
2i : T 〈 a 〉 ([b ' ] p ∧ [b" ] p )
3i : F [ c ] p
Axiom: a [b'∪b"]ϕ ⊃ [c] ε ϕ
4w1 : T ([b ' ] p ∧ [b" ] p )
5 iρ a w1
6w1 : T [b ' ] p
7 w1 : T [b" ] p
i
a
c
8w2 : Fp
9
iρ c w1
10
w1 ρ b ' ∪ b" w3
w1 ρ b ' w3
11a
12 a
w3 : Tp
11b
w1 ρ b '' w3
12 b
w3 : Tp
×
b' ( b'' )
w1
w2 ρ ε w3
w2
ε
b'∪b''
w3
×
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Decidability: some subclasses of IMLs
IMLs
〈a 〉[b]ϕ ⊃ [c ]〈 d 〉ϕ
EuLs
U
DLs
by means of the
tableau calculus
D
ConfLs
[b]ϕ ⊃ [c ]〈 d 〉ϕ
〈a 〉[b]ϕ ⊃ 〈 d 〉ϕ
〈a 〉[b]ϕ ⊃ [c ]ϕ
〈a 〉ϕ ⊃ [c ]〈 d 〉ϕ
〈a 〉ϕ ⊃ [c ]ϕ
SimLs
〈 a 〉[b]ϕ ⊃ ϕ
ϕ ⊃ [c ]〈 d 〉ϕ
GLs
U
SerLs
[b]ϕ ⊃ [c ]ϕ
〈a 〉ϕ ⊃ 〈 d 〉ϕ
[b ]ϕ ⊃ 〈 d 〉ϕ
D
U
U
by simulating
Grammar Logics
by means of the
tableau calculus
TLs
[b ]
〈 〉
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U
[ ]
〈d 〉
An Introduction to
Normal Multimodal Logics
U
by reducing the word problem to
the satisfiability problem
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27
Applications: Logic Programming Extensions
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Modal Extentions of Logic Programming
• Modal extensions of logic programming join tools for formalizing
and reasoning about temporal and epistemic knowledge with
declarative features of logic programming languages.
• They support the “context abstraction”, which allows to describe
dynamic and context-dependent properties of certain problems in
a natural and problem-oriented way.
• Goal Directed Proof Procedure: a sound and complete operational
semantics with respect to declarative semantics (filling the gap!).
• Lots of proposals [Orgun and Ma, 1994; Fisher and Owens, 1993].
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Grammar logics: NemoLOG
• beliefs, knowledges, actions, ...;
• tools for software engeneering (e.g. modularity, readability,
reusability, hierarchical dependecies, inheritance, etc.);
• parametric w.r.t. the properties of multimodal operators;
• a proof procedure that can deal in a uniform way with all
logics in the class (it uses directly the characterizing
axioms as rewriting rules ...).
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What is a program in NemoLOG ?
Goals:
[t1 ]([t2 ]a ∧ [t3 ][t4 ]b) ∧ [t5 ]c
A program is a pair:
[t 1 ]([t 2 ]a, [t 3 ][t 4 ]b), [t 5 ]c
Ds, Ax
Clauses:
[t1 ][t2 ]([t5 ]a ∧ [t6 ][t7 ]b ⊃ [t3 ][t4 ]c)
[t1 ][t2 ]([t3 ][t4 ]c : -[t5 ]a,[t6 ][t7 ]b)
Inclusion axiom clauses:
[t1 ][t2 ]
• Ds is a set of extended
clauses;
• Ax is a set of inclusion axiom
clauses.
[t3 ]
[t1][t2] [t3]
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NemoLOG:
structure knowledge and
perform epistemic
reasoning
Some
Applications
it is a framework for studying
and developing extensions of
logic programming suitable to
introduce operators for
structuring logic
programs
reason about actions
describe inheritance in a
hierarchy of classes
(modules)
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NemoLOG: an example
[export][animal]{ mode(walk).
mode(run) :- no_of_legs(X), X >= 2.
mode(gallop) :- no_of_legs(X), X >= 4. }
[ i
l]
[animal ] → [horse]
[bi d]
[export][bird]{
mode(fly).
no_of_legs(2).
covering(feather). }
[bi d ]
[export][tweety]{
owner(fred). }
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[t
t ]
[export][horse]{no_of_legs(4).
covering(hair). }
Goals:
?- [bird]mode(run).
YES!
?- [X]mode(fly).
YES! X = bird or X = tweety
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Grammar logics: DyLOG
• a multimodal language for reasoning about dynamic
domains (the effects of actions in a dinamically changing
world) in a logic programming setting;
• a ‘‘programming language Prolog-like’’ for actions: a way for
composing actions by defining conditional or iterative
actions (GOLOG, Transaction Logic);
• the procedures that define complex actions are represented
by means a set of inclusion axioms;
• a goal directed proof procedure (it uses directly the
characterizing axioms as rewriting rules ...).
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What is a program in DyLOG ?
Simple action clauses:
[always]( Fs ⊃ [a ]F )
• action laws
• precondition laws
[always]( Fs ⊃ a F )
• causal laws
[always]( Fs ⊃ F )
A program is a pair:
(Π,Obs)
• Π is a set of simple action
clau-ses
and
procedure
clauses;
• Obs is a set of initial observations.
Procedures:
p1
p2 L pn ϕ ⊃ p0 ϕ
Observations:
F
Goals:
p1
p 2 L p n Fs
Answers: a state!
a1a 2 L a m
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DyLOG: an example of program
( N > 0 ) ? m ove ( Y , X ) stack ( B − 1, Y ϕ ⊃ stack ( N − 1, Y ϕ
ϕ ⊃ stack ( 0 , X ) ϕ
pickup ( X )
putdow n ( X , Y ) ϕ ⊃ m ove ( X , Y ) ϕ
[always ]( clear ( X ) ⊃ pickup( X ) true )
[always]( on( X , Y ) ⊃ [ pickup( X )]clear ( Y ))
[always]( X ≠ Y ∧ wider ( Y , X ) ∧ clear ( Y ) ⊃ putdonw( X , Y ) true )
[always]( true ⊃ [ putdonw( X , Y )]on( X , Y ))
[ always ]( ¬ clear ( Y ) ⊃ on ( X , Y ))
[ always ]( ¬ on ( X , Y ) ⊃ clear ( Y ))
sta ck ( 2 , a ) clear ( b )
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Conclusions and future works
• Implementation vs uniformity
[Governatori, 1995; Cunningham and Pitt, 1996,Beckert and Goré,
1997]
• Extension of the tableau calculus to include dynamic logic
• Complexity of decidable classes
– Applications in Logic Programming: Epistemic reasoning;
- NemoLOG: An object-oriented logic language with state;
- DyLOG+: sensing actions and conditional plans;
• Thanks to Laura Giordano, Alberto Martelli, and Viviana Patti.
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References
•
•
•
•
•
M. Baldoni, Normal Multimodal Logics: Automatic Deduction and Logic
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Studi di Torino, Italy, 1998. Available at http://www.di.unito.it/~baldoni.
M. Baldoni. Normal Multimodal Logics with Interaction Axioms. In D. Basin, M.
D`Agostino, D. M. Gabbay, S. Matthews, and L. Vigano`, editors, Labelled
Deduction, Kluwer Accademic Publishers, May 2000.
M. Baldoni, L. Giordano, and A. Martelli. A Tableau Calculus for Multimodal Logics
and some (Un)Decidability Results. In H. de Swart, editor, Proc. of the International
Conference on Analytic Tableaux and Related Methods, TABLEAUX'98, volume
1397 of LNAI, pages 44-59. Springer-Verlag, 1998.
M. Baldoni, L. Giordano, and A. Martelli. A Modal Extension of Logic Programming:
Modularity, Beliefs and Hypothetical Reasoning. In Journal of Logic and
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NMELP’96, vol. 1216 of LNAI, pages 132-150, Springer-Verlag, 1997.
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M. Chellas. Modal Logic: an Introduction. Cambridge University Press, 1980.
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M. Fitting. Basic Modal Logic. In D. Gabbay, C. J. Hogger, J. A. Robinson, editors,
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A. Nerode. Some Lectures on Modal Logic. In F. L. Bauer, editor, Logic, Algebra,
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