Intro KC for S5 ESDs KC map Efficient representations for the modal logic S5 Alexandre Niveau Bruno Zanuttini GREYC lab, Normandie Univ., Caen, France Recent trends in knowledge compilation Dagstuhl seminar, September 2017 0 / 23
Intro KC for S5 ESDs KC map Plan Epistemic logic S5 for contingent planning 1 Compiling subjective S5 formulas 2 Epistemic splitting diagrams 3 Knowledge compilation map 4 0 / 23
Intro KC for S5 ESDs KC map Plan Epistemic logic S5 for contingent planning 1 Compiling subjective S5 formulas 2 Epistemic splitting diagrams 3 Knowledge compilation map 4 0 / 23
Intro KC for S5 ESDs KC map Introduction S5 = modal logic of mono-agent knowledge: K ( x ∨ y ) : “I know that x or y is true” K x : “I know that x is false” �≡ ¬ K x K x ∨ K x : “I know the value of x ” 1 / 23
Intro KC for S5 ESDs KC map Introduction S5 = modal logic of mono-agent knowledge: K ( x ∨ y ) : “I know that x or y is true” K x : “I know that x is false” �≡ ¬ K x K x ∨ K x : “I know the value of x ” Semantics: interpreted on knowledge states (KSs) knowledge state = set of propositional assignments = all worlds considered to be possibly the actual one KS | = K ϕ : ∀ m ∈ KS , m | = ϕ KS | = ¬ K ϕ : ∃ m ∈ KS , m �| = ϕ 1 / 23
Intro KC for S5 ESDs KC map Introduction S5 = modal logic of mono-agent knowledge: K ( x ∨ y ) : “I know that x or y is true” K x : “I know that x is false” �≡ ¬ K x K x ∨ K x : “I know the value of x ” Semantics: interpreted on knowledge states (KSs) knowledge state = set of propositional assignments = all worlds considered to be possibly the actual one KS | = K ϕ : ∀ m ∈ KS , m | = ϕ KS | = ¬ K ϕ : ∃ m ∈ KS , m �| = ϕ Note: we only consider subjective S5 formulas 1 / 23
Intro KC for S5 ESDs KC map Contingent planning Motivation: contingent planning with sensing the agent has a current KS, e.g. { xyh , xyh } ontic (nondeterministic) action toss in { xyh , xyh } → { xyh , xyh , xyh , xyh } sensing action watch in { xyh , xyh , xyh , xyh } , feedback h → { xyh , xyh } goal: reaching a state (or acquiring a certain knowledge) 2 / 23
Intro KC for S5 ESDs KC map Contingent planning Motivation: contingent planning with sensing the agent has a current KS, e.g. { xyh , xyh } ontic (nondeterministic) action toss in { xyh , xyh } → { xyh , xyh , xyh , xyh } sensing action watch in { xyh , xyh , xyh , xyh } , feedback h → { xyh , xyh } goal: reaching a state (or acquiring a certain knowledge) Planning (offline): reasoning on epistemic formulas consider all possible executions → set of KSs → S5 formula � � Ex: plan toss ; watch in { xyh , xyt } : { xyh , xyh } , { xyt , xyt } goal: positive S5 formula consistency checking, entailment checking, conjunction, forgetting 2 / 23
Intro KC for S5 ESDs KC map Plan Epistemic logic S5 for contingent planning 1 Compiling subjective S5 formulas 2 Epistemic splitting diagrams 3 Knowledge compilation map 4 2 / 23
Intro KC for S5 ESDs KC map Knowledge compilation for subjective S5 Parameterized language of Bienvenu, Fargier, Marquis [AAAI 2010]: epistemic term: T = K ϕ ∧ ¬ K ψ 1 ∧ · · · ∧ ¬ K ψ k (wlog) epistemic DNF: T 1 ∨ · · · ∨ T ℓ 3 / 23
Intro KC for S5 ESDs KC map Knowledge compilation for subjective S5 Parameterized language of Bienvenu, Fargier, Marquis [AAAI 2010]: epistemic term: T = K ϕ ∧ ¬ K ψ 1 ∧ · · · ∧ ¬ K ψ k (wlog) epistemic DNF: T 1 ∨ · · · ∨ T ℓ → general language s-S5-DNF L , L ′ ϕ restricted to language L ψ i restricted to language L ′ Idea: K ≃ ∀ → take an L that supports efficient validity checking and an L ′ that supports efficient consistency checking They considered in particular L = DNF, L ′ = CNF → EDNF 3 / 23
Intro KC for S5 ESDs KC map EDNF � � K ( ab ∨ cd ) ∧ ¬ K ( x ∨ y ) ∧ ( ¬ a ∨ z ) � K ( x ) ∧ ¬ K ( z ∨ x ) ∧ ¬ K ( a ∨ b ) ¬ K ( t ) 4 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ xyz | = ϕ 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ xyz | = ϕ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ xyz | = ϕ ? ϕ xyz | = 5 / 23
Intro KC for S5 ESDs KC map OBDD ϕ x CNF, DNF: y representations are generally large in practice z hard to minimize natural idea: L = L ′ = OBDD ⊤ ⊥ xyz | = ϕ xyz �| = ϕ 5 / 23
Intro KC for S5 ESDs KC map EBDD ∨ ∧ ∧ ¬ K K K x y y z ⊤ ⊥ ⊤ ⊥ 6 / 23
Intro KC for S5 ESDs KC map Limitations of s-S5-DNF L , L ′ Combination of K ϕ → separation between epistemic and propositional levels Epistemic splitting diagrams (ESDs) [N., Zanuttini, IJCAI 2016]: more “native” representation, viewing formulas as sets of KSs inspired from OBDDs knowledge compilation map and experimentation for the three languages (EDNFs, EBDDs, ESDs) 7 / 23
Intro KC for S5 ESDs KC map Plan Epistemic logic S5 for contingent planning 1 Compiling subjective S5 formulas 2 Epistemic splitting diagrams 3 Knowledge compilation map 4 7 / 23
Intro KC for S5 ESDs KC map Constants Model set of an S5 formula = set of KSs: � � Mod ( K ( x ↔ y )) = { xy , xy } , { xy } , { xy } , {} Mod ( ¬ K x ) = {{ x } , { x , x }} Mod ( K x ∨ K y ) = � � � � { xy , xy } , { xy } , { xy } , {} ∪ { xy , xy } , { xy } , { xy } , {} 8 / 23
Intro KC for S5 ESDs KC map Constants Model set of an S5 formula = set of KSs: � � Mod ( K ( x ↔ y )) = { xy , xy } , { xy } , { xy } , {} Mod ( ¬ K x ) = {{ x } , { x , x }} Mod ( K x ∨ K y ) = � � � � { xy , xy } , { xy } , { xy } , {} ∪ { xy , xy } , { xy } , { xy } , {} Two KSs with no variable: {} and { � ∅} → four possible formulas of empty scope → four constants: ⊥ : satisfied by no KS (Mod ( ⊥ ) = ∅ ) ⊤ : satisfied by all KSs (Mod ( ⊤ ) = {{} , { � ∅}} ) ∇ : satisfied by all nonempty KSs (Mod ( ∇ ) = {{ � ∅}} ) ∆ : satisfied only by the empty KS (Mod (∆) = {{}} ) 8 / 23
Intro KC for S5 ESDs KC map Split: base case Inspired from Shannon expansion and BDDs KS = { m ∈ KS | m | = x } ∪ { m ∈ KS | m | = x } → “Split” operator: KS = spl ( x , KS | x , KS | x ) xy = xy xy 9 / 23
Intro KC for S5 ESDs KC map Split: base case Inspired from Shannon expansion and BDDs KS = { m ∈ KS | m | = x } ∪ { m ∈ KS | m | = x } → “Split” operator: KS = spl ( x , KS | x , KS | x ) xy � � y = { xy } ∪ x · xy y xy 9 / 23
Intro KC for S5 ESDs KC map Split: base case Inspired from Shannon expansion and BDDs KS = { m ∈ KS | m | = x } ∪ { m ∈ KS | m | = x } → “Split” operator: KS = spl ( x , KS | x , KS | x ) xy � � � � y y = { xy } ∪ x · = spl ( x , { y } , ) xy y y xy 9 / 23
Intro KC for S5 ESDs KC map Split on sets of KSs What happens with sets of KSs? xy � � � � xy xy , , , { xy } xy xy xy xy 10 / 23
Intro KC for S5 ESDs KC map Split on sets of KSs What happens with sets of KSs? xy � � � � xy xy , , , { xy } xy xy xy xy 10 / 23
Intro KC for S5 ESDs KC map Split on sets of KSs What happens with sets of KSs? x · { y } � � � � xy xy � � , { xy } y , , x · xy xy y 10 / 23
Intro KC for S5 ESDs KC map Split on sets of KSs What happens with sets of KSs? � � � � � � � � y xy xy spl ( x , { y } , , { xy } ) , , y xy xy 10 / 23
Intro KC for S5 ESDs KC map Split on sets of KSs What happens with sets of KSs? � � � � � � � � y xy xy spl ( x , { y } , , { xy } ) , , y xy xy 10 / 23
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