Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion A Lightweight Epistemic Logic and its Application to Planning Elise Perrotin Joint work with Martin Cooper, Andreas Herzig, Faustine Maffre, Frédéric Maris and Pierre Régnier IRIT, Toulouse April 2nd, 2019 Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 1 / 21
Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Introductory example: learning a message Two agents are outside a room, in which there is a message m . Agents can: enter and leave the room; display the message; ask one another about the message. Possible goals: for both agents to know the message; for them to have common knowledge of the message. Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 2 / 21
Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Introductory example: learning a message Two agents are outside a room, in which there is a message m . Agents can: enter and leave the room; display the message; ask one another about the message. Possible goals: for both agents to know the message; for them to have common knowledge of the message. ◮ Typical epistemic planning problem Can we build a lightweight framework in which to model this? Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 2 / 21
Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion A lightweight epistemic planning framework Standard DEL planning is undecidable. Other approaches to simplifying epistemic planning: no common knowledge; public actions; restrict the scope of knowledge operators (e.g., allow K i . . . K j p but not K i ( p ∨ q )). ◮ In particular, K 1 K 2 ( m ∨ ¬ m ) is not allowed. Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 3 / 21
Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion A lightweight epistemic planning framework Standard DEL planning is undecidable. Other approaches to simplifying epistemic planning: no common knowledge; public actions; restrict the scope of knowledge operators (e.g., allow K i . . . K j p but not K i ( p ∨ q )). ◮ In particular, K 1 K 2 ( m ∨ ¬ m ) is not allowed. Our approach: use a visibility-based logic (inspired by DEL-PAO) and go from knowing that to knowing whether . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 3 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Visibility We have a set of observability operators OBS = { S i : i ∈ Agt } ∪ { JS } and a set of visibility atoms ATM = { σ p : σ ∈ OBS ∗ , p ∈ Prop } . We can now express “ K 1 K 2 ( m ∨ ¬ m )” as S 2 m ∧ S 1 S 2 m . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 4 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Introspection Agents should be aware of what they (indiviually and jointly) see. The set of all introspective atoms is I - ATM = { σ S i S i α : σ ∈ OBS ∗ and α ∈ ATM }∪ { σ JS α : σ ∈ OBS + and α ∈ ATM } . Atomic consequence : � either α = β, α ⇒ β iff or α = JS α ′ and β = σ α ′ for some σ ∈ OBS + Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 5 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators EL-O Language The language of EL-O is defined by the following grammar: ϕ ::= α | ¬ ϕ | ( ϕ ∧ ϕ ) where α ranges over ATM . s | = α iff α ∈ I - ATM or β ⇒ α for some β ∈ s s | = ¬ ϕ iff not ( s | = ϕ ) = ϕ ∧ ϕ ′ = ϕ ′ s | iff s | = ϕ and s | Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 6 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Relation with Classical Propositional Calculus (CPC) = CPC α iff α ∈ s s | Proposition (Expansion of states) For every state s ⊆ ATM and formula ϕ ∈ Fml EL-O , s | = ϕ if and only if s ⇒ ∪ I-ATM | = CPC ϕ . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 7 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Relation with Classical Propositional Calculus (CPC) = CPC α iff α ∈ s s | Proposition (Expansion of formulas) Define the expansion of formulas homomorphically from � ⊤ if α ∈ I-ATM Exp( α ) = . ( � α ⇐ ) otherwise Then for every state s ⊆ ATM and formula ϕ ∈ Fml EL-O , s | = ϕ iff = CPC Exp( ϕ ) . s | ◮ Using expansion, EL-O model checking problems can be polynomially reduced to classical model checking problems. Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 7 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Proposition (Axiomatization) For every formula ϕ ∈ Fml EL-O , ϕ is EL-O valid iff ϕ is provable in CPC from the following five axiom schemas: S i S i α ( Vis 1 ) JS JS α ( Vis 2 ) JS S i S i α ( Vis 3 ) JS α → S i α ( Vis 4 ) JS α → JS S i α ( Vis 5 ) Proposition (Finite model property) Let ϕ ∈ Fml EL-O be a formula and s ⊆ ATM a state. Let s ϕ = ( s ⇒ ∪ I-ATM ) ∩ ATM ( ϕ ) . Then s | = ϕ iff s ϕ | = ϕ . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 8 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Adding ‘knowing-that’ operators Definition (Accessibility relations) We associate accessibility relations to agents as follows: s ∼ i s ′ iff s and s ′ agree on every α such that s | = S i α ; s ∼ Agt s ′ iff s and s ′ agree on every α such that s | = JS α. We can extend the language of EL-O by the standard operators K i ϕ and CK ϕ , interpreted as: = K i ϕ iff s ′ | = ϕ for every s ′ such that s ∼ i s ′ ; s | = CK ϕ iff s ′ | = ϕ for every s ′ such that s ∼ Agt s ′ . s | Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 9 / 21
Introduction Visibility and introspection EL-O: Epistemic Logic of Observation EL-O Epistemic planning with conditional effects Properties Conclusion Adding ‘knowing-that’ operators Relation with standard epistemic logic | = K i α ↔ α ∧ S i α | = CK α ↔ α ∧ JS α Distributivity over disjunctions: | = K i ( p ∨ q ) → K i p ∨ K i q The fixed point axiom � � � CKp → p ∧ K i CKp i ∈ Agt is valid... ...but not the induction axiom � �� � ϕ ∧ CK � ϕ → → CK ϕ. K i ϕ i ∈ Agt Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 10 / 21
Introduction Consistent action descriptions EL-O: Epistemic Logic of Observation Simple planning tasks Epistemic planning with conditional effects Translation into classical planning Conclusion Definition (Consistent action descriptions) An action description is a pair a = � pre (a) , eff (a) � where pre (a) is the precondition of a and eff (a) are the conditional effects of a. For each conditional effect ce = � cnd ( ce ) , ceff + ( ce ) , ceff − ( ce ) � , in eff (a), cnd ( ce ) is the condition of ce , ceff + ( ce ) are the added atoms, and ceff − ( ce ) are the deleted atoms. An action description a is consistent if and only if 1 for every ce ∈ eff (a) , ceff − ( ce ) contains no introspective atoms; 2 for every ce 1 , ce 2 ∈ eff (a) , if ceff + ( ce 1 ) ∩ ( ceff − ( ce 2 )) ⇐ � = ∅ then pre (a) ∧ cnd ( ce 1 ) ∧ cnd ( ce 2 ) is unsatisfiable in EL-O . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 11 / 21
Introduction Consistent action descriptions EL-O: Epistemic Logic of Observation Simple planning tasks Epistemic planning with conditional effects Translation into classical planning Conclusion Example (Learning a message) enter i = �¬ in i , {�⊤ , { in i } , ∅�}� ; leave i = � in i , {�⊤ , ∅ , { in i }�}� ; reveal i = � in i , {�⊤ , { S i m } , ∅� , � in j , { JS m } , ∅� , �¬ in j , { S j S i m } , ∅�}� ask i , j = � ( in i ↔ in j ) ∧ ¬ S i m ∧ S j m ∧ S i S j m , {�⊤ , { JS m } , ∅�}� for i , j ∈ { 1 , 2 } and j � = i . Elise Perrotin A Lightweight Epistemic Logic and its Application to Planning 12 / 21
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