Epistemic updates on algebras: Bilattice Epistemic Action and Knowledge Zeinab Bakhtiarinoodeh (Joint work with Umberto Rivieccio) Loria, CNRS-Universite de Lorraine
Applied Logic Seminar, 14 October 2015
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Outline
Main goal Technical Aim: Obtaining a semantics and a complete axiomatization for a Bilattice-based Logic of Epistemic Action and Knowledge (BEAK) algebraic and duality-theoretic methods.
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Histoty and Motivation
”Dynamic phenomena” are best analyzed using an appropriate non-classical logic, in many contexts: which are inconsistency-tolerant, paracomplete: multiple sources of information, inconsistent/contradictory evidence where truth is procedural;
Recent works:
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Histoty and Motivation
”Dynamic phenomena” are best analyzed using an appropriate non-classical logic, in many contexts: which are inconsistency-tolerant, paracomplete: multiple sources of information, inconsistent/contradictory evidence where truth is procedural;
Recent works: Recent work of Alessandra Palmigiano and collaborators provides methods which allow one to: Define a logic of Epistemic Actions and Knowledge on a propositional basis that is weaker than classical logic, for example an intuitionistic basis. Provide a way to apply these methods to a variety of contexts where classical reasoning is not suitable.
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Dynamic Epistemic Logic (DEL)
Family of logics for multiagent interaction; describing and reasoning about information flow, how it affects epistemic setup of agents. Merging of two issues: Epistemic: what do agents know, or believe (partial knowledge, incorrect beliefs...) Dynamic: knowledge acquisition, belief updates...
giving rise to epistemic actions. Examples: Public announcements, private announcements, ...
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Epistemic Action and Knowledge(EAK)
The logic EAK was introduced by A. Baltag, L.S. Moss and S. Solecki (1999) to deal with “Public Announcements, Common Knowledge and Private Suspicions”. The language of EAK is that of modal logic (S5) expanded with dynamic operators h↵i and [↵], where ↵ is an action structure.
Intended meaning of h↵i : the action ↵ can be executed, and after execution is the case. Dually, [↵] means: if the action ↵ can be executed, then after execution holds.
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Language Language of (classical, single-agent) EAK
::= p 2 Var | ¬ |
_
| ^ | ⇤ | h↵i | [↵] ,
Where ↵ is an action structure:
↵ = (K , k , R↵ , Pre↵ : K ! Fm).
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Language Language of (classical, single-agent) EAK
::= p 2 Var | ¬ |
_
| ^ | ⇤ | h↵i | [↵] ,
Where ↵ is an action structure:
↵ = (K , k , R↵ , Pre↵ : K ! Fm). Kripke semantics For M = (W , R , v ), define M, w M, w
iff
h↵i [↵]
iff
M, w if M , w
Pre↵ (k ) and M ↵ , (w , k ) Pre↵ (k ), then M ↵ , (w , k )
where M ↵ is the updated model, after execution of ↵.
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Updated model Intermediate model (pseudo coproduct) Given ↵ := (K , k , R↵ , Pre↵ : K ! Fm) and M = (W , R , v ), let
`
K
W
W ⇥K
`
↵
M := (
(w , j )(R ⇥ R↵ )(u, i ) iff ` ` ( K v )(p ) := K v (p ).
`
K
W , R ⇥ R↵ ,
`
K
v)
wRu and jR↵ i
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Updated model Intermediate model (pseudo coproduct) Given ↵ := (K , k , R↵ , Pre↵ : K ! Fm) and M = (W , R , v ), let
`
K
W
W ⇥K
`
↵
M := (
(w , j )(R ⇥ R↵ )(u, i ) iff ` ` ( K v )(p ) := K v (p ).
The second step, M ↵ M ↵ is the submodel of
`
↵
`
K
W , R ⇥ R↵ ,
`
K
v)
wRu and jR↵ i
M with domain
W ↵ := {(w , j ) | M , w
Pre↵ (j )}.
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Epistemic updates
Epistemic change is represented in DEL as a transformation from a (relational, algebraic) model representing the current situation to a new model that represents the situation after some epistemic action has occurred.
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Epistemic updates
Epistemic change is represented in DEL as a transformation from a (relational, algebraic) model representing the current situation to a new model that represents the situation after some epistemic action has occurred. The update on the epistemic state of agents caused by an action is known as epistemic update.
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Epistemic updates
Epistemic change is represented in DEL as a transformation from a (relational, algebraic) model representing the current situation to a new model that represents the situation after some epistemic action has occurred. The update on the epistemic state of agents caused by an action is known as epistemic update. Epistemic updates are formalized on Kripke-style models via (pseudo-) co-products and sub-models, on algebras via (pseudo-) products and quotients.
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Methodology: dual characterizations Intuitionistic Alg.Semantics
Classical Alg.Semantics
Intuitionistic Rel.Semantics
Classical Rel.Semantics
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Methodology: dual characterizations Intuitionistic Alg.Semantics
Classical Alg.Semantics
Intuitionistic Rel.Semantics
Classical Rel.Semantics M ,!
`
↵M
- M↵
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Methodology: dual characterizations Intuitionistic Alg.Semantics
Classical Alg.Semantics
A⌘
Q
↵A
⇣ A↵
Intuitionistic Rel.Semantics
Classical Rel.Semantics M ,!
`
↵M
- M↵
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Methodology: dual characterizations Intuitionistic Alg.Semantics
A⌘
Q
↵A
⇣
Q
↵A
Rel.Semantics
A↵
Classical Alg.Semantics
A⌘
Intuitionistic
⇣ A↵
Classical Rel.Semantics M ,!
`
↵M
- M↵
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Methodology: dual characterizations Intuitionistic Alg.Semantics
A⌘
Q
↵A
⇣
A↵
Classical Alg.Semantics
A⌘
Q
↵A
⇣ A↵
Intuitionistic Rel.Semantics M ,!
`
↵M
- M↵
Classical Rel.Semantics M ,!
`
↵M
- M↵
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Methodology: dual characterizations Bilattice Alg.Semantics
A⌘
Q
↵A
⇣
A↵
Primary definition
Rel.Semantics M ,!
`
↵M
- M↵
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Methodology: dual characterizations Bilattice Alg.Semantics
A⌘
Q
↵A
⇣
A↵
Primary definition
Four-valued Rel.Semantics
`
M ,! ↵ M - M ↵ Obtained by dual characterization
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Modal bilattices Bimodal Boolean algebra: (A , ^, _, ⇠, ^+ , ^ , 0, 1) s.t.:
(A , ^, _, ⇠, 0, 1) is a Boolean algebra;
^+ and ^ preserve finite joins (possibly empty). Modal twist structures: A./ = (A ⇥ A , ^, _, , ¬, t, f, >, ?) s.t. A is a bimodal Boolean algebra and (a1 , a2 ) ^ (b1 , b2 ) = (a1 ^ b1 , a2 _ b2 ) (a1 , a2 ) _ (b1 , b2 ) = (a1 _ b1 , a2 ^ b2 ) (a1 , a2 ) (b1 , b2 ) = (⇠a1 _ b1 , a1 ^ b2 ) ^(a , b ) = (^+ a , ⇤+ b ^ ⇠^ a ) ¬(a , b ) = (b , a ) f = (0, 1) t = (1, 0) > = (1, 1) ? = (0, 0) 10 / 19
Intermediate structures Let A ⌘ A./ be a modal bilattice; ↵ = (K , k , R↵ , Pre↵ : K ! A) four-valued action structure over A; It means: R↵ : K ! FOUR is a four-valued relation.
Y
A := (AK , ^
↵
Q
↵
A
,⇤
Q
↵
A
)
For each f : K ! A and each j 2 K ,
(^
Q
Q
(⇤
↵
A
↵
A
f )(j ) =
f )(j ) =
_
^
{^A f (i ) | R↵ (j , i ) 2 {t, >}}
{⇤A f (i ) | R↵ (j , i ) 2 {t, >}}. 11 / 19
The modal bilattice A↵ Problem Defining b ⌘Pre↵ c
NOT a congruence.
iff
b ^ Pre↵ = c ^ Pre↵
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The modal bilattice A↵ Problem Defining b ⌘Pre↵ c
NOT a congruence.
iff
b ^ Pre↵ = c ^ Pre↵
Fact
a`
iff
v (((
f)
f)) = v (((
f)
f))
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The modal bilattice A↵ Problem Defining b ⌘Pre↵ c
iff
NOT a congruence.
b ^ Pre↵ = c ^ Pre↵
Fact
a`
iff
v (((
f)
f)) = v (((
f)
f))
Solution Define b ⌘Pre↵ c
iff
b ^ ⇠⇠Pre↵ = c ^ ⇠⇠Pre↵
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The modal bilattice A↵ Problem Defining b ⌘Pre↵ c
iff
b ^ Pre↵ = c ^ Pre↵
NOT a congruence. Fact
a`
iff
v (((
f)
f)) = v (((
f)
f))
Solution Define
Q
↵ A/⌘Pre↵ :
b ⌘Pre↵ c
iff
b ^ ⇠⇠Pre↵ = c ^ ⇠⇠Pre↵
Modal bilattice; [b ] 2
Q
^↵ [b ] := [^ ↵
Q
Q
↵ A/⌘Pre↵ , ↵A
↵
(⇠⇠ Pre↵ ^ b )]
A
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Axiomatization of BEAK
Our calculus for BEAK is defined over the language h_, ¬, ^, h↵i, f, t, >, ?i BEAK is axiomatically defined by axioms and rules of the calculus for bilattice modal logic of [3] + the following axioms and rules:
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Axiomatization of BEAK Constant axioms
h↵i f $ f
h↵i t $ ⇠⇠ Pre (↵)
h↵i> $ (Pre (↵) ^ >)
h↵i? $ ¬(Pre (↵)
?)
h↵i( _ ) $ (h↵i _ h↵i )
_ axiom axiom
h↵i(
) $ (⇠⇠ Pre (↵) ^ (h↵i
¬ axiom
h↵i¬ $ (⇠⇠ Pre (↵) ^ ¬h↵i ))
^ axiom
h↵i^ $ (⇠⇠ Pre (↵) ^
Fact preservation
h↵ip $ (⇠⇠ Pre (↵) ^ p )
The rule: from ; `
!
infer
h↵i ))
W {^h↵j i | R↵ (k , j ) 2 {t, >}})
; ` h↵i ! h↵i . 14 / 19
Algebraic semantics
For every algebraic model M = (A, v ), where A is a modal bilattice and v : Var ! A, the extension map [[·]]M : Fm ! A is defined as:
[[p ]]M [[ ]]M [[~ ]]M [[h↵i ]]M [[[↵] ]]M
= = = = =
v (p )
[[ ]]M A [[ ]]M ~A [[ ]]M [[⇠⇠ Pre (↵k )]]M ^A ⇡k ◆([[ ]]M ↵ ) [[Pre (↵k )]]M A ⇡k ◆([[ ]]M ↵ ).
◆ : [b ] 7 ! b ^ ⇠⇠ Pre↵ is an injective map that embeds
for 2 {^, _, !, . . .} for ~ 2 {^, ⇤, ¬, . . .}
Q
↵
A/⌘Pre↵ into
Q
↵
A.
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Results
Soundness of the axioms is checked w.r.t. to algebraic models.
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Results
Soundness of the axioms is checked w.r.t. to algebraic models. The proof of completeness is analogous to that of classical and intuitionistic EAK, and follows from reducibility of BEAK to bilattice modal logic and the h↵i-monotonicity axiom.
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Results
Soundness of the axioms is checked w.r.t. to algebraic models. The proof of completeness is analogous to that of classical and intuitionistic EAK, and follows from reducibility of BEAK to bilattice modal logic and the h↵i-monotonicity axiom. Soundness and completeness w.r.t. relational models follow by duality.
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Kurz, A. and A. Palmigiano, Epistemic Updates on Algebras, Logical Methods in Computer Science 9 (2013), pp. 1–28. Ma, M., Palmigiano, A. and M. Sadrzadeh, Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic, Annals of Pure and Applied Logic, 165 (2014), pp. 963–995. A. Jung and U. Rivieccio. Kripke semantics for modal bilattice logic. Proceedings of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 2013, pp. 438–447. ´ B. Kooi and U. Rivieccio. Bilattice public announcement logic. R. Gore, A. Kurucz (eds.), Advances in Modal Logic, Vol. 10, College Publications, 2014, p. 459–477.
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Thanks for your attention...
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