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Action Theory Contraction and Minimal Change

Ivan Varzinczak, profile picture
Ivan Varzinczak

The document discusses action theory contraction, which is the process of changing the laws that govern the behavior of actions in a knowledge base. It notes that observations of actions not having their expected effects, like buying something but not receiving the expected outcome, indicate a need to change the laws about how actions behave. The document outlines that it will cover preliminaries on action theories in dynamic logic, discuss semantic contraction of laws and relevant postulates, and draw a conclusion.

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Action Theory Contraction and Minimal Change Ivan Jos´ Varzinczak e Knowledge Systems Group Meraka Institute CSIR Pretoria, South Africa KR’2008 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 1 / 24
Motivation Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation Knowledge Base ‘A coffee is a hot drink’ ‘With a token I can buy coffee’ ‘After buying I have a hot drink” ... Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 4 / 24
Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
Action Theories in Dynamic Logic Possible worlds semantics: Transition Systems M = W , R W : possible worlds R : accessibility relation a1 p1 , ¬p2 p1 , p2 a2 M : a2 a1 ¬p1 , p2 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 6 / 24
Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ ± Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
Action Theories in Dynamic Logic One model of our scenario example ¬t, c, h b b M : t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 9 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 10 / 24
Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting coffee → hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting coffee → hot ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h t, c, ¬h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b t, c, h t, ¬c, h t, c, h t, ¬c, h M b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 20 / 24
Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24

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Action Theory Contraction and Minimal Change

  • 1. Action Theory Contraction and Minimal Change Ivan Jos´ Varzinczak e Knowledge Systems Group Meraka Institute CSIR Pretoria, South Africa KR’2008 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 1 / 24
  • 2. Motivation Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 3. Motivation Knowledge Base ‘A coffee is a hot drink’ ‘With a token I can buy coffee’ ‘After buying I have a hot drink” ... Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 4. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 5. Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 6. Motivation Observations ‘I have a cold coffee’ ‘I cannot buy’ ‘I bought and got no hot drink’ Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 7. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 8. Motivation ¬t, c, h c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 9. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 10. Motivation ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 11. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 12. Motivation ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Need for change the laws about the behavior of actions Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 2 / 24
  • 13. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  • 14. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  • 15. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 3 / 24
  • 16. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 4 / 24
  • 17. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  • 18. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  • 19. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  • 20. Action Theories in Dynamic Logic Dynamic Logic Well defined semantics ◮ Possible worlds models Expressive ◮ Actions, state constraints, nondeterminism Decidable ◮ exptime or pspace-complete, though Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 5 / 24
  • 21. Action Theories in Dynamic Logic Possible worlds semantics: Transition Systems M = W , R W : possible worlds R : accessibility relation a1 p1 , ¬p2 p1 , p2 a2 M : a2 a1 ¬p1 , p2 Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 6 / 24
  • 22. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  • 23. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  • 24. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  • 25. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  • 26. Action Theories in Dynamic Logic Formulas that hold in M a1 p1 , ¬p2 p1 , p2 a2 p1 ∨ p2 M : a2 a1 p1 → [a1 ]p2 p2 → a 2 ⊤ ¬p1 , p2 ¬p1 → a1 ⊤ ± Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 7 / 24
  • 27. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  • 28. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  • 29. Action Theories in Dynamic Logic Describing Laws In RAA: 3 types of laws Static Laws: ϕ ◮ coffee → hot Executability Laws: ϕ → a ⊤ ◮ token → buy ⊤ Effect Laws: ϕ → [a]ψ ◮ ¬coffee → [buy]coffee Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 8 / 24
  • 30. Action Theories in Dynamic Logic One model of our scenario example ¬t, c, h b b M : t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 9 / 24
  • 31. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 10 / 24
  • 32. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 33. Intuitions About Contraction Contracting coffee → hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 34. Intuitions About Contraction Contracting coffee → hot ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 35. Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 36. Intuitions About Contraction Contracting coffee → hot ¬t, c, ¬h ¬t, c, h t, c, ¬h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 37. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 38. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 39. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 40. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b b t, c, h t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 41. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 42. Intuitions About Contraction Contracting token → buy ⊤ ¬t, c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 43. Intuitions About Contraction Contracting laws ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 44. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 45. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 46. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 47. Intuitions About Contraction Contracting token → [buy]hot ¬t, c, h b b t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 11 / 24
  • 48. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  • 49. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  • 50. Minimal Change Choosing models Distance between models ◮ Prefer models closest to the original one ◮ Hamming distance, Dalal, etc Distance dependent on the type of law retracted ◮ Static law: look at the set of possible states (worlds) ◮ Executability law: look at the leaving arrows ◮ Effect law: look at the arriving arrows Definition Let M = W , R . M ′ = W ′ , R ′ is as close to M as M ′′ = W ′′ , R ′′ iff either W −W ′ ⊆ W −W ′′ ˙ ˙ or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′ ˙ ˙ ˙ ˙ Notation: M ′ M M ′′ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 12 / 24
  • 51. Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
  • 52. Minimal Change Choosing models: contracting ϕ Definition Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ iff − W ⊆ W′ R = R′ M′ There is w ∈ W ′ s.t. |= ϕ w Definition contract(M , ϕ) = − min{Mϕ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 13 / 24
  • 53. Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h t, c, ¬h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
  • 54. Minimal Change Choosing models: contracting coffee → hot ¬t, c, h t, c, ¬h ¬t, c, ¬h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 14 / 24
  • 55. Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
  • 56. Minimal Change Choosing models: contracting ϕ → a ⊤ Definition − Let M = W , R . M ′ = W ′ , R ′ ∈ Mϕ→ a ⊤ iff W′ = W R′ ⊆ R M If (w , w ′ ) ∈ R R ′ , then |= ϕ w M′ There is w ∈ W ′ s.t. |= ϕ → a ⊤ w Definition − contract(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 15 / 24
  • 57. Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b t, c, h t, ¬c, h t, c, h t, ¬c, h M b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
  • 58. Minimal Change Choosing models: contracting token → buy ⊤ ¬t, c, h ¬t, c, h b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 16 / 24
  • 59. Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
  • 60. Minimal Change Choosing models: contracting ϕ → [a]ψ Definition − Let M = W , R . Then M ′ = W ′ , R ′ ∈ Mϕ→[a]ψ iff W′ = W R ⊆ R′ If (w , w ′ ) ∈ R ′ R , then w ′ ∈ RelTarget(w , ¬(ϕ → [a]ψ)) M′ There is w ∈ W ′ s.t. |= ϕ → [a]ψ w Definition − contract(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 17 / 24
  • 61. Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h M b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b b Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
  • 62. Minimal Change Choosing models: contracting token → [buy]hot ¬t, c, h ¬t, c, h b b b b t, c, h t, ¬c, h t, c, h t, ¬c, h b b b ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h b Incomparable Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 18 / 24
  • 63. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  • 64. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  • 65. Quick look: Correctness of the Algorithms T an action theory Φ a law We have defined algorithms that contract Φ from T, giving a weaker T ′ ϕ a static law S ⊆ T set of static laws in T Definition (Herzig & Varzinczak, AiML 2005) T is modular iff for every static law ϕ, if T |= ϕ, then S |= ϕ PDL CPL Theorem Under modularity, the algorithms are correct w.r.t. our semantics Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 19 / 24
  • 66. Outline 1 Preliminaries Action Theories in Dynamic Logic 2 Contraction of Laws Semantic Contraction Postulates 3 Conclusion Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 20 / 24
  • 67. Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 68. Postulates Monotonicity T |= T ′ PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 69. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 70. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 71. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 72. Postulates Monotonicity T |= T ′ PDL Preservation If T |= Φ, then |= T ↔ T ′ PDL PDL Success If T |= ⊥ and |= Φ, then T ′ |= Φ PDL PDL PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 21 / 24
  • 73. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 74. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 75. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 76. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 77. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 78. Postulates Equivalences If |= T1 ↔ T2 and |= Φ1 ↔ Φ2 , then |= T1′ ↔ T2′ , for T1′ ∈ (T1 )− PDL PDL PDL Φ2 and T2′ ∈ (T2 )−1 Φ (under modularity) Recovery T ′ ∪ {Φ} |= T PDL (under modularity) Preservation of modularity If T is modular, then T ′ is modular Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 22 / 24
  • 79. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  • 80. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  • 81. Conclusion Contribution Semantics for action theory change ◮ Distance between models ◮ Minimal change Syntactic operators (algorithms) ◮ Correct w.r.t. the semantics Postulates for action theory change ◮ Modularity fruitful Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 23 / 24
  • 82. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
  • 83. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24
  • 84. Conclusion Ongoing research and future work Action theory revision ◮ Making formulas true in a model (first results: NMR’08) Contraction of general formulas ◮ not only ϕ, ϕ → a ⊤, ϕ → [a]ψ Applications in Description Logics ◮ ontology evolution/debugging Ivan Jos´ Varzinczak (KSG - Meraka) e Action Theory Contraction KR’2008 24 / 24