Linear Algebraic Representation of Knowledge State of Agent Satoshi Tojo JAIST 28 August, 2018 1 / 40
Outline Introduction 1 Linear Algebraic Semantics for Modal Logic 2 Linear Algebraic Semantics for Multi-agent Communication 3 Conclusions 4 2 / 40
Introduction One of the most important aspects of multi-agent communication is changes of agent’s knowledge or belief (cf. G¨ ardenfors 2003) Nowadays, such changes are well-discussed in terms of modal logic, as Dynamic Epistemic Logic (DEL) We show a computational tool of DEL for multi-agent communication 3 / 40
George de La Tour: Le Tricheur ` a l’as de carreau 4 / 40
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Issues In the picture, we can see many aspects of belief change of agents triggered by an informing action by others. Liar Belief revision Reliability of news source Mutual belief Channel/ Whisper /Announcement Awareness How can we formalize these troublesome situations in logic, or in an efficient, scalable and reliable computation system? 5 / 40
Outline Introduction 1 Linear Algebraic Semantics for Modal Logic 2 Linear Algebraic Semantics for Multi-agent Communication 3 Conclusions 4 6 / 40
Linear Algebraic Approach to Kripke Semantics Historical development: Frame properties (Lemmon & Scott 1977) Boolean matrix approach for ▶ bisimulation for modal logic (Fitting 2003). ▶ belief revision & fusion of belief logic (Liau 2004). ▶ DEL with communication channels (Tojo 2013, Hatano, Sano & Tojo 2015). Real-valued matrix approach for belief revision & update of belief logic (Fusaoka et al. 2007) Relational algebraic approach for modal logic of knowledge (Berghammer & Schmidt 2006). 7 / 40
Linear Algebraic Approach to Kripke Semantics Define a Kripke Model M = ( W , R , V ) by: W = { w 1 , w 2 , w 3 } , = { ( w 1 , w 1 ) , ( w 1 , w 2 ) , ( w 1 , w 3 ) , ( w 2 , w 2 ) , ( w 3 , w 3 ) } , R V ( p ) = { w 2 } . 8 / 40
Syntax & Semantics PROP = { p , q , . . . } is a finite set of propositional variables. Form ML ∋ A ::= p | ¬ A | ( A ∨ A ) | ♢ A Given any M = ( W , R , V ) and any w ∈ W , M , w | = p w ∈ V ( p ) , iff M , w | = ¬ A M , w ̸| = A , iff M , w | = A ∨ B iff M , w | = A or M , w | = B , M , w | = ♢ A for some v ∈ W : wRv and M , v | = A . iff 9 / 40
Matrix Representation of Kripke Semantics Accessibility relation R �→ a square matrix R M Valuation V ( p ) �→ a column vector ( V ( p )) M A column vector ∥ A ∥ M is defined by: ( V ( p )) M , ∥ p ∥ M := ∥¬ A ∥ M := ∥ A ∥ M , ∥ A ∨ A ∥ M := ∥ A ∥ M + ∥ A ∥ M , R M ∥ A ∥ M . ∥ ♢ A ∥ M := 1 0 1 1 1 = + = 0 1 0 1 1 0 1 0 0 0 10 / 40
Example: the column vector of ♢ p 1 1 1 0 1 ∥ ♢ p ∥ := R M ∥ p ∥ = R M ( V ( p )) M = = . 0 1 0 1 1 0 0 1 0 0 ∥ □ p ∥ = ∥¬ ♢ ¬ p ∥ = ∥ ♢ ¬ p ∥ = R ∥¬ p ∥ = R ∥ p ∥ Kripke semantics becomes an extended truth table calculation. 11 / 40
Matrix Representation of Frame Properties Name Formula Matrix Reformulation Reflexive T □ p → p R = R + E R = t R (or R = t R + R ) Symmetric B p → □♢ p R = R 2 + R Transitive 4 □ p → □□ p R t R = R t R + E (or 1 = R 1 ) Serial D □ p → ♢ p R = t RR + R Euclidean 5 ♢ p → □♢ p E : a unit square matrix 1 : a column vector of all 1s t R : the transposition of the matrix R [ 1 0 ] [ 1 ] [ 1 1 ] [ 1 0 ] ⇒ t R = E = , 1 = , R = . 0 1 1 0 1 1 1 12 / 40
Example: Verification of a Frame Property Let us check whether R is transitive ( R = R 2 + R ). 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 = 0 1 0 0 1 0 + 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 � �� � [ 1 1 1 ] 0 1 0 0 0 1 R is transitive. 13 / 40
Difficulty of The Ordinary Approach The verification of the Euclideanness property ⇒ R = t RR + R The truth of a formula with the nested modal operators. ⇒ ∥ ♢ p → □♢ p ∥ = R ∥ p ∥ + RR ∥ p ∥ = 1 14 / 40
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Spaghetti of Accessibility 16 / 40
Spaghetti of Accessibility 16 / 40
Interim Summary Our approach can cover the following topics: Matrix representation of graph of a Kripke model Computation of the truth value of a formula The validity and the satisfiability of a formula Frame properties (reflexivity, symmetricity, transitivity, seriality and Euclideanness) 17 / 40
Outline Introduction 1 Linear Algebraic Semantics for Modal Logic 2 Linear Algebraic Semantics for Multi-agent Communication 3 Conclusions 4 18 / 40
Multi-agent Communication For Multi-agent system, we propose: Logic of belief with communication channels and its dynamic operators A linear algebraic reformulation for proposed operators Dynamic change of belief by matrix calculation 19 / 40
Digression: what is communication? If you are alone in the universe ... Have you possessed language? Some claim we need language to think. 20 / 40
Further Digression: what is language? 21 / 40
CFG and RG Language of domestic finch: (( ab ) + c ) + Human language ▶ CFG - NPDA ⋆ We came to see a movie to Shibuya. ⋆ We came to Shibuya to see a movie. ⋆ (*) We came to see to Shibuya a movie. ▶ Dutch crossing ⋆ He said that A saw B help C feed the dogs. ⋆ Hij zegt dat A B C de honden zag helpen voeren. 22 / 40
Logical Studies for Multi-agent communication Historical development: DEL for public announcements (Plaza 1989 etc.) Integration of communication channels into DEL ▶ Two-dimensional approach of Facebook logic (Seligman et al. 2011, Sano & Tojo 2013). ▶ Linear algebraic approach of DEL (Tojo 2013). It is unknown whether resulting logics of two-dimensional approach is decidable. We extend our linear algebraic approach for Dynamic Logic of Relation Changers (DLRC) to handle communication channels. 23 / 40
Syntax PROP = { p , q , . . . } is a finite set of propositional variables. G = { a , b , . . . } is a finite set of agents. A ::= p | c ab | ¬ A | A ∨ B | B a A c ab : “there is a channel from agent a to agent b .” B a A : “agent a believes A .” 24 / 40
Kripke Semantics Let us extend our Kripke semantics by a channel relation. M = ( W , ( R a ) a ∈ G , ( C ab ) a , b ∈ G , V ) is a Kripke model. C ab ⊆ W is a channel relation s.t. C aa = W . Given any M = ( W , ( R a ) a ∈ G , ( C ab ) a , b ∈ G , V ) and any w ∈ W , M , w | = c ab iff w ∈ C ab M , w | = B a A for all v ∈ W : wR a v implies M , v | = A . iff The truth set � A � M is defined by: � A � M = { w ∈ W | M , w | = A } . 25 / 40
Hilbert-style Axiomatization H K c ( Taut ) A , A is a tautology ( K [B] ) B a ( A → B ) → (B a A → B a B ) ( a ∈ G) ( Selfchn ) c aa ( a ∈ G) ( MP ) From A and A → B , infer B ( Nec [B] ) From A , infer B a A ( a ∈ G) Theorem This axiomatization is decidable, sound and complete for the previous Kripke semantics. 26 / 40
Conditional Private Announcement [ A ↓ a b ] [ A ↓ a b ] : “Agent a sends a message A to agent b via a channel.” When the communication succeeds? Our assumptions: There should be a channel from a to b . Agent a believes the content of the message, to avoid Moore sentences. 27 / 40
Semantics of [ A ↓ a b ] M A ↓ a = [ A ↓ a M , w | b ] B b , w | = B iff where M A ↓ a b = ( W , ( R ′ a ) a ∈ G , ( C ab ) a , b ∈ G , V ) and ( R ′ c ) c ∈ G is defined as: If c = b , for all x ∈ W , { R b ( x ) ∩ � A � M if M , x | = c ab ∧ B a A R ′ b ( x ) := R b ( x ) otherwise. Otherwise, R ′ c := R c . Agent b restricts his/her belief by � A � M if there is a channel from a to b , agent a believes the content of the message. Other agents than b do not change beliefs. 28 / 40
Hilbert-style Axiomatization H K c[ ·↓ a b ] In addition to all the axioms and rules of H K c , we add: [ A ↓ a b ] p ↔ p , [ A ↓ a b ]c cd ↔ c cd , [ A ↓ a b ] ¬ B ¬ [ A ↓ a ↔ b ] B , [ A ↓ a [ A ↓ a b ] B ∨ [ A ↓ a b ]( B ∨ C ) ↔ b ] C , [ A ↓ a B c [ A ↓ a b ] B c B ↔ b ] B ( c ̸ = b ) [ A ↓ a ((c ab ∧ B a A ) → B b ( A → [ A ↓ a b ] B b B ↔ b ] B )) ∧ ( ¬ (c ab ∧ B a A ) → B b [ A ↓ a b ] B ) b ] ) From B , infer [ A ↓ a ( Nec [ A ↓ a b ] B Theorem This is a decidable, sound and complete axiomatization for the previous Kripke semantics. 29 / 40
PDL -extension of Our Syntax PROP = { p , q , . . . } is a finite set of propositional variables. G = { a , b , . . . } is a finite set of atomic programs. We regard each agent’s belief as an atomic program. α ::= a | ( α ∪ α ) | ( α ; α ) | ? A A ::= p | c ab | ¬ A | A ∨ A | [ α ] A [ a ] corresponds to the accessibility of agent a , that is R a . 30 / 40
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