yanjing wang and jie fan department of philosophy, peking ... · pdf filea propositional modal...

Beyond “knowing that” Conditional “knowing what” Conclusions

Conditionally knowing what

Yanjing Wang and Jie Fan

Department of Philosophy, Peking UniversityAiML 2014, Groningen

Yanjing Wang and Jie Fan: Conditionally knowing what

Beyond “knowing that”

Conditional “knowing what”

Conclusions

Epistemic Logic

A propositional modal logic that handles reasoning aboutknowledge (and belief) [von Wright 1951, Hintikka 1962]

I Language: expresses “agent i knows that φ” (Kiφ).

I Model: possibilities with equivalence relations.

I Semantics: you know that φ iff φ is true in all the situationsthat you consider possible

p ∧ ¬Kip

Epistemic Logic

p ∧ ¬Kip

Epistemic Logic

p ∧ ¬Kip

S5 system

System S5Axioms Rules

TAUT all the instances of tautologies MPφ, φ→ ψ

DISTK Ki (p → q)→ (Kip → Kiq) NECKφ

T Kip → p SUBφ

φ[p/ψ]

4 Kip → KiKip

5 ¬Kip → Ki¬Kip

TheoremS5 is sound and strongly complete for modal logic over S5 frames.

Beyond “knowing that”: motivationKnowledge is not only expressed in terms of “knowing that”:

I I know whether the claim is true.

I I know what your password is.

I I know how to go to Amsterdam.

I I know who proved this theorem.

I . . .

Linguistically: “know” takes embedded questions but “believe”doesn’t; ambiguity in concealed questions.

Philosophically: are they reducible to “knowing that”?

Logically: how to reason about those forms of knowledge?

Computationally: how to efficiently represent and do inferenceabout them?

I . . .

Beyond knowing that: research agenda

In fact, “knowing who” was briefly discussed already by Hinttikka(1962) in terms of first-order modal logic: ∃xKi (Barteld = x).

Our agenda:

I Keep the language neat and take those know-constructions asthey are (e.g., pack ∃xKi (Barteld = x) into Kw Barteld).

I Give an intuitive semantics according to some linguistic theory.

I Axiomatize the logics with (combinations of) those operators.

I Dynamify those logic with knowledge updates.

I Automate the inferences.

I Come back to philosophy and linguistics with new insights.

Our agenda:

Beyond knowing that: difficulties and some results

New operator behaves quite differently from the standard modaloperators and are usually disguised FO-modal fellows, which causesdifficulties for axiomatization and decidable machinery.

Some of our results:

I Knowing whether (non-contingency): axiomatizations andcompleteness proofs for its logic over various frame classes[Fan, Wang & van Ditmarsch AiML 14]

I Knowing what: axiomatization and decidability forconditionally knowing what logic over FO epistemic models[Wang & Fan IJCAI13, AiML14][Xiong 14]

I Knowing how: alternative non-possible-world semantics [WangLOFT12]; extend [Wang & Li AiML 12] for epistemic planning

Knowing what operator Kvi proposed by [Pla89]ELKv is defined as (where c ∈ C ):

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvic

ELKv is interpreted on M = 〈S ,D, {∼i | i ∈ I},V ,VC 〉 where D isa constant domain, VC assigns to each (non-rigid) c ∈ C a d ∈ Don each s ∈ S :

M, s � Kvic ⇐⇒ for any t1, t2 : if s ∼i t1, s ∼i t2,then VC (c , t1) = VC (c , t2).

It can express “i knows that j knows the password but i doesn’tknow what exactly it is” by KiKvjc ∧ ¬Kic .

The interaction between the two operators is crucial: it cannot betreated as KiKjp ∧ ¬Kip.

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvic

Knowing what operator Kvi proposed by [Pla89]

To handle the Sum and Product puzzle, Plaza extended ELKvwith announcement operator (call it PALKv):

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvic | 〈φ〉φ

[Pla89] proposed a system PALKVp for PALKv on top of S5.

Theorem ([WF13])

〈p〉Kvic ∧ 〈q〉Kvic → 〈p ∨ q〉Kvic is not derivable in PALKVp,thus PALKVp is not complete w.r.t. � on epistemic models.

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvic | 〈φ〉φ

Theorem ([WF13])

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvic | 〈φ〉φ

Theorem ([WF13])

Axiomatizing PALKv is indeed hard. We propose a conditionalgeneralization of Kvi operator:

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvi (φ, c)

where Kvi (φ, c) says ‘agent i knows what c is given φ’, e.g., Iknow my password for this website if it is 4-digit. More precisely,agent i would know what c is if he is informed that φ.

M, s � Kvi (φ, c) ⇔ for any t1, t2 ∈ S such that s ∼i t1 and s ∼i t2 :M, t1 � φ&M, t2 � φ implies VC (c , t1) = VC (c , t2)

Let PALKvr be:

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvi (φ, c) | 〈φ〉φ

Axiomatizing PALKv is indeed hard. We propose a conditionalgeneralization of Kvi operator:

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvi (φ, c)

where Kvi (φ, c) says ‘agent i knows what c is given φ’, e.g., Iknow my password for this website if it is 4-digit. More precisely,agent i would know what c is if he is informed that φ.

M, s � Kvi (φ, c) ⇔ for any t1, t2 ∈ S such that s ∼i t1 and s ∼i t2 :M, t1 � φ&M, t2 � φ implies VC (c , t1) = VC (c , t2)

Let PALKvr be:

φ ::= > | p | ¬φ | φ ∧ φ | Kiφ | Kvi (φ, c) | 〈φ〉φ

PALKvr looks more expressive than PALKv but in fact they areequally expressive.

Theorem ([WF13])

The comparison of the expressive power of those logics aresummarized in the following (transitive) diagram:

ELKvr ←→ PALKvr

↑ lELKv −→ PALKv

where ELKv and ELKvr are the announcement-free fragments ofPALKv and PALKvr .

We can simply forget about Plaza’s PALKv and use ELKvr !

System ELKVr

Axiom SchemasTAUT all the instances of tautologiesDISTK Ki (p → q)→ (Kip → Kiq)T Kip → p4 Kip → KiKip5 ¬Kip → Ki¬KipDISTKvr Ki (p → q)→ (Kvi (q, c)→ Kvi (p, c))Kvr4 Kvi (p, c)→ KiKvi (p, c)Kvr⊥ Kvi (⊥, c)

Kvr∨ K̂i (p ∧ q) ∧ Kvi (p, c) ∧ Kvi (q, c)→ Kvi (p ∨ q, c)

MPp, p → q

φ[p/ψ]

REψ ↔ χ

φ↔ φ[ψ/χ]

Kvi (φ, c) can be viewed as ∃xKi (φ→ c = x) where x is a rigidvariable and c is a non-rigid one.

A Kvi operator packages a quantifier, a modality, an implicationand an equality together: a blessing and a curse.

To build a suitable canonical FO-Kripke model with a constantdomain, we need to saturate each maximal consistent set with:

I counterparts of atomic formulas such as c = x

I counterparts of Ki (φ→ c = x)

By using axioms in the modal language, we need to make surethese extra bits are consistent with the maximal consistent setsand canonical relations.

LemmaEach maximal consistent set can be properly saturated with thosecounterparts.

LemmaEach saturated MCS including ♦φ has a saturated φ-successor.

LemmaEach saturated MCS including ¬Kvi (φ, c) has two saturatedφ-successors which disagree about the value of c.

Axiom Kvr∨ : K̂i (p ∧ q) ∧ Kvi (p, c) ∧ Kvi (q, c)→ Kvi (p ∨ q, c)plays an extremely important role.

TheoremELKVr is sound and strongly complete for ELKvr .

Theorem ([Xio14])

ELKvr on epistemic models is decidable.

We can axiomatize multi-agent PALKvr by adding the followingreduction axiom schemas (call the resulting system SPALKVr ):

!ATOM 〈ψ〉p ↔ (ψ ∧ p)!NEG 〈ψ〉¬φ↔ (ψ ∧ ¬〈ψ〉φ)!CON 〈ψ〉(φ ∧ χ)↔ (〈ψ〉φ ∧ 〈ψ〉χ)!K 〈ψ〉Kiφ↔ (ψ ∧ Ki (ψ → 〈ψ〉φ))!Kvr 〈φ〉Kvi (ψ, c)↔ (φ ∧ Kvi (〈φ〉ψ, c))

Corollary

SPALKVr is sound and strongly complete for PALKvr .

TheoremELKVr is sound and strongly complete for ELKvr .

Theorem ([Xio14])

ELKvr on epistemic models is decidable.

We can axiomatize multi-agent PALKvr by adding the followingreduction axiom schemas (call the resulting system SPALKVr ):

!ATOM 〈ψ〉p ↔ (ψ ∧ p)!NEG 〈ψ〉¬φ↔ (ψ ∧ ¬〈ψ〉φ)!CON 〈ψ〉(φ ∧ χ)↔ (〈ψ〉φ ∧ 〈ψ〉χ)!K 〈ψ〉Kiφ↔ (ψ ∧ Ki (ψ → 〈ψ〉φ))!Kvr 〈φ〉Kvi (ψ, c)↔ (φ ∧ Kvi (〈φ〉ψ, c))

Corollary

SPALKVr is sound and strongly complete for PALKvr .

Final words

Systematic study of “knowing-wh” constructions in logic may leadus to:

I interesting non-normal ‘modal’ operators and axioms

I discovery of new decidable (“guarded”) fragments offirst-order modal logic

I knowledge representations closer to natural language

I insights for questions in philosophy and linguistics

This is just the beginning of an interesting story!

Final words

Systematic study of “knowing-wh” constructions in logic may leadus to:

I interesting non-normal ‘modal’ operators and axioms

I discovery of new decidable (“guarded”) fragments offirst-order modal logic

I knowledge representations closer to natural language

I insights for questions in philosophy and linguistics

This is just the beginning of an interesting story!

Beyond “knowing that”: Join us!

Melvin Fitting.First-order intensional logic.Annals of Pure and Applied Logic, 127(1-3):171–193, 2004.

J. A. Plaza.Logics of public communications.In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, and Z. W. Ras,editors, Proceedings of the 4th International Symposium onMethodologies for Intelligent Systems, pages 201–216, 1989.

Yanjing Wang and Jie Fan.Knowing that, knowing what, and public communication:Public announcement logic with Kv operators.In Proceedings of IJCAI, pages 1139–1146, 2013.

Yanjing Wang and Jie Fan.Conditionally knowing what.in proceedings of AiML14, April 2014.

Shihao Xiong.Decidability of ELKv r .Bachelaor thesis, May 2014.

yanjing wang and jie fan department of philosophy, peking ... · pdf filea propositional modal...

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