A Lightweight Epistemic Logic and its Application to Planning Elise - - PowerPoint PPT Presentation

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.

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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?

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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 Ki . . . Kjp but not Ki(p ∨ q)).

◮ In particular, K1K2(m ∨ ¬m) is not allowed.

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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 Ki . . . Kjp but not Ki(p ∨ q)).

◮ In particular, K1K2(m ∨ ¬m) is not allowed.

Our approach: use a visibility-based logic (inspired by DEL-PAO) and go from knowing that to knowing whether.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties Adding ‘knowing-that’ operators

Visibility

We have a set of observability operators OBS = {Si : i ∈ Agt} ∪ {JS} and a set of visibility atoms ATM = {σ p : σ ∈ OBS∗, p ∈ Prop}. We can now express “K1K2(m ∨ ¬m)” as S2 m ∧ S1 S2 m.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties 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 Si Si α : σ ∈ OBS∗ and α ∈ ATM}∪ {σ JS α : σ ∈ OBS+ and α ∈ ATM}. Atomic consequence: α ⇒ β iff

• either α = β,
• r α = JS α′ and β = σ α′ for some σ ∈ OBS+

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties 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) s | = ϕ ∧ ϕ′ iff s | = ϕ and s | = ϕ′

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties Adding ‘knowing-that’ operators

Relation with Classical Propositional Calculus (CPC)

s | =CPC α iff α ∈ s Proposition (Expansion of states) For every state s ⊆ ATM and formula ϕ ∈ FmlEL-O, s | = ϕ if and

• nly if s⇒ ∪ I-ATM |

=CPC ϕ.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties Adding ‘knowing-that’ operators

Relation with Classical Propositional Calculus (CPC)

s | =CPC α iff α ∈ s Proposition (Expansion of formulas) Define the expansion of formulas homomorphically from Exp(α) =

• ⊤

if α ∈ I-ATM ( α⇐)

• therwise

. Then for every state s ⊆ ATM and formula ϕ ∈ FmlEL-O, s | = ϕ iff s | =CPC Exp(ϕ). ◮ Using expansion, EL-O model checking problems can be polynomially reduced to classical model checking problems.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties Adding ‘knowing-that’ operators

Proposition (Axiomatization) For every formula ϕ ∈ FmlEL-O, ϕ is EL-O valid iff ϕ is provable in CPC from the following five axiom schemas: Si Si α (Vis1) JS JS α (Vis2) JS Si Si α (Vis3) JS α → Si α (Vis4) JS α → JS Si α (Vis5) Proposition (Finite model property) Let ϕ ∈ FmlEL-O be a formula and s ⊆ ATM a state. Let sϕ = (s⇒ ∪ I-ATM) ∩ ATM(ϕ). Then s | = ϕ iff sϕ | = ϕ.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties 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 | = Si 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 Kiϕ and CKϕ, interpreted as: s | = Kiϕ iff s′ | = ϕ for every s′ such that s ∼i s′; s | = CKϕ iff s′ | = ϕ for every s′ such that s ∼Agt s′.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Visibility and introspection EL-O Properties Adding ‘knowing-that’ operators

Relation with standard epistemic logic

| = Kiα ↔ α ∧ Si α | = CKα ↔ α ∧ JS α Distributivity over disjunctions: | = Ki(p ∨ q) → Kip ∨ Kiq The fixed point axiom CKp → p ∧

i∈Agt

KiCKp

• is valid...

...but not the induction axiom

• ϕ ∧ CK

ϕ →

• i∈Agt

Kiϕ

• → CKϕ.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

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 ce1, ce2 ∈ eff (a), if ceff +(ce1) ∩ (ceff −(ce2))⇐ = ∅

then pre(a) ∧ cnd(ce1) ∧ cnd(ce2) is unsatisfiable in EL-O.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Example (Learning a message) enter i = ¬ini, {⊤, {ini}, ∅}; leavei = ini, {⊤, ∅, {ini}}; reveali = ini, {⊤, {Si m}, ∅, inj, {JS m}, ∅, ¬inj, {Sj Si m}, ∅} aski,j = (ini ↔ inj) ∧ ¬Si m ∧ Sj m ∧ Si Sj m, {⊤, {JS m}, ∅} for i, j ∈ {1, 2} and j = i.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Example (Calls in the original gossip problem) Calli

j = pre(Calli j), eff (Calli j) with pre(Calli j) = ⊤ and:

eff (Calli

j) = {Si s1 ∨ Sj s1, {Si s1, Sj s1}, ∅,

. . . , Si sn ∨ Sj sn, {Si sn, Sj sn}, ∅}.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Definition (Semantics) We define the relation Ra by: sRas′ iff s | = pre(a) and s′ =

    s \

• ce∈eff (a),

s| =cnd(ce)

ceff −(ce) ⇐      ∪

• ce∈eff (a),

s| =cnd(ce)

ceff +(ce).

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Definition (Simple planning tasks) A simple epistemic planning task is a triple P = Act, s0, Goal where Act is a finite set of consistent action descriptions, s0 ∈ 2ATM is a finite state (the initial state) and Goal ∈ FmlEL-O is a boolean formula. It is solvable if at least one state s such that s | = Goal is reachable from s0 via some sequence of actions from Act.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Example (Learning a message) P = Act, s0, Goal with: Act = {enter i, leavei, reveali, aski,j : i, j ∈ {1, 2}, i = j} s0 = {m} Goal = ¬in1 ∧ ¬in2 ∧ JS m Possible solutions: enter 1, enter 2, reveal1, leave1, leave2 enter 2, reveal2, leave2, ask1,2

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Example (Generalized gossip problem of depth k) Gk = ActGk, sGk

0 , GoalGk where:

sGk = {Si si : i ∈ Agt} ∪ {si : i ∈ Agt} GoalGk =

σ ∈{Si : i∈Agt}≤k

• j∈Agt σ sj ,

ActGk = {Calli

j : i, j ∈ Agt, i = j}, where

Calli

j = ⊤, {Si σmsr ∨ Sj σmsr,

{σ σmsr : σ ∈ {Si, Sj}≤k−m}, ∅ : m < k, σm ∈ {Si : i ∈ Agt}m, r ∈ Agt}

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Translation into classical planning

Definition We define the relations Rclass

a

by: sRclass

a

s′ iff s | =CPC pre(a) and s′ =

   s \

• ce∈eff (a),

s| =CPCcnd(ce)

ceff −(ce)

    ∪

• ce∈eff (a),

s| =CPCcnd(ce)

ceff +(ce).

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Definition (Expansion of simple epistemic planning tasks) Consider the simple epistemic planning task P = Act, s0, Goal. Its expansion is defined as Exp(P) = {Exp(pre(a)), Exp(eff (a)) : pre(a), eff (a) ∈ Act}, s0, Exp(Goal), where

Exp(eff (a)) =

• Exp(cnd(ce)), ceff +(ce),
• ceff −(ce)

⇐ : ce ∈ eff (a)

• .

Proposition Let a be an action description. Then Ra = Rclass

Exp(a).

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion Consistent action descriptions Simple planning tasks Translation into classical planning

Proposition Let P be a simple planning task. Then P is solvable iff its expansion Exp(P) is classically solvable. Proposition The problem of deciding solvability of a simple epistemic planing task is PSpace complete.

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Introduction EL-O: Epistemic Logic of Observation Epistemic planning with conditional effects Conclusion

Using a visibility logic, we’ve designed a method for epistemic planning in which solvability of planning tasks is decidable (as

• pposed to DEL), and deciding is PSpace complete.

Our method is less restrictive than other approaches (joint vision, private announcements, knowing-whether) Limitation: we can’t model problems such as the muddy children problem, where agents know disjunctions Future work: distributed planning; joint vision restricted to groups

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