Valuing Others’ Opinions: Preference, Belief and Reliability Dynamics

Sujata Ghosh

1

and Katsuhiko Sano

2

1

Indian Statistical Institute, Chennai, India

2

Department of Philosophy, Graduate School of Letters, Hokkaido University, Sapporo, Japan

Keywords:

Preference, Belief, Reliability, Hybrid Logic, Public Announcement Logic, Propositional Dynamic Logic.

Abstract:

Deliberation often leads to changes in preferences and beliefs of an agent, inﬂuenced by the opinions of

others, depending on how reliable these agents are according to the agent under consideration. Sometimes,

it also leads to changes in the opposite direction, that is, reliability over agents gets updated depending on

their preferences and/or beliefs. There are various formal studies of preference and belief change based on

reliability and/or trust, but not the other way around − this work contributes to the formal study of the latter

aspect, that is, on reliability change based on agent preferences. In process, some policies of preference

change based on agent reliabilities are also discussed. A two-dimensional hybrid language is proposed to

describe such processes, and axiomatisations and decidability are discussed.

1 INTRODUCTION

Deliberation forms an important component in any

decision-making process. It is basically a conversa-

tion through which individuals provide their opinions

regarding certain issues, give preferences among pos-

sible choices, justify these preferences. This process

may lead to changes in their opinions, because they

are inﬂuenced by one another. A factor that some-

times plays a big role in enforcing such changes, is

the amount of reliability the agents have on one an-

other’s opinions. Such reliabilities may change as

well through this process of deliberation, e.g. on hear-

ing someone else’s preferences about a certain issue,

one can start or stop relying on that person’s opin-

ion. One may tend to unfriend certain friends hearing

about their preferences regarding certain issues (e.g.

Helen De Cruz’s recent remarks in her article ‘Being

Friends with a Brexiter?’ in the Philosophers On se-

ries of the Daily Nous blog

1

).

Formal studies on preferences (cf. (Arrow et al.,

2002; Endriss, 2011)) and trust (cf. (Liau, 2003; De-

molombe, 2004; Herzig et al., 2010)) abound in the

literature on logic in artiﬁcial intelligence. Recently,

there has been work on relating the notions of be-

lief and trust, e.g. about agents changing their be-

liefs based on another agent’s announcement depend-

ing on how trustworthy that agent is about the issue

1

http://dailynous.com/2016/06/28/philosophers-on-

brexit/#DeCruz

in question (e.g. see (Lorini et al., 2014)). And,

also on relating preference and reliability, e.g. about

agents changing their preferences based on another

agent’s preferences, on whom he or she relies the

most (Ghosh and Vel

´

azquez-Quesada, 2015a; Ghosh

and Vel

´

azquez-Quesada, 2015b). A pertinent issue

that arises in this context is: an agent’s assessment of

another individual’s reliability might change as well.

How would one model that? This work precisely pro-

vides a way to answer this question. We focus on re-

liability changes based on (public) announcement of

individual preferences and we provide formal frame-

works to describe such changes. In process, we also

provide some policies of preference change as well.

Note that the notion of reliability considered here is

not topic-based (in contrast to the notion of trust de-

scribed in (Lorini et al., 2014)) but deals with only

comparative judgements about agents (cf. Section 2

for details). The following provides an apt example

of the situations we would like to model:

Our Running Example: Consider three ﬂat-

mates Isabella, John and Ken discussing about

redecorating their house and they were won-

dering whether to put a print of Monet’s pic-

ture on the left wall or on the right wall of the

living room. Isabella and Ken prefer to put it

on the right wall, while John wants to put it on

the left. Isabella has more faith in John’s taste

than on hers and Ken’s, and John has more

faith in Isabella’s taste than on his’ and Ken’s.

Ghosh S. and Sano K.

Valuing Othersâ

˘

A

´

Z Opinions: Preference, Belief and Reliability Dynamics.

DOI: 10.5220/0006204806150623

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 615-623

ISBN: 978-989-758-220-2

Copyright

c

2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

615

Ken has full faith in his own taste. As long

as Isabella’s and John’s preferences are diﬀer-

ent and each think that the other’s taste is bet-

ter (by taking their preferences into consider-

ation), the three ﬂatmates would never reach

an agreement. But it so happens that on hear-

ing about John’s and Ken’s choices, Isabella

starts relying more on Ken, whereas even af-

ter hearing about Isabella’s and Ken’s choices

John’s reliability attribution to Isabella does

not change.

To model such situations we introduce a two-

dimensional hybrid logic framework extending the

basic logic proposed in (Ghosh and Vel

´

azquez-

Quesada, 2015a; Ghosh and Vel

´

azquez-Quesada,

2015b) in the line of those developed in (Gargov et al.,

1987; Sano, 2010; Seligman et al., 2013). We add

dynamic operators to model preference and reliabil-

ity changes. The main novelty of this work is that

reliability changing policies based on agent prefer-

ences are introduced and studied formally, which has

not been dealt with before. In addition, reliabilities

are modelled (more naturally) as total pre-orders (in-

stead of total orders (Ghosh and Vel

´

azquez-Quesada,

2015a; Ghosh and Vel

´

azquez-Quesada, 2015b)), and

preference changing policies are modiﬁed accord-

ingly. The proposed logic is expressive enough to deal

with both these kinds of changes.

2 TWO-DIMENSIONAL HYBRID

LOGIC

Let us ﬁrst motivate our assumptions on preference

and reliability orders that we make below in the lines

of (Ghosh and Vel

´

azquez-Quesada, 2015a). As men-

tioned earlier, we are modelling situations akin to

joint deliberation where agents announce their prefer-

ences. Each agent can change her preferences upon

getting information about the other agents’ prefer-

ences, inﬂuenced by her reliability over agents (in-

cluding herself, so she might consider herself as more

reliable than some agents but also as less reliable than

some others). Agents can also change their opinions

regarding how reliable they think the other agents are

in comparison to themselves, inﬂuenced by the an-

nounced preferences of those agents.

The agents’ preferences are represented by binary

relations (as in (Arrow et al., 2002; Gr

¨

une-Yanoﬀ

and Hansson, 2009) and further references therein),

which is typically assumed to be reﬂexive and tran-

sitive. This paper also follows this ordinary assump-

tion, and so, we note that this assumption do allow the

possibility of incomparable worlds.

The notion of reliability is related to that of trust,

a well-studied concept (e.g., (Falcone et al., 2008)),

with several proposals for its formal representation,

e.g. an attitude of an agent who believes that an-

other agent has a given property (Falcone and Castel-

franchi, 2001). One also says that “an agent i trusts

agent j’s judgement about ϕ” (called “trust on cred-

ibility” in (Demolombe, 2001)). Trust can also be

deﬁned in terms of other attitudes, such as knowl-

edge, beliefs, intentions and goals (e.g., (Demolombe,

2001; Herzig et al., 2010)), or as a semantic primi-

tive, typically by means of a neighbourhood function

(Liau, 2003). Some others (e.g., (Lorini et al., 2014))

deal with graded trust.

Reliability as discussed here is closer to the notion

of trust in (Holliday, 2010), where it is understood as

an ordering among sets of sources of information (cf.

the discussion in (Goldman, 2001)). Such a notion

of reliability does not yield any absolute judgements

(“i relies on j’s judgement [about ϕ]”), but only com-

parative ones (“for i, agent j

0

is at least as reliable as

agent j”). For the purposes of this work, similar to

(Ghosh and Vel

´

azquez-Quesada, 2015a), such com-

parative judgements suﬃce.

In contrast to (Ghosh and Vel

´

azquez-Quesada,

2015a), our reliability relation is assumed to be a re-

ﬂexive, transitive and total relation. Reﬂexivity and

transitivity are, more often than not, natural require-

ments for an ordering and totality disallows incompa-

rability, as before. The changes in reliability for an

agent depend on the information assimilated (similar

to approaches like (Rodenh

¨

auser, 2014)), in particu-

lar, about the other agents’ preferences.

The focus of this work is joint deliberation, so let A

be a ﬁnite non-empty set of agents (|A| = n ≥ 2).

Deﬁnition 1. A PR (preference/reliability) frame F is

a tuple (W, {≤

i

, 4

i

}

i∈A

) where (1) W is a ﬁnite non-

empty set of worlds; (2) ≤

i

⊆ W × W is a preorder

(i.e., a reﬂexive and transitive relation), agent i’s pref-

erence relation among worlds in W (u ≤

i

v is read as

“world v is at least as preferable as world u for agent

i”); (3) 4

i

⊆ A × A is a total pre-order (i.e., a connected

pre-order), agent i’s reliability relation among agents

in A ( j 4

i

k is read as “agent k is at least as reliable

as agent j for agent i”). Let mr(i) denote the set of all

maximally reliable agents for i.

We deﬁne u <

i

v (“u is less preferred than v for

agent i”) as u ≤

i

v and v

i

u, and u '

i

v (“u and v

are equally preferred for agent i”) as u ≤

i

v and v ≤

i

u.

Moreover, j ≺

i

k (“ j is less reliable than k for agent

i”) is deﬁned as j 4

i

k and k $

i

j, and j ≈

i

k (“ j and k

are equally reliable for agent i”) as j 4

i

k and k 4

i

j.

Example 1. Recall the example in Section 1. Put A

= {i, j, k}, where i, j, and k represent Isabella, John,

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

616

and Ken, respectively. By denoting with w

x

the world

where ‘Monet’s picture is at wall x’ (x = l, r), the ex-

ample’s situation can be represented by a PR frame

F

exp

= ({w

l

, w

r

}, {≤

y

, 4

y

}

y∈A

) in which the preference

orders are given by: w

l

<

i

w

r

, w

r

<

j

w

l

and w

l

<

k

w

r

,

and the reliability orders are given by: i ≈

i

k ≺

i

j,

j ≈

j

k ≺

j

i and j ≈

k

i ≺

k

k.

In (Ghosh and Vel

´

azquez-Quesada, 2015a),

Ghosh and Vel

´

azquez-Quesada propose a language

to talk about the preference changes and their ef-

fects. Following the semantic idea of (Seligman et al.,

2013), we extend their syntax for the static language

into a two-dimensional syntax with the help of de-

pendent product of two hybrid logics (Sano, 2010).

Let P be a countable inﬁnite set of propositional vari-

ables, N

1

= { a, b, c, . . . } be a countable inﬁnite set of

world-nominals (syntactic names for worlds) and let

N

2

= { i, j, k, . . . } be a countable inﬁnite set of agent-

nominals (syntactic names for agents).

Deﬁnition 2 (Language H L). Formulas ϕ,ψ,...

(read ϕ as “the current agent satisﬁes the property ϕ

in the current state” or indexically as “I am ϕ in the

current state”) and relational expressions (or program

terms) π, ρ, . . . of the language HL are given by

ϕ, ψ ::= p | a | i | ¬ϕ | ϕ ∨ ψ | @

i

ϕ | @

a

ϕ | hπiϕ,

π, ρ ::= 1

W

| ≤ | ≥ | 1

A

| v

k

| w

k

| − α |

π ∪ π

0

| π ∩ π

0

| (π, j) u

i

(π

0

, k)| ?(ϕ, ψ),

where p ∈ P, a ∈ N

1

, i,j, k ∈ N

2

, α ∈ { 1

W

, ≤, ≥ } ∪

{

1

A

, v

k

, w

k

| k ∈ N

2

}

. Propositional constants (>, ⊥),

other Boolean connectives (∧, →, ↔) and the dual

modal universal operators [π] are deﬁned as usual,

e.g. [π] ϕ := ¬hπi¬ϕ. Moreover, we deﬁne h<iϕ as

h≤ ∩− ≥iϕ and h@

k

iϕ as hv

k

∩ − w

k

iϕ, respectively.

Finally, ?ϕ is deﬁned as the program term ?(ϕ,ϕ).

We note that @

a

ϕ is read as “the current agent sat-

isﬁes ϕ in the world named by a”and @

i

ϕ as “agent

i satisﬁes ϕ in the current world.” The set of rela-

tional expressions contains the constants 1

W

, 1

A

(the

global relations, whose corresponding operators mean

“for all states” and “for all agents”, respectively), the

preference and reliability relations (≤, v

k

), their re-

spective converse relations (≥, w

k

; cf. (Burgess, 1984;

Goldblatt, 1992)), all the complements of the atomic

relations, and an additional construct of the forms

(π, j) u

i

(π

0

, k) (needed for deﬁning distributed prefer-

ence later in Section 3, explained below) and ?(ϕ, ψ)

(a generalization of the test operator in (Harel et al.,

2000), also explained below), and it is closed under

union and intersection operations over relations.

The formulas are interpreted in terms of world-

agent pairs below, and we may read [≤]ϕ as “in all

states which the current agent considers as least as

good as the current state, the current agent satisﬁes

ϕ”. Moreover, we may read hv

k

iϕ as “there is a more

or equally reliable agent j than the current agent such

that j satisﬁes ϕ, from agent k’s perspective.” For ex-

ample, @

i

hv

k

ij can be read as “agent j is more or

equally reliable than agent i from agent k’s perspec-

tive.” hw

k

iϕ is read as “there is a less or equally reli-

able agent j than the current agent such that j satisﬁes

ϕ, from agent k’s perspective.”

We note that the program construction ?(ϕ,ψ)

(check if the ﬁrst element of a given pair of states sat-

isﬁes ϕ and if the second does ψ) is a generalization

of the test operator in the standard (regular) proposi-

tional dynamic logic (Harel et al., 2000). So ?ϕ :=

?(ϕ, ϕ) enables us to check if both elements of a given

pair satisﬁes ϕ. Moreover, the program construction

(π, j) u

i

(π

0

, k) enables us to deﬁne, as agent i’s re-

lation between states, the distributed preference be-

tween agents j and k, i.e., the intersection of j’s pref-

erence and k’s preference. Together with the other

program constructions, it is useful for providing the

axiom system for the preference and reliability chang-

ing operations to be introduced in Section 3. The fol-

lowing two deﬁnitions establish what a model is and

how formulas of HL are interpreted over them.

Deﬁnition 3 (PR model). A PR model is a tuple M =

(F, V) where F = (W, {≤

i

, 4

i

}

i∈A

) is a PR-frame and V

is a valuation function from P ∪ N

1

∪ N

2

to P(W × A)

assigning a subset of the form { w} × A to a world-

nominal a ∈ N

1

and a subset of the form W × {i} to

an agent-nominal i ∈ N

2

. Throughout the paper, we

denote the unique element in the ﬁrst coordinate of

V(a) = { w } × A and the second coordinate of V(i) =

W × { a } by a and i, respectively.

Deﬁnition 4 (Truth deﬁnition). Given a PR-model M,

a satisfaction relation M, (w, i) ϕ , and relations R

π

⊆

(W × A)

2

are deﬁned by simultaneous induction by:

M,(w, i) p iﬀ (w, i) ∈ V(p),

M,(w, i) a iﬀ w = a,

M,(w, i) i iﬀ i = i,

M,(w, i) ¬ϕ iﬀ M,(w, i) 1 ϕ,

M,(w, i) ϕ ∨ ψ iﬀ M,(w, i) ϕ or M,(w, i) ψ,

M,(w, i) @

a

ϕ iﬀ M,(a, i) ϕ,

M,(w, i) @

i

ϕ iﬀ M,(w, i) ϕ,

M,(w, i) hπiψ iﬀ For some (v, j) ∈ W × A,

(w, i)R

π

(v, j) and M, (v, j) ψ,

(w, i)R

1

W

(v, j) iﬀ w, v ∈ W and i = j,

(w, i)R

≤

(v, j) iﬀ w ≤

i

v and i = j,

(w, i)R

≥

(v, j) iﬀ v ≤

i

w and i = j,

Valuing Othersâ

˘

A

´

Z Opinions: Preference, Belief and Reliability Dynamics

617

(w, i)R

− α

(v, j) iﬀ ((w, i),(v, i)) < R

α

and i = j

(α ∈ { 1

W

, ≤,≥ }),

(w, i)R

1

A

(v, j) iﬀ w = v and i, j ∈ A,

(w, i)R

v

k

(v, j) iﬀ w = v and i 4

k

j,

(w, i)R

w

k

(v, j) iﬀ w = v and j 4

k

i,

(w, i)R

− β

(v, j) iﬀ w = v and ((w,i), (v, i)) < R

β

(β ∈

{

1

A

, v

k

, w

k

| k ∈ N

2

}

),

(w, i)R

π∪ρ

0

(v, j) iﬀ (w, i)R

π

(v, j) or (w, i)R

ρ

(v, j),

(w, i)R

π∩ρ

(v, j) iﬀ (w, i)R

π

(v, j) and (w, i)R

ρ

(v, j),

(w, i)R

(π,j)u

i

(π

0

,k)

(v, j) iﬀ i = j = i and (w,j)R

π

(v, j)

and (w, k)R

ρ

(v, k)

(w, i)R

?(ϕ,ψ)

(v, j) iﬀ M, (w,i) ϕ and M, (v, j) ψ.

We say that ϕ is valid in a PR-model M (written: M ϕ)

if M, (w,i) ϕ for all pairs (w, i) in M.

The logic HL is so expressive that we can formal-

ize the notion of belief as well as our preference and

reliability dynamics introduced in the later sections.

For example, following the idea found in (Boutilier,

1994), we can deﬁne the conditional belief opera-

tor B(ψ,ϕ) (read “under the condition that the cur-

rent agent satisﬁes ψ, the current agent believes that

she satisﬁes ϕ” or “the current agent desires (or has a

goal) that she satisﬁes ϕ under the condition that she

satisﬁes ψ”) by

B

ψ

ϕ := [1

W

]((ψ ∧ ϕ) → h≤i(ψ ∧ ϕ ∧ [≤](ψ → ϕ))).

Then the unconditional belief operator Bϕ is deﬁned

as B(>, ϕ), which read as “the current agent believes

that she satisﬁes ϕ” or “in the most preferred states for

the current agent, she satisﬁes ϕ.” We can also deﬁne

the conditional reliability operator R

k

(ψ, ϕ) (read “the

most reliable ψ-agents for agent k satisfy ϕ.”) by

R

k

(ψ, ϕ) := [1

A

](ψ → hv

k

i(ψ ∧ [v

k

](ψ → ϕ))),

where we can simplify the clause because of connect-

edness as noted in (Boutilier, 1994). The uncondi-

tional version R

k

ϕ of R

k

(ψ, ϕ) is deﬁned as R

k

(>, ϕ)

which read as “the most reliable agents for agent k

satisfy ϕ.” We may also deﬁne the “diamond”-version

of R

k

ϕ as ¬R

k

¬ϕ to denote hR

k

iϕ. Then hR

k

ij means

that agent j is one of the most reliable agents for k.

Example 2. Let us represent “the current agent

likes to put Monet’s picture on wall x” by a state-

nominal a

x

in the setting of Example 1. On the

PR-frame F

exp

of Example 1, we deﬁne V(a

x

) =

{ (w

x

, i), (w

x

, j), (w

x

, k) } where x = l or r. We use i,

j, k as syntactic names (i.e., agent nominals) for i, j

and k, where we interpret, e.g., i = i in terms of our

valuation function V. Deﬁne M

exp

:= (F

exp

, V). For

example, the preference w

l

<

i

w

r

can be formalized

as a formula @

i

@

a

l

h<ia

r

, which is valid on M

exp

.

We can formalize Isabella’s reliability of i ≈

i

k ≺

i

j as

@

i

hv

i

ik∧@

k

hv

i

ii∧@

k

h@

i

ij, which is valid on M

exp

.

Moreover, @

i

Ba

x

formalizes “Isabella believes that

she likes to put Monet’s picture on wall x” and, when

x = r, @

i

Ba

r

is valid on M

exp

. Similarly, @

j

Ba

l

and

@

k

Ba

r

are also valid in M

exp

. We can see that, from

Isabella’s perspective, Ken is one of the most reliable

agents who believes that a

r

. This can be formalized

as R

i

(Ba

r

, k).

The static axiom systems HPR and HPR

(m,n)

are

given as in Table 1, where uniform substitution means

a substitution that uniformly replaces propositional

variables by formulas and nominals from N

i

by nom-

inals from N

i

(i = 1 or 2).

Theorem 1 (Soundness and completeness). ϕ is valid

in all (possibly inﬁnite) PR-models iﬀ ϕ is derivable

in HPR. Moreover, ϕ is valid in all PR-models with

ﬁxed m worlds and ﬁxed n agents iﬀ ϕ is derivable in

HPR

(m,n)

. Therefore, HPR

(m,n)

is decidable.

We note that the, as far as the authors know, decid-

ability is still unknown for HPR, even the fragment

of HPR without program constructions (cf. (Sano,

2010)). So related computational properties of such

fragment has not been yet well-studied (for purely

bimodal logic fragment with a slightly diﬀerent se-

mantics, the reader is referred to (Marx and Mikul

´

as,

2001)).

3 PREFERENCE DYNAMICS

Intuitively, a public announcement of the agents’ in-

dividual preferences might induce an agent i to ad-

just her own preferences according to what has been

announced and the reliability ordering she assigns

to the set of agents.

2

For example, an agent might

adopt the preferences of the set of agents on whom

she relies the most, or might use the strict prefer-

ences of her most reliable agents for ‘breaking ties’

among her equally-preferred zones. In (Ghosh and

Vel

´

azquez-Quesada, 2015a) the authors introduced

the general lexicographic upgrade operation, which

creates a preference ordering following a priority list

of orderings. We generalize those operations in the

following, where we consider the reliability orderings

to be pre-orders, rather than being total orders (that is,

also anti-symmetric and connected) as they are in the

earlier work, which was quite an artiﬁcial assumption

2

Note that this work, in line with its predecessor, (Ghosh

and Vel

´

azquez-Quesada, 2015a), also does not focus on the

formal representation of such announcement, but rather on

the formal representation of its eﬀects.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

618

Table 1: Axiomatizations HPR and HPR

(m,n)

.

Bi-Hybrid Logical Axioms of HPR

All classical tautologies (Dual

π

) hπip ↔ ¬[π]¬p

(K

π

) [π](p → q) → ([π]p → [π]q)

Let n ∈ N

i

and (n, m) ∈ N

2

i

(i = 1, 2) below in this group

(K

@

) @

n

(p → q) → (@

n

p → @

n

q)

(SelfDual

@

) ¬@

n

p ↔ @

n

¬p (Ref) @

n

n

(Intro) n ∧ p → @

n

p (Agree) @

n

@

m

p → @

m

p

(Back) hπi@

a

@

i

p → @

a

@

i

p

Inference Rules of HPR

Modus Ponens, Uniform Substitutions,

Necessitation Rules for [π], @

i

and @

a

(Name) From n → ϕ infer ϕ,

where n ∈ N

1

∪ N

2

is fresh in ϕ

(BG

π

) From @

a

@

i

hπi(b ∧ j) → @

b

@

j

ϕ infer @

a

@

i

[π]ϕ,

where b and j are fresh in @

a

@

j

[π]ϕ

Interaction Axioms of HPR

(Com@) @

i

@

a

p ↔ @

a

@

i

p

(Red@

1

) a ↔ @

i

a (Red@

2

) i ↔ @

a

i

(DcomhWi@

2

) @

i

hαip ↔ @

i

hαi@

i

p (α ∈ { 1

W

, ≤, ≥})

(ComhAi@

1

) @

a

hβip ↔ hβi@

a

p (β ∈ { 1

A

, v

k

, w

k

})

Axioms for Atomic Programs of HPR

(U

W

) @

a

h1

W

ib (Cnv

≤

) @

a

h≤ib ↔ @

b

h≥ia

(U

A

) @

i

h1

A

ij (Cnv

v

) @

i

hv

k

ij ↔ @

j

hw

k

ii

(Eq

v

) @

i

j → ([v

i

]p ↔ [v

j

]p)

Axioms for Compounded Programs of HPR

(∪) hπ ∪ ρip ↔ hπip ∨ hρip

(?) h?(ϕ, ψ)ip ↔ ϕ ∧ h1

A

ih1

W

i(ψ ∧ p)

(∩) @

a

@

i

hπ ∩ ρi(b ∧ j) ↔ @

a

@

i

(hπi(b ∧ j) ∧ hρi(b ∧ j))

(−

W

) @

a

h− αib ↔ @

a

¬hαib (α ∈ {1

W

, ≤, ≥})

(−

A

) @

i

h− βij ↔ @

i

¬hβij (β ∈ {1

A

, v

k

, w

k

})

(u

i

) @

a

@

k

h(π, j) u

i

(π

0

, j

0

)i(b ∧ k

0

)

↔ @

i

(k ∧ k

0

) ∧ @

a

@

j

hπi(b ∧ j) ∧ @

a

@

j

0

hπ

0

i(b ∧ j

0

))

Axioms for PR-frames of HPR

(4

≤

) @

a

h≤ib ∧ @

b

h≤ic → @

a

h≤ic

(4

v

) @

j

hv

i

ik ∧ @

k

hv

i

il → @

j

hv

i

il

(Ref

≤

) @

a

h≤ia (Cmp

v

) @

j

hv

i

ik ∨ @

k

hv

i

ij

Additional Axioms and Rules for HPR

(m,n)

(|W| ≤ m)

W

0≤k,l≤m

@

a

k

a

l

(|A| ≤ n)

W

0≤k,l≤n

@

i

k

i

l

(|W| ≥ m) From

V

1≤k,l≤m

¬@

a

k

a

l

→ ψ infer ψ,

where a

k

s are fresh in ψ

(|A| ≥ n) From

V

1≤k,l≤n

¬@

i

k

i

l

→ ψ infer ψ,

where i

k

s are fresh in ψ.

on agents’ reliabilities. Agent i’s preference ordering

after an announcement, ≤

0

i

, can be deﬁned in terms

of the just announced preferences (the agents’ prefer-

ences before the announcement, ≤

1

, . . . , ≤

n

) and how

much i relied on each agent (i’s reliability before the

announcement, 4

i

): ≤

0

i

:= f (≤

1

, . . . , ≤

n

, 4

i

) for some

function f . Here are some such functions inspired by

(van Benthem, 2007; Ghosh and Vel

´

azquez-Quesada,

2015a).

Deﬁnition 5. Given a set X ⊆ A of agents, u <

X

v if

u <

k

v holds for all agents k ∈ X. Moreover, u

X

v

is used to mean u <

X

v or v <

X

u and dom(

X

) :=

{u ∈ A|u

X

v for some v ∈ A }.

Note that dom(

X

) allows us to specify the connected

components by the relation

X

. Recall that mr(i) de-

notes the set of all maximally reliable agents for i.

Deﬁnition 6 (Conservative Upgrade). Agent i takes

the strict preference ordering of her most reliable

agents, and leaves the rest undecided (equipreferable).

More precisely, the upgraded ordering ≤

0

i

is deﬁned

by: u ≤

0

i

v iﬀ (u <

mr(i)

v or u = v) or (u, v < dom(

X

)).

Deﬁnition 7 (Radical Upgrade). Agent i takes the

strict preference ordering of her most reliable agents,

and in the remaining disjoint zones she uses her old

ordering. More precisely, the upgraded ordering ≤

0

i

is deﬁned by: u ≤

0

i

v iﬀ (u <

mr(i)

v or u = v) or (u, v <

dom(

X

) and u ≤

i

v).

Note that both the conservative and radical upgrades

preserve preorders (and thus upgraded models belong

to our class of semantic models).

3.1 Expressing the Preference Dynamics

To formalize preference dynamics from the previous

section, we add the following dynamic operators to

the static syntax H L. First of all, we regard all the

agents involved in our two preference upgrade above

as agent nominals (syntactic names of agents) and so

let us denote agent i’s syntactic name as i of boldface

and the set of all syntactic names in mr(i) as mr(i).

HL

{ pu }

is deﬁned to be an expansion of HL with all

operators hpu

i

R

i, where i be an agent-nominal and R

is a list of sets of agent-nominals deﬁned as R = mr(i)

(conservative upgrade) or R = (mr(i);{i}) (radical up-

grade).

Deﬁnition 8 (Operators). A formula Req(R), repre-

senting requirements for the list R is deﬁned as the

conjunction

V

j,k∈mr(i)

¬@

j

k (mutual disjointness of

agents involved in mr(i)) and

V

j∈mr(i)

hR

i

ij (mr(i) is

the set of maximally reliable agents for i). Given a

Valuing Othersâ

˘

A

´

Z Opinions: Preference, Belief and Reliability Dynamics

619

PR-model M = (W, {≤

i

, 4

i

}

i∈A

, V), deﬁne:

M,(w, j) hpu

i

R

iϕ iﬀ M, (w, j) Req(R)

and pu

i

R

(M),(w, j) ϕ,

where pu

i

R

(M) is the same model as M except ≤

i

is

replaced by ≤

R

where R = mr(i) or R = (mr(i);{i})

and corresponding ≤

R

’s are given by Deﬁnitions 6 and

7, respectively.

For an axiom system for the modality hpu

i

R

i, we

will provide recursion axioms: valid formulas and

validity-preserving rules indicating how to translate a

formula with the new modality into a provably equiv-

alent one without them. In this case, the modalities

can take the form of any relational expression. So

we provide a ‘matching’ relational expression in the

original model M by deﬁning relational transformers

similar to those in (Ghosh and Vel

´

azquez-Quesada,

2015a; Ghosh and Vel

´

azquez-Quesada, 2015b), in

spirit of the program transformers of (van Benthem

et al., 2006).

Before going into the notion of relational trans-

former, we have two observations. Firstly, when π :=

?j∩ ≤, we note that (w, i)R

π

(v, k) is equivalent to the

conjunction of i = k = j and w ≤

j

v. Similarly, when

we put π

0

:= ?¬j∩ ≤, we remark that (w, i)R

π

0

(v, k) is

equivalent to the conjunction of w ≤

i

v and i = k and

i , j. Secondly, to reﬂect the relation <

X

in Deﬁnition

5, we need our program construction (π, j) u

i

(ρ, k) to

taking the intersection of (strict) preference relations

of the possibly diﬀerent agents than i. These observa-

tions allow us to capture the idea behind Deﬁnitions 6

and 7 syntactically in the following deﬁnition.

Deﬁnition 9 (Relational transformer). Let us

introduce the following abbreviations for re-

lational expressions: We deﬁne <

mr(i)

:=

i

{

(≤ ∩ − ≥, j) | j ∈ mr(i)

}

. Then >

mr(i)

is similarly

deﬁned and

mr(i)

is deﬁned to be <

mr(i)

∪ >

mr(i)

.

Moreover, a formula d(

mr(i)

) is deﬁned as h

mr(i)

i>.

A relational transformer T u

i

R

is a function from re-

lational expressions to relational expressions deﬁned

as follows. When R = mr(i) (conservative upgrade),

T u

i

R

(α) := α (α ∈

{

1

A

, 1

W

, v

k

, v

k

| k ∈ N

2

}

),

T u

i

R

(≤) :=

?i ∩ (<

mr(i)

∪1

A

∪?¬d(

mr(i)

)

∪ (?¬i∩ ≤)

T u

i

R

(≤) :=

?i ∩ (>

mr(i)

∪1

A

∪?¬d(

mr(i)

)

∪ (?¬i∩ ≥)

T u

i

R

(π ∪ ρ) := T u

i

R

(π) ∪ T u

i

R

(ρ),

T u

i

R

(π ∩ ρ) := T u

i

R

(π) ∩ T u

i

R

(ρ),

T u

i

R

(?(ϕ, ψ)) :=?(hpu

i

R

iϕ, hpu

i

R

iψ).

T u

i

R

((π, k) u

j

(ρ, l)) := ((T u

i

R

(π), k) u

j

(T u

i

R

(ρ), l))

T u

i

R

(−β) := −T u

i

R

(β),

where β ∈ { 1

W

, ≤, ≥} ∪

{

1

A

, v

k

, w

k

| k ∈ N

2

}

. When

R = (mr(i); { i }), we replace the occurrence of

“?¬d(

mr(i)

)” in T u

i

R

(≤) or T u

i

R

(≥) with

“?¬d(

mr(i)

)i>∩ ≤

00

or “?¬d(

mr(i)

)∩ ≤,

00

respectively.

Theorem 2. The axioms and rules below together

with those of HPR (or, those of HPR

(m,n)

) provide

sound and complete axiom systems for HL

{ pu }

with

respect to possibly inﬁnite PR models (or, PR models

with m worlds and n agents, respectively).

hpu

i

R

ip ↔ Req(R) ∧ p,

hpu

i

R

i(ϕ ∨ ψ) ↔ hpu

i

R

iϕ ∨ hpu

i

R

iψ,

hpu

i

R

i¬ϕ ↔ Req(R) ∧ ¬hpu

i

R

iϕ

hpu

i

R

ij ↔ Req(R) ∧ j, hpu

i

R

ia ↔ Req(R) ∧ a,

hpu

i

R

i@

j

ϕ ↔ Req(R) ∧ @

j

hpu

i

R

iϕ,

hpu

i

R

i@

a

ϕ ↔ Req(R) ∧ @

a

hpu

i

R

iϕ

hpu

i

R

ihπiϕ ↔ Req(R) ∧ hTu

i

R

(π)ihpu

i

R

iϕ,

From ϕ → ψ, we may infer hpu

i

R

iϕ → hpu

i

R

iψ.

Proof. Soundness of the new axioms are straightfor-

ward. Completeness follows from the completeness

of the static system HPR (cf. Chapter 7 of (van Dit-

marsch et al., 2008), for an extensive explanation of

this technique).

Example 3. In our running example of Section 1,

each agent is regarded to employ conservative up-

grades to change his or her preference. Let us write

the corresponding upgrade operators of { i, j, k } by

hpu

i

R

i

i and hpu

j

R

j

i, hpu

k

R

k

i, respectively. Then, three

ﬂatmates did not reach an agreement after conserva-

tive upgrades of all agents, i.e.,

@

i

Ba

r

∧ @

j

Ba

l

∧ @

k

Ba

r

∧

hpu

i

R

i

ihpu

j

R

j

ihpu

k

R

k

i(@

i

Ba

l

∧ @

j

Ba

r

∧ @

k

Ba

r

).

is valid in M

exp

, because upgraded preferences are

given by w

r

<

0

i

w

l

, w

l

<

0

j

w

r

and w

l

<

0

k

w

r

.

4 RELIABILITY DYNAMICS

A public announcement of the agents’ individual pref-

erences may change the agents’ reliability attributions

as well: for example, an agent may consider more

reliable those agents whose preferences coincide (or,

for some reason, diﬀer) from her own. In such cases,

agent i’s new reliability ordering, 4

0

i

, can be given in

terms of the agents’ current preferences, ≤

1

, . . . , ≤

n

,

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

620

and i’s current reliability ordering, 4

i

. Thus, ≤

0

i

:=

g(≤

1

, . . . , ≤

n

, 4

i

) for some function g. We now provide

formal deﬁnitions of some such possibilities.

4.1 Reliability Change Operations

The notion of “matching preference orders” will form

the basis for the reliability dynamics. The idea is that

two preference orderings match each other to a certain

extent if the orderings are identical on some subset of

the state space. A full match indicates that the order-

ings coincide on the whole domain; a partial match

indicates that they coincide up to some proper subset

of the domain.

Deﬁnition 10 (Matching preferences). Let F be a PR

frame given by F = (W,{≤

i

, 4

i

}

i∈A

) and let i ∈ A be

an agent. If ≤

i

is identical with ≤

j

on W

0

⊆ W, then

W

0

is said to be a set of match for i and j (notation:

≤

i

∼

W

0

≤

j

).

• Preference orders ≤

i

and ≤

j

are said to fully match

each other iﬀ ≤

i

∼

W

≤

j

.

3

FullMat(i) denotes the

set of agents in A \ {i} having full match with i.

• Preference orders ≤

i

and ≤

j

have zero match with

each other iﬀ there is no W

0

⊆ W with |W

0

| ≥ 2

such that ≤

i

∼

W

0

≤

j

.

4

ZeroMat(i) denotes the set

of agents in A \ {i} having zero match with i.

With these deﬁnitions, we can deﬁne some opera-

tions for reliability change.

Deﬁnition 11 (Full, Zero matching upgrade). Agent

i puts those agents that have full/zero match with

her own preference ordering above those that do not,

keeping her old reliability ordering within each of the

two zones. More precisely, if 4

i

is agent i’s current

reliability ordering, then her new reliability ordering

4

0

i

is deﬁned by:

j 4

0

i

k iﬀ

j, k ∈ V and j 4

i

k

or

k ∈ V and j < V

or

j, k < V and j 4

i

k

.

Here V = FullMat(i) ∪ {i}, ZeroMat(i), respectively.

Once again, we can consider more generalized deﬁ-

nitions for upgrade policies as well, but we just stick

to simple deﬁnitions to give the main idea. Note that

both the full matching and zero matching upgrades

preserve total preorders (and thus upgraded models

belong to our class of semantic models).

3

Note how, by the ﬁniteness of W (the reﬂexivity of the

preference relations), there is always a maximal X ⊆ W

such that ≤

i

∼

X

≤

j

for every agent i, j.

4

For the same reason, there is always a minimal X ⊆ W such

that ≤

i

∼

X

≤

j

for every agent i, j.

4.2 Expressing the Reliablity Dynamics

To describe reliability dynamics from the previous

section, the following dynamic operators are added

to the static syntax of HL. H L

{ rc }

is deﬁned to be an

expansion of HL with all operators hrc

i

E

i, where i is

an agent-nominal and E a pair of HL-formulas of the

form @

a

χ (recall: a is a world-nominal). An underly-

ing semantic intuition for hrc

i

E

i is: Given a PR-model

M, the pair E = (@

a

1

χ

1

, @

a

2

χ

2

) can be regarded as a

partition (i.e., an equivalence relation on agents) in the

sense that (

n

i ∈ A | M, (i, a

k

) χ

k

o

)

1≤k≤2

forms a parti-

tion of A, and the reliability ordering 4

i

of the original

PR model M is rewritten into the updated reliability

ordering 4

0

i

as in Deﬁnition 11 of the Section 4.1.

Deﬁnition 12 (Operators). Given any pair E =

(ϕ

1

, ϕ

2

) of formulas of the form @

a

χ, a formula

Eq(E) is deﬁned as the conjunction of [1

A

](ϕ

1

∨

ϕ

2

) (exhaustiveness for agents) and ¬h1

A

i(ϕ

1

∧ ϕ

2

)

(pairwise disjointness for agents). Given a pair E

= (@

a

1

χ

1

, @

a

2

χ

2

) and a PR-model M = (W, {≤

i

, 4

i

}

i∈A

, V), deﬁne:

M,(w, j) hrc

i

E

iϕ iﬀ M, (w, j) Eq(E)

and rc

i

E

(M),(w, j) ϕ,

where rc

i

E

(M) is the same model as M except 4

i

is

replaced by 4

0

i

of Deﬁnition 11.

Deﬁnition 13 (Relational transformer). Let E =

(ϕ

1

, ϕ

2

) be a pair. A relational transformer T r

i

E

is a

function from relational expressions to relational ex-

pressions deﬁned as follows.

T r

i

E

(α) := α (α ∈ { 1

A

, 1

W

, ≤, ≥ }),

T r

i

E

(v

i

) :=

(

v

i

∩(?ϕ

1

∪?ϕ

2

)

)

∪ (1

A

∩?(ϕ

1

, ϕ

2

)),

T r

i

E

(w

i

) :=

(

w

i

∩(?ϕ

1

∪?ϕ

2

)

∪ (1

A

∩?(ϕ

1

, ϕ

2

)),

T r

i

E

(v

k

) := (?@

i

k ∩ T r

i

E

(v

i

)) ∪ (?¬@

i

k∩ v

k

) (k , i),

T r

i

E

(w

k

) := (?@

i

k ∩ T r

i

E

(w

i

)) ∪ (?¬@

i

k∩ w

k

) (k , i),

T r

i

E

(π ∪ ρ) := T r

i

E

(π) ∪ T r

i

E

(ρ),

T r

i

E

(π ∩ ρ) := T r

i

E

(π) ∩ T r

i

E

(ρ),

T r

i

E

(?(ϕ, ψ)) :=?(hrc

i

E

iϕ, hrc

i

E

iψ).

T r

i

E

((π, k) u

j

(ρ, k)) := (T r

i

E

(π), k) u

j

(T r

i

E

(ρ), k),

T r

i

E

(−α) := −T r

i

E

(α),

where α ∈ { 1

W

, ≤, ≥ } ∪

{

1

A

, v

k

, w

k

| k ∈ N

2

}

).

When k , i, i.e., k and i are syntactically distinct agent

nominals, the reader may wonder why we should have

generalized test operators “?@

i

k” and “?¬@

i

k” in the

deﬁnitions T r

i

E

(v

k

) and T r

i

E

(v

k

). This is because the

Valuing Othersâ

˘

A

´

Z Opinions: Preference, Belief and Reliability Dynamics

621

same agent might have two distinct (syntactic) names.

Based on a similar strategy for Theorem 2, we can

now prove the following theorem.

Theorem 3. The axioms and rules below together

with those of HPR (or, those of HPR

(m,n)

) provide

sound and complete axiom systems for HL

{ rc }

with

respect to possibly inﬁnite PR models (or, PR models

with m worlds and n agents, respectively).

hrc

i

E

ip ↔ Eq(E) ∧ p, hrc

i

E

i(ϕ ∨ ψ) ↔ hrc

i

E

iϕ ∨ hrc

i

E

iψ,

hrc

i

E

i¬ϕ ↔ Eq(E) ∧ ¬hrc

i

E

iϕ

hrc

i

E

ij ↔ Eq(E) ∧ j, hrc

i

E

ia ↔ Eq(E) ∧ a,

hrc

i

E

i@

j

ϕ ↔ Eq(E) ∧ @

j

hrc

i

E

iϕ,

hrc

i

E

i@

a

ϕ ↔ Eq(E) ∧ @

a

hrc

i

E

iϕ

hrc

i

E

ihπiϕ ↔ Eq(E) ∧ hT r

i

E

(π)ihrc

i

E

iϕ,

From ϕ → ψ, we may infer hrc

i

E

iϕ → hrc

i

E

iψ.

Example 4. After Isabella and John know others’

preferences, we regard, in our running example, that

Isabella uses full-match reliability change hrc

i

E

i

i and

John employs zero-match reliability change hrc

j

E

j

i.

Unlike Example 3, let us ﬁrst consider reliability

changes of Isabella and John and then take the con-

servative upgrades of all agents. This process and the

resulting agreements among agents are formalized as

hrc

i

E

i

ihrc

j

E

j

ihpu

i

R

i

ihpu

j

R

j

ihpu

k

R

k

i

(@

i

Ba

r

∧ @

j

Ba

r

∧ @

k

Ba

r

),

which is valid in M

exp

, because Isabella’s reliability

is changed into i ≺

0

i

j ≺

0

i

k and John’s reliability does

not change.

We note here that while the main focus of the work

is to model joint deliberation in form of simultaneous

preference and reliability upgrades, the model opera-

tions and modalities of Sections 3.1 and 4.2 deal with

single agent upgrades. This presentation style has

been chosen in order to simplify notation and read-

ability, but the provided deﬁnitions can be easily ex-

tended in order to match our goals. In particular, the

model operations of Deﬁnitions 8 and 12 can be ex-

tended to simultaneous upgrades by asking for a list R

of lexicographic lists (with R

i

the list for agent i), and

asking for a list E of partition lists (with E

i

the list for

agent i), respectively. Then the corresponding modal-

ities, hpu

i

R

i and hrc

i

E

i can still be axiomatised by the

presented system with some simple modiﬁcations.

5 CONCLUSION

This work continues the line of study in (Ghosh and

Vel

´

azquez-Quesada, 2015a; Ghosh and Vel

´

azquez-

Quesada, 2015b) and provides a further interplay be-

tween the preferences that the agents have about the

world around and the reliability attributions they have

with respect to one another. We deal with both

preference change based on reliability, and reliabil-

ity change based on preferences, and propose two-

dimensional dynamic hybrid logics to express such

changes. The main technical results that we have

are sound and complete axiomatizations which lead

to decidability (provided the numbers of agents and

of states are ﬁxed ﬁnite numbers) as well. In process,

we also discuss about agent beliefs in such situations,

e.g. relating reliability attributions with the notions

of belief (cf. the running example in the text). The

novel contribution of the work is the study of change

in reliability attribution of agents based on their pref-

erences.

To conclude, let us provide some pointers towards

future work: (1) What other reasonable preference

and reliability upgrade policies can there be and how

to model them? (2) How to investigate the role of

knowledge in such changes, especially if manipula-

tion comes into play? (3) What would be the char-

acterizing conditions for reaching consensus in such

deliberative processes? We endeavor to provide an-

swers to such questions in future.

5

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