Knowing Values and Public Inspection
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1 Knowing Values and Public Inspection Malvin Gattinger with Jan van Eijck & Yanjing Wang arxiv.org/abs/ slides at w4eg.de/malvin , LIRa ILLC, Amsterdam
2 Introduction Knowing that announcing that: [!ψ]k i (ϕ) (ψ K i [!ψ]ϕ)
3 Introduction Knowing that announcing that: [!ψ]k i (ϕ) (ψ K i [!ψ]ϕ) What about knowing what and announcing what?
4 Example Student Subject Assessment 1 Mathematics good 2 Mathematics very good 3 Logic good 4 Computer Science bad
5 Example Student Subject Assessment 1 Mathematics good 2 Mathematics very good 3 Logic good 4 Computer Science bad M, 3 [Subject]Kv(Assessment)
6 Knowing what Kv i (c)
7 Knowing what Kv i (c) Semantics: M, s Kv i (c) t 1, t 2 : s i t 1 s i t 2 V c (t 1 ) = V c (t 2 )
8 Knowing what Kv i (c) Semantics: M, s Kv i (c) t 1, t 2 : s i t 1 s i t 2 V c (t 1 ) = V c (t 2 ) Considered already by (Plaza 2007) in combination with PAL, recently by (Wang and Fan 2013,Wang and Fan (2014)). Latest news: we can make it normal (Gu and Wang 2016) generalize all the way (Baltag 2016)
9 Announcing what How about announcing what? Other words: revealing, telling the value, inspecting,...
10 Announcing what How about announcing what? Other words: revealing, telling the value, inspecting,... PIL := Public Inspection Logic
11 Contents 1. Single Agent PIL Syntax Semantics Example Proof System Armstrong Axioms Completeness via Canonical Graph 2. Multi-Agent PIL 3. Comparison with Dependence Logic 4. Related and Future Work
12 Single Agent PIL Let c range over some set of variables C. 1. Language: ϕ ::= ϕ ϕ ϕ Kv(c) [c]ϕ
13 Single Agent PIL Let c range over some set of variables C. 1. Language: ϕ ::= ϕ ϕ ϕ Kv(c) [c]ϕ 2. Models: M = S, V where V : (C S) D for some D. Write s = c t iff V (c, s) = V (c, t).
14 Single Agent PIL Let c range over some set of variables C. 1. Language: ϕ ::= ϕ ϕ ϕ Kv(c) [c]ϕ 2. Models: M = S, V where V : (C S) D for some D. Write s = c t iff V (c, s) = V (c, t). 3. Interpretation: M, s Kv(c) for all t S : s = c t M, s [c]ϕ M s c, s ϕ where M s c is S, V C S with S = {t S s = c t}.
15 Example Model M = S, V S = {s, t, u} V as follows, e.g. V (s, d) = 1. c d e s t u M, s Kv(t) M Kv(t) M, s [e]kv(d) M, s [d]kv(e)
16 Proof system for PIL Tautologies propositional tautologies Distribution [c](ϕ ψ) ([c]ϕ [c]ψ) Learning [c]kv(c) No Forgetting Kv(c) [d]kv(c) Determinacy c ϕ [c]ϕ Commutativity [c][d]ϕ [d][c]ϕ Irrelevance Kv(c) ([c]ϕ ϕ) Modus Ponens: ϕ, ϕ ψ ψ Necessitation: ϕ [c]ϕ
17 Dependency For any two finite sets C, D C, let Kv(C, D) := [c 1 ]... [c n ](Kvd 1... Kvd m ) We get: M, s Kv(C, D) for all t S : if s = C t then s = D t
18 Dependency For any two finite sets C, D C, let We get: Kv(C, D) := [c 1 ]... [c n ](Kvd 1... Kvd m ) M, s Kv(C, D) for all t S : if s = C t then s = D t Lemma ϕ ::= ϕ ϕ ϕ Kv(C, C) has the same expressive power. Proof. Push sensing operator through negations and conjunctions: [c]( Kvd [e]kvf ) [c] Kvd [c][e]kvf Kv(c, d) Kv({c, e}, f )
19 Hints from Database Theory: Armstrong s Axioms Lemma The (Armstrong 1974) axioms are valid and provable: Projectivity Kv(C, D) for any D C Transitivity Kv(C, D) Kv(D, E) Kv(C, E) Additivity Kv(C, D) Kv(C, E) Kv(C, D E)
20 Hints from Database Theory: Armstrong s Axioms Lemma The (Armstrong 1974) axioms are valid and provable: Projectivity Kv(C, D) for any D C Transitivity Kv(C, D) Kv(D, E) Kv(C, E) Additivity Kv(C, D) Kv(C, E) Kv(C, D E)
21 Completeness Theorem (Strong Completeness) For all L 1 and all ϕ L 1 : If ϕ, then also ϕ.
22 Completeness Theorem (Strong Completeness) For all L 1 and all ϕ L 1 : If ϕ, then also ϕ. Proof Strategy: Suppose ϕ. Then { ϕ} is consistent Take a maximally consistent set Γ { ϕ}. Build a model M Γ such that for some world C we have M Γ, C Γ which implies ϕ.
23 Completeness via the Canonical Dependency Graph Canonical Graph Let G Γ := (P(C), ) where A B iff Kv(A, B) Γ. Example Γ = {Kv(c, d), Kv(e),... }. {d, e} {c, d, e} {e} {e, c} {d} {c, d} {c}
24 Lemma G Γ is projective, transitive and additive. Call s C closed under G Γ iff whenever A s and A B according to G Γ, then also B s.
25 Lemma G Γ is projective, transitive and additive. Call s C closed under G Γ iff whenever A s and A B according to G Γ, then also B s. Canonical model: M Γ = (S, V ) where S := {s C s is closed under G Γ } and V s (c) = { 0 if c s 1 otherwise Truth Lemma M Γ, C ϕ iff ϕ Γ
26 Example continued {d, e} {c, d, e} {e} {e, c} {d} {c, d} {c} V c d e s = {e} t = {d, e} u = C = {c, d, e} M, u Kv(c, d) Kv(e)...
27 Multi-Agent PIL Language: ϕ ::= ϕ ϕ ϕ Kv i c [c]ϕ Add to the models: i S S Interpretation: M, s Kv i c t S : s i t s = c t Proof system with restricted Irrelevance: Kv i c ([c]ϕ ϕ) where ϕ does not mention any agent besides i Strong completeness proof: S = P(C), use a canonical graph for each agent to define i.
28 Comparison with Dependence Logic Our following validity does not translate to the Armstrong framework: [c](kvd Kve) ([c]kvd [c]kve)
29 Comparison with Dependence Logic Our following validity does not translate to the Armstrong framework: [c](kvd Kve) ([c]kvd [c]kve) Pointed models convey more information than a team: c d e After inspecting c we know d or we know e. [c](kvd Kve) After inspecting c we know d or after inspecting c we know e. ([c]kvd [c]kve)
30 Comparison with Dependence Logic Our following validity does not translate to the Armstrong framework: [c](kvd Kve) ([c]kvd [c]kve) Pointed models convey more information than a team: c d e After inspecting c we know d or we know e. [c](kvd Kve) After inspecting c we know d or after inspecting c we know e. ([c]kvd [c]kve) False: c globally determines d or c globally determines e Armstrong s system can not express [c](kvd Kve).
31 Related and Future Work p Kϕ [!ϕ]ϕ (Plaza 2007) p Kϕ Kv(c) [!ϕ]ϕ (Plaza 2007) p Kϕ Kv(c) Kv(ϕ, c) [!ϕ]ϕ (Wang and Fan 2013) (Wang and Fan 2014) (Gu and Wang 2016) Kv(c) [c]ϕ this p Kϕ Kv(c) Kv(ϕ, c) [c]ϕ [!ϕ]ϕ (Baltag 2016) What does it mean to know a function? Application to Security Protocols?
32 Summary Knowing what Announcing what
33 Bonus Slide: Provable Stuff c (seriality) Kv(c) (ϕ [c]ϕ) (Irrelevance*) [c](ϕ ψ) [c]ϕ [c]ψ (Distribution*) [c 1 ]... [c n ](ϕ ψ) ([c 1 ]... [c n ]ϕ [c 1 ]... [c n ]ψ) (multi-distribution) [c 1 ]... [c n ](ϕ ψ) [c 1 ]... [c n ]ϕ [c 1 ]... [c n ]ψ (multi-distribution*) [c 1 ]... [c n ](Kv(c 1 )... Kv(c n )) (multi-learning) (Kv(c 1 ) Kv(c n )) [d 1 ]... [d n ](Kv(c 1 ) Kv(c n )) (multi-no Forgetting) (Kv(c 1 ) Kv(c n )) ([c 1 ]... [c n ]ϕ ϕ) (multi-irrelevance)
34 References Armstrong, William Ward Dependency Structures of Data Base Relationships. In IFIP Congress, 74: Geneva, Switzerland. Baltag, Alexandru To Know Is to Know the Value of a Variable. Advances in Modal Logic 11: Gu, Tau, and Yanjing Wang Knowing Value Logic as a Normal Modal Logic. Advances in Modal Logic 11: Plaza, Jan Logics of Public Communications. Synthese 158 (2): doi: /s Wang, Yanjing, and Jie Fan Knowing That, Knowing What, and Public Communication: Public Announcement Logic with Kv Operators. In IJCAI 13, Conditionally Knowing What. Advances in Modal Logic 10:
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