Short Introduction: Belief Revision Nina Gierasimczuk Institute for Logic, Language and Computation University of Amsterdam NASSLLI Austin, Texas June 17th, 2012 Nina Gierasimczuk Day 1: Belief Revision The problem Belief revision is a topic of much interest in theoretical computer science and logic, and it forms a central problem in research into artificial intelligence. In simple terms: how do you update a database of knowledge in the light of new information? What if the new information is in conflict with something that was previously held to be true? G¨ardenfors, Belief Revision Nina Gierasimczuk Day 1: Belief Revision History: Data Bases, Theories and Beliefs Computer science: updating databases (Doyle 1979 and Fagin et al. 1983) Philosophy (epistemology): scientific theory change and revisions of probability assignments; belief change (Levi 1977, 1980, Harper 1977) and its rationality. Nina Gierasimczuk Day 1: Belief Revision AGM Belief Revision Model Names: Carlos Alchourr´on,Peter G¨ardenfors, and David Makinson. 1985 paper in the Journal of Symbolic Logic. Starting point of belief revision theory. Nina Gierasimczuk Day 1: Belief Revision Belief Representation in AGM belief := sentence belief := sentence in some formal language beliefs of an agent := a set of such sentences (belief set) Language of Beliefs in AGM Beliefs are expressed in propositional logic: propositions p; q; r;::: connectives: negation (:), conjunction (^), disjunction (_), implication (!), and equivalence ($). Nina Gierasimczuk Day 1: Belief Revision Belief Representation in AGM belief := sentence belief := sentence in some formal language beliefs of an agent := a set of such sentences (belief set) Language of Beliefs in AGM Beliefs are expressed in propositional logic: propositions p; q; r;::: connectives: negation (:), conjunction (^), disjunction (_), implication (!), and equivalence ($). Nina Gierasimczuk Day 1: Belief Revision Belief Representation in AGM belief := sentence belief := sentence in some formal language beliefs of an agent := a set of such sentences (belief set) Language of Beliefs in AGM Beliefs are expressed in propositional logic: propositions p; q; r;::: connectives: negation (:), conjunction (^), disjunction (_), implication (!), and equivalence ($). Nina Gierasimczuk Day 1: Belief Revision Belief Representation in AGM belief := sentence belief := sentence in some formal language beliefs of an agent := a set of such sentences (belief set) Language of Beliefs in AGM Beliefs are expressed in propositional logic: propositions p; q; r;::: connectives: negation (:), conjunction (^), disjunction (_), implication (!), and equivalence ($). Nina Gierasimczuk Day 1: Belief Revision The Properties of a Belief Set Real vs. Ideal The properties of belief set lead to assumptions about the agent (and vice versa) Example (What beliefs do the beliefs lead to?) 1 John is a bachelor. John is handsome. John is a handsome bachelor. 2 If we charge high fees for university, only the rich enroll. We charge high fees for university. Only the rich enroll. 3 The barber is male. The barber shaves only those men in town who do not shave themselves. The barber is female. Nina Gierasimczuk Day 1: Belief Revision The Properties of a Belief Set Real vs. Ideal The properties of belief set lead to assumptions about the agent (and vice versa) Example (What beliefs do the beliefs lead to?) 1 John is a bachelor. John is handsome. John is a handsome bachelor. 2 If we charge high fees for university, only the rich enroll. We charge high fees for university. Only the rich enroll. 3 The barber is male. The barber shaves only those men in town who do not shave themselves. The barber is female. Nina Gierasimczuk Day 1: Belief Revision The Properties of a Belief Set Real vs. Ideal The properties of belief set lead to assumptions about the agent (and vice versa) Example (What beliefs do the beliefs lead to?) 1 John is a bachelor. John is handsome. John is a handsome bachelor. 2 If we charge high fees for university, only the rich enroll. We charge high fees for university. Only the rich enroll. 3 The barber is male. The barber shaves only those men in town who do not shave themselves. The barber is female. Nina Gierasimczuk Day 1: Belief Revision The Properties of a Belief Set Real vs. Ideal The properties of belief set lead to assumptions about the agent (and vice versa) Example (What beliefs do the beliefs lead to?) 1 John is a bachelor. John is handsome. John is a handsome bachelor. 2 If we charge high fees for university, only the rich enroll. We charge high fees for university. Only the rich enroll. 3 The barber is male. The barber shaves only those men in town who do not shave themselves. The barber is female. Nina Gierasimczuk Day 1: Belief Revision Logical Consequence Definition For any set A of sentences, Cn(A) is the set of logical consequences of A. Cn is a function from sets of sentences to sets of sentences that satisfies the following three conditions: A ⊆ Cn(A) (inclusion); If A ⊆ B, then Cn(A) ⊆ Cn(B) (monotony); Cn(A) = Cn(Cn(A)) (iteration). If p can be derived from A by classical propositional logic, then p 2 Cn(A). Nina Gierasimczuk Day 1: Belief Revision Belief Set In AGM every sentence that follows from the belief set...is already in it. Definition B is a belief set just in case B = Cn(B). Discussion: Logical omniscience; belief sets are theories. \Belief set is not what you actually believe, but what you are committed to believe" (Levi 1991). Belief bases instead of belief sets. BbJ = fp; p $ qg, BbN = fp; qg; :p. Nina Gierasimczuk Day 1: Belief Revision Belief Set In AGM every sentence that follows from the belief set...is already in it. Definition B is a belief set just in case B = Cn(B). Discussion: Logical omniscience; belief sets are theories. \Belief set is not what you actually believe, but what you are committed to believe" (Levi 1991). Belief bases instead of belief sets. BbJ = fp; p $ qg, BbN = fp; qg; :p. Nina Gierasimczuk Day 1: Belief Revision Three Ways of Taking in New Information What can I do to my belief set? If you give priority to the new information: 1 Revision: B ∗ p; p is added and other things are removed, so that the resulting new belief set B0 is consistent. 0 2 Contraction: B ÷ p; p is removed from B giving a new belief set B . 0 3 Expansion: B + p; p is added to B giving a new belief set B . Levi identity: If B ÷ p denotes the contraction of B by p, then: B ∗ p := (B ÷ :p) + p: Nina Gierasimczuk Day 1: Belief Revision Contraction Remainders Obviously, we want B ÷ p is a subset of B that does not imply p. But carefully: no unnecessary removals from B! Definition For any set A and sentence p the remainder for A and p is any set C such that: C is a subset of A, C does does not imply p, and there is no set C 0 such that C 0 does not imply p and C ⊂ C 0 ⊆ A. There can be many different remainders Belief set: B = fp; q; p ! qg, incoming information: :q. Remainders: B0 = fp ! qg, B00 = fpg. Nina Gierasimczuk Day 1: Belief Revision Contraction Remainders Definition For any set A and sentence p the remainder set A?p is the set of inclusion-maximal subsets of A that do not imply p. Formally: a set C is an element of A?p just in case C ⊂ A, C 6` p, and there is no set C 0 such that C 0 6` p and C ⊂ C 0 ⊆ A. So, the result of contraction could be chosen to be just one of the remainders. But: For every theory A and any p, A revised with p will be complete when p is inconsistent with A. Fact: when A is a theory with p 2 A, then for every proposition y, either (p _ y) 2 A ÷ p or (p _:y) 2 A ÷ p, where ÷ is contraction of the above kind. Nina Gierasimczuk Day 1: Belief Revision Contraction Remainders Definition For any set A and sentence p the remainder set A?p is the set of inclusion-maximal subsets of A that do not imply p. Formally: a set C is an element of A?p just in case C ⊂ A, C 6` p, and there is no set C 0 such that C 0 6` p and C ⊂ C 0 ⊆ A. So, the result of contraction could be chosen to be just one of the remainders. But: For every theory A and any p, A revised with p will be complete when p is inconsistent with A. Fact: when A is a theory with p 2 A, then for every proposition y, either (p _ y) 2 A ÷ p or (p _:y) 2 A ÷ p, where ÷ is contraction of the above kind. Nina Gierasimczuk Day 1: Belief Revision Contraction Remainders Definition For any set A and sentence p the remainder set A?p is the set of inclusion-maximal subsets of A that do not imply p. Formally: a set C is an element of A?p just in case C ⊂ A, C 6` p, and there is no set C 0 such that C 0 6` p and C ⊂ C 0 ⊆ A. So, the result of contraction could be chosen to be just one of the remainders. But: For every theory A and any p, A revised with p will be complete when p is inconsistent with A. Fact: when A is a theory with p 2 A, then for every proposition y, either (p _ y) 2 A ÷ p or (p _:y) 2 A ÷ p, where ÷ is contraction of the above kind.