COOPERATIVE REPLIES TO UNBELIEVABLE ASSERTIONS

A Dialogue Protocol based on Logical Interpolation

M. Nyk¨anen

School of Computing, University of Eastern Finland, Kuopio, Finland

S. Eloranta, O. Niinivaara

Department of Computer Science, University of Helsinki, University of Helsinki, Helsinki, Finland

R. Hakli

Helsinki Institute for Information Technology, Helsinki, Finland

Keywords:

Belief revision, Dialogue protocols, Convictions, Integrity constraints, Argumentation.

Abstract: We propose a dialogue protocol for situations in which an agent makes to another agent an assertion that

the other agent ﬁnds impossible to believe. In this interaction, unbelievable assertions are rejected using

explanations formed by logical interpolation and new assertions are being made such that all previous rebuttals

are taken into account.

1 INTRODUCTION

When two agents carry out a conversation with each

other, one of them may well assert something which

the other cannot believe for some reason. Witness the

following example (Hansson, 1991):

Conversation 1.

Amy. Last summer I saw a three-toed woodpecker

just outside my window. I could clearly see its

red forehead and its red rump.

Bob. You must be mistaken. A three-toed wood-

pecker does not have a red forehead or a red rump.

Amy. You make me uncertain. Thinking about it, the

only thing I am certain of is that the bird had a red

forehead.

We study here such conversations: Both agents

have beliefs that they are certain of and that they are

not willing to give up during the conversation. Here,

Bob’s ornithological knowledge is one example, and

Amy’s certainty of seeing a bird with a red forehead

is another. When hearing an unbelievable assertion,

Bob faces the task of helping Amy by offering infor-

mative rebuttals. Amy then faces the task of generat-

ing another assertion while taking Bob’s rebuttal into

account. This interaction continues until either Amy

comes up with an assertion which Bob can consider

possible or she concludes that they have irreconcil-

able differences, at least as far as this conversation is

concerned.

We consider these conversations in the context of

belief revision (Alchourr´on et al., 1985) in the pres-

ence of what we call convictions. By these convic-

tions we mean those beliefs the agent refuses to give

up, at least during the current conversation. In several

ﬁelds there are important concepts that can be inter-

preted as convictions: In computer science, integrity

constraints (Reiter, 1988) are needed to ensure con-

sistency of databases. In philosophy, the properties

of knowledge differ from those of belief (Hintikka,

1962), and people take a different stand on what they

take to know and not merely believe. In nonpriori-

tized belief revision, core beliefs are immune to revi-

sion (Hansson, 1999, gives a survey). In theories of

argumentation, agents have dark-side commitments,

which are their fundamental commitments that they

ﬁnd extremely hard to retract once stated in a conver-

sation (Walton and Krabbe, 1995, pp. 11–12).

We encounterthe problem of what an agent should

do when another agent asserts something that con-

ﬂicts with his convictions. In this paper we propose

a solution, in which the agents carry out a conversa-

tion as an interactive preparatory phase before belief

revision. In this phase they seek together a ﬁnal asser-

tion which does not conﬂict with either agent’s con-

victions. In Conversation 1, Amy’s second assertion

might serve as something which they both might be

able to believe. We do not consider what happens

after this preparatory phase, that is, we do not con-

cern ourselves whether the agents actually revise their

245

Nykänen M., Eloranta S., Niinivaara O. and Hakli R..

COOPERATIVE REPLIES TO UNBELIEVABLE ASSERTIONS - A Dialogue Protocol based on Logical Interpolation.

DOI: 10.5220/0003181702450250

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 245-250

ISBN: 978-989-8425-41-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

epistemic states or not.

Focusing on these assertion-rebuttalconversations

raises immediately three questions: First, how can an

agent form his rebuttal to the unbelievable assertion?

Second, how should the other agent form her

assertion on the basis of her epistemic state while tak-

ing into account the new information in the rebuttal?

And third, how can the conversation stay focused on

its original subject?

Our answer to the ﬁrst question is to use logi-

cal interpolation, since it gives a formula which is

entailed by the convictions of the agent and entails

the negation of the unbelievable assertion. Moreover,

it can be read as a description how or an explana-

tion why (Hintikka and Halonen, 1999) the assertion

conﬂicted with the convictions of the agent, thereby

bringing some currently relevant part of his convic-

tions into light. Our answer to the second question is

to use hypothetical thinking, as if the agent thought:

“if I were to believe this rebuttal, then I would have

this belief about my topic instead of the one I ex-

pressed before”. She will form her next assertion as

the least disbelieved alternative to the topic given the

new information. Our answer to the third question is

to require that their utterances remain relevant to it

in the letter-sharing sense (Makinson, 2009, Deﬁni-

tion 1.1), which is guaranteed by interpolation.

In related work, there are some approaches in

which agents have convictions but do not use inter-

action for conﬂict resolution. These include such ap-

proaches to nonprioritized belief revision that secure

some beliefs from revision. For instance, in Accom-

modative Belief Revision (Eloranta et al., 2008), the

agent tries to guess what the other would have said,

had she had his knowledge. In our solution, the agent

does not have to guess what the other agent would be-

lieve, instead he gives her a chance to tell it.

Then there are approaches in which interaction is

used as a preprocessing step before belief revision,

but the possibility of agents having their private con-

victions is not considered. These include such merg-

ing approaches as mutual belief revision (Jin et al.,

2007) and belief negotiation (Booth, 2006) in which

all the agents’ beliefs are weakened until they no

longer contradict each other. As opposed to that, our

solution is asymmetric. We have one agent, who is ea-

ger to inform another agent about some of her beliefs,

whereas the other agent is willing to reply and share

some of his convictions in case he ﬁnds the original

assertion unbelievable. Application areas with such

We adhere to the convention that the asserting agent

(such as Amy in Conversation 1) is female, whereas the re-

butting agent (such as Bob) is male, and refer to them as

“she” and “he” as well as by name.

a setting include knowledge base systems in which

some agents (either human beings or software agents)

collect information and send it to one agent acting as

a knowledge base with integrity constraints.

Certain types of argumentation-based dialogues

(Walton and Krabbe, 1995; Parsons et al., 2003) can

also be viewed as preparatory phases for belief revi-

sion: They aim at ﬁnding out whether a particular as-

sertion should be believed by exchanging information

about arguments that either support or undermine it.

In our approach, however, the goal is to ﬁnd out what

could be believed about the topic when the agents’

convictions are taken into account, not whether a par-

ticular proposition should be believed or not.

For example, van Veenen and Prakken (2006) in-

cluded asking “Why did you rebut my assertion?”

among the moves in their negotiation protocol as an

embedded persuasion game. However, their idea is

to bring the grounds for the rebuttal to light so that

they too can be subject to further scrutiny by the other

agent within this conversation. In contrast, the pur-

pose of our dialogues is not to persuade the other

agent to accept the original assertion, but to ﬁnd an

alternative assertion that is acceptable.

Our aim is that the agents’ assertions in the dia-

logues satisfy the Cooperative Principle presented by

Grice (1989, Chapters 2 and 3) to govern conversa-

tions between cooperative agents. These maxims rule

out the naive extremities to deal with an unacceptable

input, that is, either to terminate the dialogue or to re-

ply with all one knows about the subject. Something

more is needed, thus we will propose the use of logi-

cal interpolation as a cooperative reply to an assertion

that an agent is convinced to be false.

The paper is organized as follows. In section 2,

we will introduce our notations and present the inter-

polation principle as a tool for generating cooperative

replies to unbelievable assertions. In section 3, we

will propose guidelines for generating a new modi-

ﬁed assertion based on this reply. Section 4 presents

the conversation protocol driven by these interpolants

and shows that the conversations will always end with

a rational outcome. In section 5, we will give conclu-

sions and propose some directions for future research.

2 COOPERATIVE REPLIES

We will consider dialogues, that is, conversations be-

tween two agents. We assume that these two agents,

named A and B, such as Amy or Bob in Conversa-

tion 1, have epistemic states, which we denote with A

and B correspondingly. These states contain the be-

lief sets consisting of all the beliefs they currently

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

246

hold; these sets we denote with B(A ) and B(B ). In

these belief sets, beliefs are expressed with formulas

of classical propositional logic; that is, they are be-

liefs about the actual state of affairs, and not about for

instance beliefs about each other’s beliefs. On the one

hand, each agent may be willing to give up some of

these beliefs given new evidence to the contrary. On

the other hand, (s)he may regard some of them as con-

victions which (s)he will hold on to, regardless of any

such new evidence. We denote the sets of convictions

for agents A and B with C(A) and C(B ) correspond-

ingly. We assume that the sets of beliefs and the sets

of convictions are non-contradictory and deductively

closed, that is, B(A) = Cn(B(A)), etc. We also as-

sume that what an agent is convinced of, (s)he also

believes, that is, C(A ) ⊆ B(A ) and C(B) ⊆ B(B).

Agent A is the initiator of the conversation. We are

focusing entirely on the situation, in which agent B is

convinced that the assertion made by A is false. We

are not concerned whether agent B will givepriority to

the assertion if it does not contradict his convictions.

We use lower-case Greek letters to denote propo-

sitional formulas. Agent A initiates the conversation

with her initial assertion ϕ. Whenever the assertion is

acceptable by agent B, the dialogue ends. Therefore

we shall presume this initial assertion to be unbeliev-

able, that is, C(B) |= ¬ϕ. As already mentioned in

Section 1, we propose for agent B to use the inter-

polant to this entailment as a cooperative reply in this

situation, since (i) it is entailed by C(B) and (ii) it

entails ¬ϕ and (iii) it only employs vocabulary that

appears in the original assertion (in terms of proposi-

tional variables).

Let V denote all the propositional variables and

Voc(α) ⊂ V those appearing in the formula α. Let us

recall what interpolation is:

Theorem 1 (Craig interpolation, for propositional

logic). Given two propositional formulas α and β,

if α |= β, then there is some interpolant θ such that

(i) α |= θ, (ii) θ |= β, and (iii) Voc(θ) ⊆ Voc(α) ∩

Voc(β).

In the full paper (Nyk¨anen et al., 2011), we prove

this theorem for the system G3cp of sequent calculus

(Negri and von Plato, 2001, Chapter 3.1) in a way

which provides an explicit algorithmic construction

for the interpolant θ given the formulas α and β.

In Conversation 1, α is a formula representing

(some relevant part of) Bob’s convictions C(B ), β is

the negation ¬ϕ of Amy’s initial assertion ϕ, and θ is

a suggestion for an explanation why α rules out β.

Example 1. Consider Conversation 1, and let the

propositional variable p stand for “Amy saw a three-

toed woodpecker”, q stand for “Amy saw a bird with

a red forehead”, r stand for “Amy saw a bird with

a red rump”, s stand for “Amy saw a lark”. The

conversation starts when Amy asserts p ∧ q∧ r. As-

sume Bob’s convictions include (p∨s) → (¬q∧ ¬r).

Then (p ∨ s) → (¬q ∧ ¬r) entails an interpolant p →

¬q ∧ ¬r, which again entails the negation of Amy’s

assertion, ¬(p∧ q∧ r).

Let us consider how well this suggestion fares

in light of Grice’s Maxims (Grice, 1989, Chapters 2

and 3). These maxims elaborated his general Cooper-

ative Principle into more speciﬁc conversational rules

which the participants can be expected to observe:

Maxim of Quantity. (i) Make your contribution as

informative as required (for the current purposes

of the exchange). (ii) Do not make your contribu-

tion more informative than is required.

Maxim of Quality. Try to make a contributionwhich

is true. More speciﬁcally: (i) Do not say what you

believe to be false. (ii) Do not say that for which

you lack adequate evidence.

Maxim of Relevance. Be relevant.

Maxim of Manner. Be clear.

Using an interpolant as a reply conforms to part (i)

of the maxim of Quantity, because it conveysinforma-

tion that agent A supposedly was not aware of, since it

entails the negation of what she said. In part (ii), the

amount of informativeness can be controlled by the

selection of a suitable interpolation formula.

Using an interpolant as the reply conforms also

to the maxim of Quality: not only does agent B re-

ply with a belief of his, but with a conviction, and we

assume here that a rational agent does not obtain con-

victions without proper evidence.

According to the maxim of Relevance, a reply

should somehow be related to the preceding conver-

sation. A natural syntactic concept is letter-sharing:

two formulas are relevant to each other, if they share

some propositional variable (Makinson, 2009, Deﬁni-

tion 1.1). In this regard, an interpolant is an extremely

relevant reply, since it consists only of variables in

both C(B) and the current suggestion by agent A, by

property (iii) of Theorem 1. Makinson (2009) notes

that although letter-sharing is not wholly unproblem-

atic as a notion of relevance, it does have its uses in

computational contexts. Hence we deﬁne here the

topic of the conversation to be the variables Voc(ϕ)

about which agent A wants to have a conversation

with agent B.

Another more reﬁned concept of relevance could

be to split the beliefs of an agent into disjoint parts,

where each part consists of the agent’s beliefs about

a particular subject matter (Parikh, 1999). Since

Theorem 1 can be extended to such split theories

COOPERATIVE REPLIES TO UNBELIEVABLE ASSERTIONS - A Dialogue Protocol based on Logical Interpolation

247

(Kourosias and Makinson, 2007, Theorem 1.1), we

anticipate our approach to apply in this setting as well.

There are also more semantic accounts of rele-

vance. In the theory of Sperber and Wilson (2004),

the main determinant of relevance of a reply is how

much positive cognitive effect (such as learning, set-

tling doubt, or correcting mistaken assumptions) it

creates in the recipient. In this respect too, a reply by

interpolant fares well since it contradicts with what

agent A supposedly believes and should thus invoke a

process of belief revision.

Regarding the maxim of Manner, one could ar-

gue that it does not concern conversations such as

ours, where the messages exchanged are formulated

in logic instead of in a natural language. We note,

however, that even in our conversations agent B can

tailor the form of his interpolant to enhance its clarity

to agent A.

3 ON GENERATING NEW

ASSERTIONS

Let us contemplate on what agent A should do when

she learns a formula θ that tells her why her previous

assertion was unbelievable.

If θ conﬂicts with the convictions of agent A, we

take that this dialogue should fail. If the rebuttal is not

unbelievable to agent A, then she can either (i) accept

the input as a conviction (since it was B’s conviction),

(ii) accept the input as a belief, or (iii) treat the in-

put conditionally. The treatment may depend, e.g., on

how reliable the agent considers the other agent. By

the maxim of Quality, agent A then continues the di-

alog with a new assertion ψ, which she accordingly

(i) believes (now that she is convinced that θ), (ii) be-

lieves (now that she has come to believe that θ), or

(iii) would believe, if she were to believe that θ.

Denoting the doxastic conditional “if I were to

believe θ, then I would also believe ψ” as θ◦→ ψ,

our requirement of the maxim of Quality becomes

A |= θ◦→ ψ, where this entailment is deﬁned through

the Ramsey test: if agent A were to revise her epis-

temic state A with θ, then would ψ be believed in the

resulting state A ◦ γ? Thus in all three alternatives, A

may anwer ψ if and only if ψ ∈ B(A ◦ θ). Note that

this revision might be only tentative: the actual epis-

temic state of agent A might still be A.

Now let us assume that the revision operator ’◦’

that agent A uses (either when revising her epistemic

state or when evaluating conditionals) satisﬁes the ba-

sic rationality criteria (R1)–(R4) for belief revision

(Alchourr´on et al., 1985) and the rationality crite-

rion (IR1) for iterated belief revision (Darwiche and

Pearl, 1997). That is:

α ∈ B(A ◦ α). (R1)

If ¬α 6∈ B(A ) then

B(A ◦ α) = Cn(B(A) ∪ {α}).

(R2)

If α is satisﬁable then B(A ◦ α) is consistent. (R3)

If α ≡ β then B(A ◦ α) = B(A ◦ β). (R4)

If α |= β then B(A ◦ α) = B((A ◦ β) ◦ α). (IR1)

Postulate (R1) says that the new piece of informa-

tion is accepted, that is, the insertion succeeds. Pos-

tulate (R2) says that if the new piece of information

is compatible with the old beliefs, neither is any of

them discarded nor is anything not entailed by the old

beliefs and the new information added to the belief

set. Postulate (R3) says that adding a satisﬁable for-

mula to the belief set must not make it inconsistent.

Postulate (R4) calls for syntax independence. Pos-

tulate (IR1) says that if α |= β, then the beliefs in

the epistemic state obtained when learning ﬁrst β and

then α are the same as when learning just α in the ﬁrst

place.

By the maxim of Relevance, agent A must bear in

mind (and take into account) all the rebutting formu-

las during the dialog. Let γ denote the conjunction of

those formulas. Thus in the scenarios, we must use γ

instead of θ.

By the basic postulates for belief revision, we have

A |= θ◦→ ψ if and only if A ◦ θ |= θ◦→ ψ. Now

assume that after n rebuttals, γ

= θ

∧ θ

∧ . . . ∧ θ

Then by postulate (IR1), we have A |= γ

◦→ ψ if and

only if A ◦ γ

◦ γ

◦ . . . ◦ γ

|= γ

◦→ ψ. Thus the truth

value of the conditional does not depend on whether

the agent has actually revised her epistemic state on

the way or not: all three alternatives for A’s actions

remain equivalent in this respect.

Our framework allows for several methods for

constructing a formula ψ satisfying this require-

ment. Some methods are discussed in the full paper

(Nyk¨anen et al., 2011).

4 A CONVERSATION PROTOCOL

We will now give a conversation protocol in which

agent B uses interpolation to create rebuttals. In our

protocol, agent A starts with an assertion ϕ, which

also ﬁxes the topic of the conversation. When agent B

receives an assertion which conﬂicts with C(B), he

answers with an interpolant θ. In the protocol, ψ con-

tains the most recent assertion made by agent A, and γ

is the conjunction of all the rebuttals made by agent B

to her previous assertions, as above. The protocol is

depicted as follows:

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

248

CONVERSATION PROTOCOL

1 ψ ← ϕ; γ ← ⊤

2 A asserts ψ

3 while C(B) |= ¬ψ with some interpolant θ

4 do B replies that he is convinced of θ

5 γ ← γ ∧ θ

6 if C(A ) |= ¬γ

7 then A says that their convictions

conﬂict with each other

8 return FAIL

9 ψ ← some formula chosen by A such

that A |= γ ◦→ ψ

10 A asserts ψ

11 B replies that he too considers this ψ believable

12 return SUCCESS with ψ

Selecting the next assertion ψ on line 9 is possi-

ble if and only if C(A ) ∪ {γ} is consistent, and this

is guaranteed by line 6. If our conversation protocol

terminates successfully (on line 12) then we do have

an agreement: both agents could believe the ﬁnal as-

sertion ψ. For agent B, this follows by line 3. For

agent A, this is an invariant of the while loop: it holds

before the loop by line 1 and the maxim of Quality (i),

and it continues to hold after each execution of the

loop body by line 9.

Notice the algorithm uses the current epistemic

states of the agents. As to the convictions the agents

have, this causes no problems, because the agents do

not give up their convictions. The beliefs of agent B

are not used in the protocol. As to the beliefs of

agent A, whether she has actually revised her epis-

temic state with γ or not does not affect the truth value

of the conditional, as discussed in section 3.

The protocol could generate a conversation like

the following:

Conversation 2.

Amy. I saw a bird with a red forehead and a red rump.

It was a three-toed woodpecker!

Bob. A three-toed woodpecker does not have a red

forehead.

Amy (thinking to herself). In that case I would have

to give up believing either that I saw a three-toed

woodpecker or a red forehead. I prefer to keep

believing the former. Hence I can take Bob’s re-

buttal into account by giving up the latter. I can

also keep believing in the red rump, since he did

not challenge that part.

(aloud to Bob). In that case I think I saw a three-

toed woodpecker, which had a red rump but no

red forehead.

Bob. A three-toed woodpecker does not have a red

rump either.

Amy (after similar thinking). Well, in that case I

think I saw a three-toed woodpecker, but it had

neither a red rump nor a red forehead.

Bob. Now that is something I could believe!

If our protocol terminates with failure instead (on

line 8) then this outcome is warranted as well: On the

one hand, C(B ) |= γ by property (i) of Theorem 1 and

lines 3 and 5. On the other hand, C(A ) |= ¬γ by line 6.

Agent A can even explain this conﬂict between their

convictions with an interpolant corresponding to the

entailment on line 6, thereby expressing her own con-

victions about the topic. This could be useful if the

agents attempt to reconcile their convictions some-

how, but we do not consider such attempts here.

If we assume that agent A keeps her assertions ψ

relevantto the topic in our chosen letter-sharing sense,

then our protocol becomes ﬁnite, albeit O(2

|Voc(ϕ)|

) in

the worst case.

Theorem 2 (Finiteness). Assume that the asser-

tions ψ by agent A in our conversation protocol sat-

isfy Voc(ψ) ⊆ Voc(ϕ). Then the maximum number of

times its while loop can be executed is

|{w ↾ Voc(ϕ): w ∈ Mod(C(A)) \ Mod(C(B))}|.

Proof. See the full paper (Nyk¨anen et al., 2011).

However, we expect actual conversations to ter-

minate in much fewer rounds than the pessimal upper

bound in Theorem 2, since the more precise explana-

tions θ agent B gives for his rejections, the fewer as-

sertions agent A needs. And indeed, maxim of Quan-

tity (i) directs agent B towards such precise explana-

tions θ. However, we leave constructing precise inter-

polants θ for later study.

5 CONCLUSIONS AND FUTURE

WORK

We considered dialogues as a preparatory phase for

belief revision and we presented a dialog protocol for

resolving conﬂicts resulting from unbelievable asser-

tions. Depending on the result of the dialogue, the

agents either have found out that their convictions are

in conﬂict with each other, or they have found a for-

mula that neither of them ﬁnds unbelievable. During

the dialogue, the unbelievable assertions do not cause

the rebutting agent to change his epistemic state, but

the asserting agent might (or might not) change her

Here f ↾ D = {hx, f (x)i : x ∈ D} denotes the restriction

of the function f into the domain D.

COOPERATIVE REPLIES TO UNBELIEVABLE ASSERTIONS - A Dialogue Protocol based on Logical Interpolation

249

epistemic state due to the rebuttals. In general, we al-

low the agents to change their epistemic states during

the dialog.

Our protocol can terminate successfully even

when there is a conﬂict, since it may never surface

during the protocol. Suppose for instance that A as-

serts some a∨b, where A can believe the ﬁrst disjunct

but not the second disjunct, and vice versa for B. We

leave such pseudoagreements to further study, since

avoiding them would require continuing the conver-

sation further even after ﬁnding this ﬁrst mutually be-

lievable formula.

The work can be extended to several directions,

such as adding the possibility to extend the topic

with new literals, adding the possibility to agree to

restrict the topic, adding new utterance types to the

agents (for instance, for making the protocol symmet-

ric), and considering more expressive languages as is

done e.g. in cooperative query answering (Gaaster-

land et al., 1992).

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