In Praise of Belief Bases: Doing Epistemic Logic without Possible - PowerPoint PPT Presentation

In Praise of Belief Bases: Doing Epistemic Logic without Possible Worlds Emiliano Lorini CNRS-IRIT, Universit´ e Paul Sabatier, Toulouse Workshop on “Doxastic Agency and Epistemic Logic” Bochum, December 2017 1

Explicit beliefs, implicit beliefs and belief bases “...a sentence is explicitly believed when it is actively held to be true by an agent and implicitly believed when it follows from what is believed” (Levesque 1984, p. 198). The concept of explicit belief is tightly connected with the concept of belief base (Nebel 1992; Makinson 1985; Hansson 1993; Rott 1998): belief base ≈ set of facts explicitly believed by an agent Existing logical formalizations of explicit and implicit beliefs (Levesque 1984; Fagin & Halpern 1987) do not clearly account for this connection 2

Contribution of the paper Multi-agent logic capturing the distinction between: explicit belief, as a fact in an agent’s belief base, and implicit belief, as a fact that is deducible from the agent’s explicit beliefs, given the agents’ common ground Main difference with Kripke-style semantics for epistemic logic: Kripkean semantics: notions of possible world and doxastic/epistemic alternative are primitive Our semantics: notion of doxastic alternative is defined from — and more generally grounded on — the concept of belief base 3

Motivation Explicit representation of agents’ belief bases is crucial in order to facilitate the task of designing intelligent systems: robotic agents conversational agents Extensional semantics for epistemic logic, whose most representative example is the Kripkean semantics, is too abstract and too far from the agent specification 4

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 5

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 6

Grammar A countably infinite set of atomic propositions Atm = { p , q , . . . } A finite set of agents Agt = { 1 , . . . , n } Language L 0 ( Atm ): ::= p | ¬ α | α 1 ∧ α 2 | E i α α where p ranges over Atm and i ranges over Agt Language L ( Atm ): ::= α | ¬ ϕ | ϕ 1 ∧ ϕ 2 | I i ϕ ϕ where α ranges over language L 0 ( Atm ) 7

Reading of the operators E i α : agent i explicitly (actually) believes α I i ϕ : agent i implicity (potentially) believes ϕ 8

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 9

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 10

Multi-agent belief base Definition (Multi-agent belief base) A multi-agent belief base is a tuple B = ( B 1 , . . . , B n , V ) where: for every i ∈ Agt , B i ⊆ L 0 is agent i ’s belief base, V ⊆ Atm is the actual state. 11

Satisfaction relation Definition (Satisfaction relation) Let B = ( B 1 , . . . , B n , V ) be a multi-agent belief base. Then: B | = p ⇐ ⇒ p ∈ V B | = ¬ α ⇐ ⇒ B �| = α B | = α 1 ∧ α 2 ⇐ ⇒ B | = α 1 and B | = α 2 B | = E i α ⇐ ⇒ α ∈ B i 12

Doxastic alternatives Definition (Doxastic alternatives) Let B = ( B 1 , . . . , B n , V ) and B ′ = ( B ′ 1 , . . . , B ′ n , V ′ ) be two multi-agent belief bases. Then, B R i B ′ if and only if, for every α ∈ B i , B ′ | = α . B R i B ′ : B ′ is a doxastic alternative for agent i at B 13

Multi-agent belief model Definition (Multi-agent belief model) A multi-agent belief model (MAB) is a pair ( B , Cxt ), where B is a multi-agent belief base and Cxt is a set of multi-agent belief bases. The class of MABs is denoted by MAB . Cxt represents the agents’ common ground (or context) 14

Satisfaction relation Definition (Satisfaction relation (cont.)) Let ( B , Cxt ) ∈ MAB . Then: ( B , Cxt ) | = α ⇐ ⇒ B | = α ∀ B ′ ∈ Cxt : if B R i B ′ then( B ′ , Cxt ) | ( B , Cxt ) | = I i ϕ ⇐ ⇒ = ϕ 15

Consistent multi-agent belief model Definition (Consistent MAB) ( B , Cxt ) is a consistent MAB (CMAB) if and only if, for every B ′ ∈ Cxt ∪ { B } , there exists B ′′ ∈ Cxt such that B ′ R i B ′′ . The class of CMABs is denoted by CMAB . 16

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 17

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 18

Notional doxastic model Definition (Notional doxastic model plus satisfaction relation) A notional doxastic model (NDM) is a tuple M = ( W , D , N , V ) where: W is a set of worlds, → 2 L 0 is a doxastic function, D : Agt × W − → 2 W is a notional function, and N : Agt × W − → 2 W is a valuation function, V : Atm − and that satisfies the following conditions for all i ∈ Agt and w ∈ W : (C1) N ( i , w ) = � α ∈D ( i , w ) || α || M , (C2) there exists v ∈ W such that v ∈ N ( i , w ), with: ( M , w ) | = p ⇐ ⇒ w ∈ V ( p ) ( M , w ) | = ¬ ϕ ⇐ ⇒ ( M , w ) �| = ϕ ( M , w ) | = ϕ ∧ ψ ⇐ ⇒ ( M , w ) | = ϕ and ( M , w ) | = ψ ( M , w ) | ⇐ ⇒ α ∈ D ( i , w ) = E i α ( M , w ) | = I i ϕ ⇐ ⇒ ∀ v ∈ N ( i , w ) : ( M , v ) | = ϕ and || α || M = { v ∈ W : ( M , v ) | = α } . The class of notional doxastic models is denoted by NDM . 19

Quasi-notional doxastic model Definition (Quasi-notional doxastic model) A quasi-notional doxastic model (quasi-NDM) is a tuple M = ( W , D , N , V ) where W , D , N and V are as in a NDM except that Condition C1 is replaced by the following condition, for all i ∈ Agt and w ∈ W : (C1 ∗ ) N ( i , w ) ⊆ � α ∈D ( i , w ) || α || M . The class of quasi-notional doxastic models is denoted by QNDM . A NDM/quasi-NDM M = ( W , A , D , N , V ) is finite if and only if W , D ( i , w ) and V − 1 ( w ) are finite for all i ∈ Agt and w ∈ W . 20

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 21

Outline 1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda 22

Relations between semantics NDMs quasi-NDMs Theorem 3 CMABs Theorem 1 finite NDMs finite quasi-NDMs Theorem 2 Figure: An arrow means that satisfiability relative to the first class of structures implies satisfiability relative to the second class. Dotted arrows denote relations that follow straightforwardly given the inclusion between classes. 23

Equivalence between quasi-NDMs and finite quasi-NDMs Theorem 1 Let ϕ ∈ L . Then, if ϕ is satisfiable for the class of quasi-NDMs, if and only if it is satisfiable for the class of finite quasi-NDMs. Idea of the proof. Filtration argument: A (possibly infinite) quasi-NDM M = ( W , D , N , V ) and finite Σ ⊆ L Define filtrated model M Σ = ( W Σ , D Σ , N Σ , V Σ ) with: W Σ = {| w | Σ : w ∈ W } for all p ∈ Atm : V Σ ( p ) = {| w | Σ : ( M , w ) | = p } if p ∈ Atm (Σ) V Σ ( p ) = ∅ otherwise for all | w | Σ , D Σ ( i , | w | Σ ) = D ( i , w ) ∩ Σ for all | w | Σ , N Σ ( i , | w | Σ ) = {| v | Σ : v ∈ N ( i , w ) } Prove that M Σ is a finite quasi-NDM and that ( M , w ) | = ϕ iff ( M Σ , | w | Σ ) | = ϕ for all ϕ ∈ Σ Apply the construction to Σ = sub ( ϕ ) � 24

Equivalence between finite NDMs and finite quasi-NDMs Theorem 2 Let ϕ ∈ L . Then, ϕ is satisfiable for the class of finite NDMs if and only if ϕ is satisfiable for the class of finite quasi-NDMs. Idea of the proof. Finite quasi-NDM M = ( W , D , N , V ) satisfying ϕ Define terminology of model M , T ( M ) = ∪ w ∈ W , i ∈ Agt Atm ( D ( i , w )) Define injection (naming function): f : Agt × W − → Atm \ ( T ( M ) ∪ Atm ( ϕ )) Injection f exists since Atm is infinite while T ( M ) is finite (since M is finite) Define new model M ′ = ( W ′ , D ′ , N ′ , V ′ ) with: W ′ = W , N ′ = N , for every i ∈ Agt and w ∈ W : D ′ ( i , w ) = D ( i , w ) ∪ { f ( i , w ) } , for every p ∈ Atm : V ′ ( p ) = V ( p ) if p ∈ T ( M ) ∪ Atm ( ϕ ) , V ′ ( p ) = N ( i , w ) if p = f ( i , w ) , V ′ ( p ) = ∅ otherwise. Prove that M ′ is a finite NDM and that M ′ satisfies ϕ � 25