SLIDE 1 An analyti al de omp
trust in terms
mental attitudes (w
in p rogress) Rob ert Demolomb e 1 1 Institut de Re her he en Info rmatique de T
June 2012 Demolomb e An analyti al de omp
trust in terms
mental attitudes (w
in p rogress)
SLIDE 2 Assumption ( la Castelfran hi et al.): the truster b elieves that if he has some pa rti ula r goal, then this goal will b e rea hed Analysis
p
supp
fo r trust:
◮
Empiri al
◮
truster's
◮
info rmation from trusted sour es
◮
Analyti al
◮
trust in something an b e supp
b y trust in
things 2
SLIDE 3 Obje tive
this w
systemati analysis
the p
so ial relationships b et w een the truster and the trustees Metho d:
◮
fo rmalization in mo dal logi
◮
no analysis
the logi al p rop erties
the involved mo dalities: b elief, a tion, intention,
...
◮
trust has the from
p rop erties 3
SLIDE 4 Mo dalities and
erato rs
φ ⇒ ψ
: φ entails ψ Bel iφ : i b elieves φ Goal iφ : i 's goal is φ A ttempt iφ : i attempts to b ring it ab
that φ Int iφ : i 's intention is to b ring it ab
that φ Obgφ : it is
ry that φ F
: it is fo rbidden that φ F
def
=
Obg¬φ Ask i, jφ : i asks j to b ring it ab
that φ Commit i(φ | ψ) : i has
himself to b ring it ab
that φ if ψ holds
φ
: φ holds no w and alw a ys in the future
♦φ
: φ will hold at some moment in the future
♦φ
def
= ¬¬φ
4
SLIDE 5 Logi al p rop erties Bel i
eys a K system Prop erties
the
(su ient) (EQUIV) Si ⊢ φ ↔ φ′ et ⊢ ψ ↔ ψ′ , alo rs ⊢ (φ ⇒ ψ) → (φ′ ⇒ ψ′) (PROPG) (φ 1 ∧ φ 2 ⇒ φ 3) → (φ 1 ∧ φ 2 ⇒ φ 1 ∧ φ 3) (TRANS) (φ 1 ⇒ φ 2) ∧ (φ 2 ⇒ φ 3) → (φ 1 ⇒ φ 3) (DIST) (φ 1 ∧ (φ 1 ⇒ φ 2) ∧ ψ) ⇒ (φ 2 ∧ ψ) (MA TIMP) (φ ⇒ ψ) → (φ → ψ) 5
SLIDE 6 Initial fo rm
trust i 's goal: to rea h a state
aairs Example: i b elieves that if he has no ash (¬φ) and his goal is to get ash (♦φ), then he will get ash (♦φ) (F1) Bel i(¬φ ∧ Goal i♦φ ⇒ ♦φ) 6
SLIDE 7 Initial fo rm
trust i 's goal: to rea h a state
aairs Example: i b elieves that if he has no ash (¬φ) and his goal is to get ash (♦φ), then he will get ash (♦φ) (F1) Bel i(¬φ ∧ Goal i♦φ ⇒ ♦φ) i 's goal: to maintain a state
aairs Example: i b elieves that if his a r w
w ell (φ) and his goal is that it still w
w ell (φ), then it still w
w ell (φ) (M1) Bel i(φ ∧ Goal iφ ⇒ φ) 7
SLIDE 8 Analysis
trust supp
to rea h If there is some agent j who holds some p rop ert y Prop( j, φ) su h that: (F2) Bel i((¬φ ∧ Goal i♦φ ⇒ ∃ jProp( j, φ)) ∧ (∃ jProp( j, φ) ⇒ ♦φ)) (F2) is a supp
fo r (F1) (F1) Bel i(¬φ ∧ Goal i♦φ ⇒ ♦φ) b e ause (F2) entails (F1) 8
SLIDE 9 Analysis
trust supp
to rea h If there is some agent j who holds some p rop ert y Prop( j, φ) su h that: (F2) Bel i((¬φ ∧ Goal i♦φ ⇒ ∃ jProp( j, φ)) ∧ (∃ jProp( j, φ) ⇒ ♦φ)) (F2) is a supp
fo r (F1) (F1) Bel i(¬φ ∧ Goal i♦φ ⇒ ♦φ) b e ause (F2) entails (F1) to maintain If there is no agent who holds some p rop ert y Prop′( j, φ) su h that: (M2) Bel i((φ ∧ Goal iφ ⇒ ¬∃ jProp′( j, φ)) ∧ (¬∃ jProp′( j, φ) ⇒ φ)) (M2) is a supp
fo r (M1) i 's assumption: the
w a y to hange φ is that there is an agent j su h that Prop′( j, φ) 9
SLIDE 10 T rust in abilit y (F2) Bel i((¬φ ∧ Goal i♦φ ⇒ ∃ jProp( j, φ)) ∧ (∃ jProp( j, φ) ⇒ ♦φ)) Abilit y Able jφ def
=
A ttempt jφ ⇒ ♦φ T rust in abilit y: Bel i Able jφ Prop 1( j, φ) def
=
A ttempt jφ ∧ ( A ttempt jφ ⇒ ♦φ) 10
SLIDE 11 T rust in abilit y (F2) Bel i((¬φ ∧ Goal i♦φ ⇒ ∃ jProp( j, φ)) ∧ (∃ jProp( j, φ) ⇒ ♦φ)) Abilit y Able jφ def
=
A ttempt jφ ⇒ ♦φ T rust in abilit y: Bel i Able jφ Prop 1( j, φ) def
=
A ttempt jφ ∧ ( A ttempt jφ ⇒ ♦φ) i 's b elief:
∃ j(
A ttempt jφ ∧ Able jφ) ⇒ ♦φ If Prop( j, φ) is Prop 1( j, φ) , (F2) holds 11
SLIDE 12 T rust in abilit y (M2) Bel i((φ ∧ Goal iφ ⇒ ¬∃ jProp′( j, φ)) ∧ (¬∃ jProp′( j, φ) ⇒ φ)) Abilit y Prop′ 1( j, φ) def
=
A ttempt j¬φ ∧ ( A ttempt j¬φ ⇒ ♦¬φ) 12
SLIDE 13 T rust in abilit y (M2) Bel i((φ ∧ Goal iφ ⇒ ¬∃ jProp′( j, φ)) ∧ (¬∃ jProp′( j, φ) ⇒ φ)) Abilit y Prop′ 1( j, φ) def
=
A ttempt j¬φ ∧ ( A ttempt j¬φ ⇒ ♦¬φ) Logi al p rop erties:
⊢ ∃ jProp′
1( j, φ) → ♦¬φ
⊢ φ → ¬∃ jProp′
1( j, φ) i 's b elief:
¬∃ j(
A ttempt j¬φ ∧ Able j¬φ) ⇒ φ If Prop′( j, φ) is Prop′ 1( j, φ) , (M2) holds 13
SLIDE 14 A tive j 's intention triggers j 's a tion A tive jφ def
=
Int jφ ⇒ A ttempt jφ Prop 2( j, φ) def
=
Int jφ ∧ ( Int jφ ⇒ A ttempt jφ) ∧ Able jφ 14
SLIDE 15 A tive j 's intention triggers j 's a tion A tive jφ def
=
Int jφ ⇒ A ttempt jφ Prop 2( j, φ) def
=
Int jφ ∧ ( Int jφ ⇒ A ttempt jφ) ∧ Able jφ Logi al p rop ert y:
⊢ ∃ jProp
2( j, φ) → ♦φ i 's b elief:
∃ j(
Int jφ ∧ A tive jφ ∧ Able jφ) ⇒ ♦φ 15
SLIDE 16 A tive Prop′ 2( j, φ) def
=
Int j¬φ ∧ ( Int j¬φ ⇒ A ttempt j¬φ) ∧ Able j¬φ Logi al p rop ert y:
⊢ ∃ jProp′
2( j, φ) → ♦¬φ i 's b elief:
¬∃ j(
Int j¬φ ∧ A tive j¬φ ∧ Able j¬φ) ⇒ φ 16
SLIDE 17 Intention adoption No rms fulllment T
If j b elieves that he is
to do something, then he intends to do that thing Ob ey jφ def
=
Bel j ObgInt jφ ⇒ Int jφ Prop
j, φ) def
=
Bel j ObgInt jφ ∧ ( Bel j ObgInt jφ ⇒ Int jφ)∧ A tive jφ ∧ Able jφ 17
SLIDE 18 Intention adoption No rms fulllment T
If j b elieves that he is
to do something, then he intends to do that thing Ob ey jφ def
=
Bel j ObgInt jφ ⇒ Int jφ Prop
j, φ) def
=
Bel j ObgInt jφ ∧ ( Bel j ObgInt jφ ⇒ Int jφ)∧ A tive jφ ∧ Able jφ Logi al p rop ert y:
⊢ ∃ jProp
j, φ) → ♦φ i 's b elief:
∃ j(
ObgInt jφ ∧ Ob ey jφ ∧ A tive jφ ∧ Able jφ) ⇒ ♦φ 18
SLIDE 19 No rms fulllment and institutional p
er If i asks j to b ring it ab
that φ, then j b elieves that it is
ry that he adopts the intention to b ring it ab
that φ Example: p
i asks j to stop his a r InstP
er i, jφ def
=
Ask i, jφ ⇒ Bel j ObgInt jφ Prop
j, φ) def
=
Ask i, jφ ∧ ( Ask i, jφ ⇒ Bel j ObgInt jφ)∧ Ob ey jφ ∧ A tive jφ ∧ Able jφ 19
SLIDE 20 No rms fulllment and institutional p
er If i asks j to b ring it ab
that φ, then j b elieves that it is
ry that he adopts the intention to b ring it ab
that φ Example: p
i asks j to stop his a r InstP
er i, jφ def
=
Ask i, jφ ⇒ Bel j ObgInt jφ Prop
j, φ) def
=
Ask i, jφ ∧ ( Ask i, jφ ⇒ Bel j ObgInt jφ)∧ Ob ey jφ ∧ A tive jφ ∧ Able jφ Logi al p rop ert y:
⊢ ∃ jProp
j, φ) → ♦φ i 's b elief:
∃ j(
Ask i, jφ ∧ Ob ey jφ ∧ InstP
er i, jφ ∧ A tive jφ ∧ Able jφ) ⇒ ♦φ 20
SLIDE 21 No rms fulllment T
Prop′
j, φ) def
=
Bel j ObgInt j¬φ ∧ ( Bel j ObgInt j¬φ ⇒ Int j¬φ)∧ A tive j¬φ ∧ Able j¬φ 21
SLIDE 22 No rms fulllment T
Prop′
j, φ) def
=
Bel j ObgInt j¬φ ∧ ( Bel j ObgInt j¬φ ⇒ Int j¬φ)∧ A tive j¬φ ∧ Able j¬φ Logi al p rop ert y:
⊢ ∃ jProp′
j, φ) → ♦¬φ i 's b elief:
¬∃ j(
Bel j ObgInt j¬φ ∧ Ob ey j¬φ ∧ A tive j¬φ ∧ Able j¬φ) ⇒ φ
¬∃ jProp′
j, φ) : no agent who fullls the no rms b elieves that ObgInt j¬φ Prop′′
j, φ) def
=
Bel j F
j¬φ ∧ ( Bel j F
j¬φ ⇒ Int j¬φ)∧ A tive j¬φ ∧ Able j¬φ
¬∃ jProp′
j, φ) : no agent who violates the no rms b elieves that F
j¬φ 22
SLIDE 23 Contra t T
If i asks to a taxi driver j to
himself to b ring i at the airp
in a
where i
himself to pa y the taxi driver, then the taxi driver j adopts the intention to b ring i at the airp
Contra t i, j(φ, ψ) def
=
Ask i, j Commit j(φ | Commit iψ) ⇒ Int jφ Prop
j, φ) def
=
Ask i, j Commit j(φ | Commit iψ) ∧
(
Ask i, j Commit j(φ | Commit iψ) ⇒ Int jφ)∧ A tive jφ ∧ Able jφ 23
SLIDE 24 Contra t T
If i asks to a taxi driver j to
himself to b ring i at the airp
in a
where i
himself to pa y the taxi driver, then the taxi driver j adopts the intention to b ring i at the airp
Contra t i, j(φ, ψ) def
=
Ask i, j Commit j(φ | Commit iψ) ⇒ Int jφ Prop
j, φ) def
=
Ask i, j Commit j(φ | Commit iψ) ∧
(
Ask i, j Commit j(φ | Commit iψ) ⇒ Int jφ)∧ A tive jφ ∧ Able jφ Logi al p rop ert y:
⊢ ∃ jProp
j, φ) → ♦φ i 's b elief:
∃ j(
Ask i, j Commit j(φ | Commit iψ) ∧ Contra t i, j(φ, ψ) ∧ A tive jφ ∧ Able jφ) ⇒ ♦φ 24
SLIDE 25 T
Prop′
j, φ) def
= ∃ k(
Ask j, k Commit k(¬φ | Commit jψ) ∧
(
Ask j, k Commit k(¬φ | Commit jψ) ⇒ Int k¬φ)∧ A tive k¬φ ∧ Able k¬φ Example: i 's goal is not b e killed and j asks to some maa memb er to kill i 25
SLIDE 26 T
Prop′
j, φ) def
= ∃ k(
Ask j, k Commit k(¬φ | Commit jψ) ∧
(
Ask j, k Commit k(¬φ | Commit jψ) ⇒ Int k¬φ)∧ A tive k¬φ ∧ Able k¬φ Example: i 's goal is not b e killed and j asks to some maa memb er to kill i Logi al p rop ert y:
⊢ ∃ jProp′
j, φ) → ♦¬φ i 's b elief:
¬∃ jProp′
j, φ) ⇒ φ 26
SLIDE 27 Altruism If j b elieves that i 's goal is φ, then j adopts the intention to b ring it ab
that φ Example: i is an
man who w ants to nd some help to ross the road and j is a w a re
i 's goal Altruis j, iφ def
=
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int jφ Prop
j, φ) def
=
Bel j(¬φ ∧ Goal i♦φ)∧
(
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int jφ)∧ A tive jφ ∧ Able jφ 27
SLIDE 28 Altruism If j b elieves that i 's goal is φ, then j adopts the intention to b ring it ab
that φ Example: i is an
man who w ants to nd some help to ross the road and j is a w a re
i 's goal Altruis j, iφ def
=
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int jφ Prop
j, φ) def
=
Bel j(¬φ ∧ Goal i♦φ)∧
(
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int jφ)∧ A tive jφ ∧ Able jφ Logi al p rop ert y:
⊢ ∃ jProp
j, φ) → ♦φ i 's b elief:
∃ j(
Bel j(¬φ ∧ Goal i♦φ) ∧ Altruis j, iφ ∧ A tive jφ ∧ Able jφ) ⇒ ♦φ i b elieves that if there is some altruist agent, his goal φ will b e rea hed 28
SLIDE 29 P erversion P ervert j, iφ def
=
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int j¬φ j intends to p revent i to rea h his goal Prop′
j, φ) def
=
Bel j(¬φ ∧ Goal i♦φ)∧
(
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int j¬φ)∧ A tive j¬φ ∧ Able j¬φ 29
SLIDE 30 P erversion P ervert j, iφ def
=
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int j¬φ j intends to p revent i to rea h his goal Prop′
j, φ) def
=
Bel j(¬φ ∧ Goal i♦φ)∧
(
Bel j(¬φ ∧ Goal i♦φ) ⇒ Int j¬φ)∧ A tive j¬φ ∧ Able j¬φ Logi al p rop ert y:
⊢ ∃ jProp′
j, φ) → ♦¬φ i 's b elief:
¬∃ j(
Bel j(¬φ∧ Goal i♦φ)∧ P ervert j, iφ∧ A tive j¬φ∧ Able j¬φ) ⇒ ♦φ i b elieves that if there is no p ervert agent the state
aairs φ will b e maintained 30
SLIDE 31 Summa ry Initial trust denition:
◮
to rea h a state
aairs
◮
to maintain a state
aairs Analysis
p
trust supp
◮
A tion and abilit y
◮
Intention and "a tiveness"
◮
Obligation and
edien e (no rms fulllment)
◮
Request and
fulllment (mutual interest)
◮
Altruism Dualit y : to rea h vs to maintain
◮
there is a "go
agent
◮
there is no "bad" agent 31
SLIDE 32 F
The p rop
axiomati
is su ient to p rove that ea h p rop ert y Prop 1 , Prop 2 , Prop 3 , ... logi ally entails Prop 32
