The Logic of Secrets LAMAS 2020, 8 May 2020 Thomas gotnes - PowerPoint PPT Presentation

The Logic of Secrets LAMAS 2020, 8 May 2020 Thomas Å gotnes University of Bergen, Norway Southwest University (SWU), China Zuojun Xiong, SWU Yuzhi Zhang, SWU

Secrets • Of fundamental importance in, e.g., • safety and security • cryptography • authentication • access control • … • (and in business and politics and romance and..)

What is a secret? • “ a piece of knowledge that is hidden and intended to be kept hidden” (Wiktionary) • “ a piece of information that is only known by one person or a few people and should not be told to others” (Cambridge Dictionary) • “ something that is kept or meant to be kept unknown or unseen by others” (Oxford English Dictionary) • “ something kept from the knowledge of others” (Merriam- Webster)

Fundamentally about What is a secret? knowledge and ignorance • “ a piece of knowledge that is hidden and intended to be kept hidden” (Wiktionary) • “ a piece of information that is only known by one person or a few people and should not be told to others” (Cambridge Dictionary) • “ something that is kept or meant to be kept unknown or unseen by others” (Oxford English Dictionary) • “ something kept from the knowledge of others” (Merriam- Webster)

In this paper we • Formalise secrets (more precisely: secretly knowing) • Using the standard framework for reasoning about knowledge and ignorance: modal epistemic logic • Key question: what are the (epistemic) properties of secretly knowing? • Introduce a modality for secretly knowing and study its properties S a ϕ a secretly knows ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ (2) any other agent b does not know ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ V K a b 6 = a ¬ K b ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ V K a b 6 = a ¬ K b ϕ (2”) a knows that any other agent b does not know whether ϕ

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ V K a b 6 = a ¬ K b ϕ (2”) a knows that any other agent b does not know whether ϕ V b 6 = a ( ¬ K b ϕ ∧ ¬ K b ¬ ϕ ) K a

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ V K a b 6 = a ¬ K b ϕ (2”) a knows that any other agent b does not know whether ϕ V b 6 = a ( ¬ K b ϕ ∧ ¬ K b ¬ ϕ ) K a

Necessary epistemic conditions for secretly knowing a secretly knows ϕ (1) a knows ϕ K a ϕ V b 6 = a ¬ K b ϕ (2) any other agent b does not know ϕ (2’) a knows that any other agent b does not know ϕ V K a b 6 = a ¬ K b ϕ (2”) a knows that any other agent b does not know whether ϕ V b 6 = a ( ¬ K b ϕ ∧ ¬ K b ¬ ϕ ) K a V K a ϕ ∧ K a b 6 = a ¬ K b ϕ

The secretly-knowing modality L SK : ϕ ::= p | ¬ ϕ | ( ϕ ∧ ϕ ) | K a ϕ | S a ϕ

The secretly-knowing modality L SK : ϕ ::= p | ¬ ϕ | ( ϕ ∧ ϕ ) | K a ϕ | S a ϕ ∼ a ⊆ W × W eq. rel., V : W → 2 Prop Epistemic model: M = ( W, ∼ , V )

The secretly-knowing modality L SK : ϕ ::= p | ¬ ϕ | ( ϕ ∧ ϕ ) | K a ϕ | S a ϕ ∼ a ⊆ W × W eq. rel., V : W → 2 Prop Epistemic model: M = ( W, ∼ , V ) M, w | = p i ff w 2 V ( p ). M, w | = ¬ ϕ i ff M, w 6 | = ϕ . M, w | = ϕ ^ ψ i ff M, w | = ϕ and M, w | = ψ . 8 w 0 2 W , if w ⇠ a w 0 , then M, w 0 | M, w | = K a ϕ i ff = ϕ . 8 w 0 ⇠ a w M, w 0 | M, w | = S a ϕ i ff = ϕ and 8 b 6 = a , 9 u ⇠ b w 0 M, u | = ¬ ϕ .

The secretly-knowing modality L SK : ϕ ::= p | ¬ ϕ | ( ϕ ∧ ϕ ) | K a ϕ | S a ϕ ∼ a ⊆ W × W eq. rel., V : W → 2 Prop Epistemic model: M = ( W, ∼ , V ) M, w | = p i ff w 2 V ( p ). M, w | = ¬ ϕ i ff M, w 6 | = ϕ . M, w | = ϕ ^ ψ i ff M, w | = ϕ and M, w | = ψ . 8 w 0 2 W , if w ⇠ a w 0 , then M, w 0 | M, w | = K a ϕ i ff = ϕ . 8 w 0 ⇠ a w M, w 0 | M, w | = S a ϕ i ff = ϕ and 8 b 6 = a , 9 u ⇠ b w 0 M, u | = ¬ ϕ . Have that: M, w | = S a ϕ ⇔ M, w | V b 6 = a ¬ K b ϕ = K a ϕ ∧ K a

The secretly-knowing modality L S : ψ ::= p | ¬ ψ | ( ψ ∧ ψ ) | S a ψ ∼ a ⊆ W × W eq. rel., V : W → 2 Prop Epistemic model: M = ( W, ∼ , V ) M, w | = p i ff w 2 V ( p ). M, w | = ¬ ϕ i ff M, w 6 | = ϕ . M, w | = ϕ ^ ψ i ff M, w | = ϕ and M, w | = ψ . 8 w 0 2 W , if w ⇠ a w 0 , then M, w 0 | M, w | = K a ϕ i ff = ϕ . 8 w 0 ⇠ a w M, w 0 | M, w | = S a ϕ i ff = ϕ and 8 b 6 = a , 9 u ⇠ b w 0 M, u | = ¬ ϕ .

Properties of secretly knowing: interaction axioms Interaction axioms for S a and K a ⇣V ⌘ (S) Def. of S a S a ϕ $ K a ϕ ^ K a b 6 = a ¬ K b ϕ (4SK) Positive secret S a ϕ ! K a S a ϕ knowledge introspection (5SK) Negative secret ¬ S a ϕ ! K a ¬ S a ϕ knowledge introspection (P) S a ϕ ! ( K a ϕ ^ ¬ K b ϕ ) Secret privacy (NKS) Secret unknowability ¬ K b S a ϕ (NSK1) Knowledge is no secret ¬ S a K b ϕ (NSK2) Ignorance is no secret ¬ S a ¬ K b ϕ (NC) Secret neg. completeness K a S a ϕ _ K a ¬ S a ϕ ( a 6 = b )

Properties of secretly knowing: interaction axioms between agents Interaction axioms for S a and S b (Ex1) Secret exclusivity S a ϕ → ¬ S b ϕ (Ex2) Higher-order secret exclusivity S a ¬ S a ϕ → ¬ S b ¬ S b ϕ (N1) No secret secrets ¬ S a S b ϕ (N2) No secret non-secrets ¬ S a ¬ S b ϕ

Properties of secretly knowing: basic principles Axioms for S a (K) S a ( ϕ ! ψ ) ! ( S a ϕ ! S a ψ ) Secret distribution (T) Secret veridicality S a ϕ ! ϕ (4) Secret introspection S a ϕ ! S a S a ϕ (C) ( S a ϕ ^ S a ψ ) ! S a ( ϕ ^ ψ ) Secret combination (D) Secrets partiallity S a ϕ ! ¬ S a ¬ ϕ ( > ) No tautological secrets ¬ S a > ( ? ) No contradictory secrets ¬ S a ? Rules for S a (RE) From ϕ $ ψ infer S a ϕ $ S a ψ Replacement of equivalents (Nnec) From ϕ infer ¬ S a ϕ Negative necessitation (Dnec) From ϕ infer ¬ S a ¬ ϕ Diamond necessitation

Properties of secretly knowing: basic principles Axioms for S a (K) S a ( ϕ ! ψ ) ! ( S a ϕ ! S a ψ ) Secret distribution (T) Secret veridicality S a ϕ ! ϕ (4) Secret introspection S a ϕ ! S a S a ϕ (C) ( S a ϕ ^ S a ψ ) ! S a ( ϕ ^ ψ ) Secret combination (D) Secrets partiallity S a ϕ ! ¬ S a ¬ ϕ ( > ) No tautological secrets ¬ S a > ( ? ) No contradictory secrets ¬ S a ? Rules for S a (RE) From ϕ $ ψ infer S a ϕ $ S a ψ Replacement of equivalents (Nnec) From ϕ infer ¬ S a ϕ Negative necessitation (Dnec) From ϕ infer ¬ S a ¬ ϕ Diamond necessitation

Properties of secretly knowing: basic principles Axioms for S a (K) S a ( ϕ ! ψ ) ! ( S a ϕ ! S a ψ ) Secret distribution (T) Secret veridicality S a ϕ ! ϕ (4) Secret introspection S a ϕ ! S a S a ϕ (C) ( S a ϕ ^ S a ψ ) ! S a ( ϕ ^ ψ ) Secret combination (D) Secrets partiallity S a ϕ ! ¬ S a ¬ ϕ ( > ) No tautological secrets ¬ S a > ( ? ) No contradictory secrets ¬ S a ? Rules for S a (RE) From ϕ $ ψ infer S a ϕ $ S a ψ Replacement of equivalents (Nnec) From ϕ infer ¬ S a ϕ Negative necessitation (Dnec) From ϕ infer ¬ S a ¬ ϕ Diamond necessitation (5) ¬ S a ϕ → S a ¬ S a ϕ

Properties of secretly knowing: basic principles Axioms for S a (K) S a ( ϕ ! ψ ) ! ( S a ϕ ! S a ψ ) Secret distribution (T) Secret veridicality S a ϕ ! ϕ (4) Secret introspection S a ϕ ! S a S a ϕ (C) ( S a ϕ ^ S a ψ ) ! S a ( ϕ ^ ψ ) Secret combination (D) Secrets partiallity S a ϕ ! ¬ S a ¬ ϕ ( > ) No tautological secrets ¬ S a > ( ? ) No contradictory secrets ¬ S a ? Rules for S a (RE) From ϕ $ ψ infer S a ϕ $ S a ψ Replacement of equivalents (Nnec) From ϕ infer ¬ S a ϕ Negative necessitation (Dnec) From ϕ infer ¬ S a ¬ ϕ Diamond necessitation (5) 6 | ¬ S a ϕ → S a ¬ S a ϕ =