Confluence Operators Negotiation as Pointwise Merging S ebastien - PowerPoint PPT Presentation

Confluence Operators – Negotiation as Pointwise Merging – S´ ebastien Konieczny Ram´ on Pino P´ erez CNRS - Centre de Recherche en Informatique de Lens (CRIL) Universit´ e d’Artois, Lens, France konieczny@cril.fr Departamento de Matem´ aticas, Facultad de ciencias Universidad de Los Andes, M´ erida, Venezuela pino@ula.ve 1 / 24

Belief change operators Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. Merging Belief merging is the process of defining the beliefs of a group of agents. So the question is: Given a set of agents that have their own beliefs, what can be considered as the beliefs of the group? 2 / 24

Motivation Update Revision Merging 3 / 24

Motivation Update Revision Merging 3 / 24

Motivation Update Revision Merging 3 / 24

Motivation Update Revision Merging ? 3 / 24

Motivation Update Revision Merging Confluence 3 / 24

Framework • Propositional logic: A formula ϕ is build from ◮ A set P of propositional symbols ( a , b , . . . ) ◮ And logical connectives ( ¬ , ∧ , ∨ , → , . . . ) An interpretation ω is a function from P to { 0 , 1 } mod ( ϕ ) = { ω ∈ W | ω | = ϕ } A formula is complete if it has a unique model • A base ϕ is a (finite set of) propositional formula • A profile Ψ is a multi-set of bases : Ψ = { ϕ 1 , . . . , ϕ n } 4 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 5 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 3 principles: • Primacy of update • Coherence • Minimal change 5 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 3 principles: • Primacy of update • Coherence • Minimal change 5 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 3 principles: • Primacy of update • Coherence • Minimal change 5 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 3 principles: • Primacy of update • Coherence • Minimal change 5 / 24

Revision Revision Belief revision is the process of accomodating a new piece of evidence that is more reliable than the current beliefs of the agent. In belief revision the world is static, it is the beliefs of the agents that evolve. 3 principles: • Primacy of update • Coherence • Minimal change 5 / 24

Revision ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) µ = b 6 / 24

Revision ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } 6 / 24

Revision ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } ϕ ◦ µ = b ∧ ¬ m 6 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update Update In belief update the new piece of evidence denotes a change in the world. The world is dynamic, and these (observed) changes modify the beliefs of the agent. 7 / 24

Update ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } 8 / 24

Update ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } ϕ ⋄ µ = ( b ∧ ¬ m ) 8 / 24

Update ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } ϕ ⋄ µ = ( b ∧ ¬ m ) 8 / 24

Update ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } ϕ ⋄ µ = ( b ∧ ¬ m ) ∨ ( b ∧ m ) 8 / 24

Update ϕ = ( b ∧ ¬ m ) ∨ ( ¬ b ∧ m ) mod ( ϕ ) = { 10 , 01 } µ = b mod ( µ ) = { 10 , 11 } ϕ ⋄ µ = ( b ∧ ¬ m ) ∨ ( b ∧ m ) 8 / 24

Merging Merging Belief merging is the process of defining the beliefs of a group of agents. So the question is: Given a set of agents that have their own beliefs, what can be considered as the beliefs of the group? 9 / 24

Merging Merging Belief merging is the process of defining the beliefs of a group of agents. So the question is: Given a set of agents that have their own beliefs, what can be considered as the beliefs of the group? ϕ 1 ϕ 2 ϕ 3 a, b → c a, b ¬ a △ ( { ϕ 1 , ϕ 2 , ϕ 3 } ) = 9 / 24

Merging Merging Belief merging is the process of defining the beliefs of a group of agents. So the question is: Given a set of agents that have their own beliefs, what can be considered as the beliefs of the group? ϕ 1 ϕ 2 ϕ 3 a, b → c a, b ¬ a △ ( { ϕ 1 , ϕ 2 , ϕ 3 } ) = b → c , b 9 / 24

Merging Merging Belief merging is the process of defining the beliefs of a group of agents. So the question is: Given a set of agents that have their own beliefs, what can be considered as the beliefs of the group? ϕ 1 ϕ 2 ϕ 3 a, b → c a, b ¬ a △ ( { ϕ 1 , ϕ 2 , ϕ 3 } ) = b → c , b , a 9 / 24

Confluence g : german car e : expensive car s : sport car ϕ 1 = ¬ g ∧ ¬ e ∧ s mod ( ϕ 1 ) = { 001 } ϕ 2 = ( g ∧ e ∧ s ) ∨ ( ¬ g ∧ ¬ e ∧ s ) mod ( ϕ 2 ) = { 001 , 111 } µ = ¬ ( g ∧ ¬ e ∧ s ) mod ( µ ) = W \ { 101 } • Belief/Goal Merging: △ µ ( { ϕ 1 , ϕ 2 } ) = ¬ g ∧ ¬ e ∧ s 10 / 24

Confluence g : german car e : expensive car s : sport car ϕ 1 = ¬ g ∧ ¬ e ∧ s mod ( ϕ 1 ) = { 001 } ϕ 2 = ( g ∧ e ∧ s ) ∨ ( ¬ g ∧ ¬ e ∧ s ) mod ( ϕ 2 ) = { 001 , 111 } µ = ¬ ( g ∧ ¬ e ∧ s ) mod ( µ ) = W \ { 101 } • Belief/Goal Merging: △ µ ( { ϕ 1 , ϕ 2 } ) = ¬ g ∧ ¬ e ∧ s 10 / 24

Confluence g : german car e : expensive car s : sport car ϕ 1 = ¬ g ∧ ¬ e ∧ s mod ( ϕ 1 ) = { 001 } ϕ 2 = ( g ∧ e ∧ s ) ∨ ( ¬ g ∧ ¬ e ∧ s ) mod ( ϕ 2 ) = { 001 , 111 } µ = ¬ ( g ∧ ¬ e ∧ s ) mod ( µ ) = W \ { 101 } • Belief/Goal Merging: △ µ ( { ϕ 1 , ϕ 2 } ) = ¬ g ∧ ¬ e ∧ s • Confluence: ♦ µ ( { ϕ 1 , ϕ 2 } ) = ( ¬ g ∧ ¬ e ∧ s ) ∨ ( ¬ g ∧ e ∧ s ) = ¬ g ∧ s 001 011 10 / 24

Confluence g : german car e : expensive car s : sport car ϕ 1 = ¬ g ∧ ¬ e ∧ s mod ( ϕ 1 ) = { 001 } ϕ 2 = ( g ∧ e ∧ s ) ∨ ( ¬ g ∧ ¬ e ∧ s ) mod ( ϕ 2 ) = { 001 , 111 } µ = ¬ ( g ∧ ¬ e ∧ s ) mod ( µ ) = W \ { 101 } • Belief/Goal Merging: △ µ ( { ϕ 1 , ϕ 2 } ) = ¬ g ∧ ¬ e ∧ s • Confluence: ♦ µ ( { ϕ 1 , ϕ 2 } ) = ( ¬ g ∧ ¬ e ∧ s ) ∨ ( ¬ g ∧ e ∧ s ) = ¬ g ∧ s 001 011 10 / 24