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 ∈ 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 = {p, p ↔ q}, BbN = {p, q}; ¬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 = {p, p ↔ q}, BbN = {p, q}; ¬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 = {p, q, p → q}, incoming information: ¬q. Remainders: B0 = {p → q}, B00 = {p}.

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 ∈ A, then for every proposition y, either (p ∨ y) ∈ A ÷ p or (p ∨ ¬y) ∈ 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 ∈ A, then for every proposition y, either (p ∨ y) ∈ A ÷ p or (p ∨ ¬y) ∈ 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 ∈ A, then for every proposition y, either (p ∨ y) ∈ A ÷ p or (p ∨ ¬y) ∈ A ÷ p, where ÷ is contraction of the above kind.

Nina Gierasimczuk Day 1: Belief Revision Partial Meet Contraction

Solution: Let B ÷ p be the intersection of some of the remainders!

Definition A selection function for B is a function γ such that if B⊥p is non-empty, then γ(B⊥p) is a non-empty subset of B⊥p.

The outcome of the partial meet contraction is equal to the intersection of the set of selected elements of B⊥p, i.e., B ÷ p = ∩γ(B⊥p).

Nina Gierasimczuk Day 1: Belief Revision Selection Function

Nina Gierasimczuk Day 1: Belief Revision Partial Meet Contraction

Nina Gierasimczuk Day 1: Belief Revision Properties of Partial Meet Contraction AGM ‘Rationality’ Postulates of Contraction

1 Closure: B ÷ p = Cn(B ÷ p) the outcome is logically closed 2 Success: If p ∈/ Cn(∅), then p ∈/ Cn(B ÷ p) the outcome does not contain p 3 Inclusion: B ÷ p ⊆ B the outcome is a subset of the original set 4 Vacuity: If p ∈/ Cn(B), then B ÷ p = B if the incoming sentence is not in the original set then there is no effect 5 Extensionality: If p ↔ q ∈ Cn(∅), then B ÷ p = B ÷ q. the outcomes of contracting with equivalent sentences are the same 6 Recovery: B ⊆ (B ÷ p) + p. contraction leads to the loss of as few previous beliefs as possible

Nina Gierasimczuk Day 1: Belief Revision The Characterization of Partial Meet Contraction

Theorem ÷ is an operator of partial meet contraction for a belief set B if and only if it satisfies the six above-mentioned postulates.

Nina Gierasimczuk Day 1: Belief Revision From Selection Functions to Orders

A selection function indicates “best” outcomes. possibly according to some well-behaved plausibility relation.

Definition A selection function γ for a belief set B is relational if and only if there is a binary relation R such that for all sentences p, if B⊥p is non-empty, then

γ(B⊥p) = {A ∈ B⊥p | R(A, C) for all C ∈ B⊥p}.

Furthermore, if R is transitive, then γ and the partial meet contraction that it gives rise to are transitively relational.

AGM Supplementary Postulates on Contraction: 7 Conjunctive inclusion: If p ∈/ B ÷ (p ∧ q), then B ÷ (p ∧ q) ⊆ B ÷ p 8 Conjunctive overlap: (B ÷ p) ∩ (B ÷ q) ⊆ B ÷ (p ∧ q)

Theorem ÷ is a transitively relational partial meet contraction for a belief set B if and only if it satisfies (1-6) and, in addition, both 7 and 8.

Nina Gierasimczuk Day 1: Belief Revision From Selection Functions to Orders

A selection function indicates “best” outcomes. possibly according to some well-behaved plausibility relation.

Definition A selection function γ for a belief set B is relational if and only if there is a binary relation R such that for all sentences p, if B⊥p is non-empty, then

γ(B⊥p) = {A ∈ B⊥p | R(A, C) for all C ∈ B⊥p}.

Furthermore, if R is transitive, then γ and the partial meet contraction that it gives rise to are transitively relational.

AGM Supplementary Postulates on Contraction: 7 Conjunctive inclusion: If p ∈/ B ÷ (p ∧ q), then B ÷ (p ∧ q) ⊆ B ÷ p 8 Conjunctive overlap: (B ÷ p) ∩ (B ÷ q) ⊆ B ÷ (p ∧ q)

Theorem ÷ is a transitively relational partial meet contraction for a belief set B if and only if it satisfies (1-6) and, in addition, both 7 and 8.

Nina Gierasimczuk Day 1: Belief Revision AGM Revision Postulates AGM ’Rationality Postulates’

If ÷ is a partial meet contraction, then the revision ∗ defined via Levi identity:

B ∗ p = (B ÷ ¬p) + p

has the following properties: 1 Closure: B ∗ p = Cn(B ∗ p) 2 Success: p ∈ B ∗ p 3 Inclusion: B ∗ p ⊆ B + p 4 Vacuity: If ¬p ∈/ B, then B ∗ p = B + p 5 Consistency: B ∗ p is consistent if p is consistent. 6 Extensionality: If (p ↔ q) ∈ Cn(∅), then B ∗ p = B ∗ q. And if ÷ is a transitive relational partial meet contraction, then ∗ satisfies also: 7 Superexpansion: B ∗ (p ∧ q) ⊆ (B ∗ p) + q 8 Subexpansion: If ¬q ∈/ B ∗ p, then (B ∗ p) + q ⊆ B ∗ (p ∧ q).

Nina Gierasimczuk Day 1: Belief Revision Possible Worlds Modeling of Beliefs

Alternative models of belief states as sets of possible worlds. A possible world is a maximal consistent subset of the language. A proposition is meant a set of possible worlds. For any set A of sentences, let [A] denote the set of possible worlds that make A true, and similarly for any sentence p let [p] be the set of possible worlds that make p true.

Nina Gierasimczuk Day 1: Belief Revision A Belief Revision

The outcome of revising B by p is a B0 such that 0 1 [B ] ⊆ [p]; 0 2 [B ] is non-empty if [p] is non-empty; 0 3 [B ] is equal to [B] ∩ [p] if [B] ∩ [p] is non-empty.

The above rule for revision corresponds to partial meet revision.

Nina Gierasimczuk Day 1: Belief Revision Sphere (Onion) models Grove 1988

[B] can be thought of as surrounded by a system of concentric spheres. Each sphere represents a degree of closeness or similarity to [B]. The outcome of revising B with p should be the intersection of [p] with the narrowest sphere around [B] that has a non-empty intersection with [p].

Sphere-based revision corresponds to transitively relational partial meet revision.

Nina Gierasimczuk Day 1: Belief Revision Sphere (Onion) models Grove 1988

Nina Gierasimczuk Day 1: Belief Revision Sphere (Onion) models: Revision with p Grove 1988

Nina Gierasimczuk Day 1: Belief Revision Onion models and AGM: Contraction with ¬p

Nina Gierasimczuk Day 1: Belief Revision Onion models and AGM: Contraction with ¬p

Nina Gierasimczuk Day 1: Belief Revision Onion models and AGM: Revision

Nina Gierasimczuk Day 1: Belief Revision Degrees of Implausibility Spohn 1988

The epistemic state as a mapping r from possible worlds to degrees of implausibility.

Definition If B is consistent with the domain of r, define

0 rmin(B) = min{r(w ) ∈ dom(r) ∩ B}

and the conditional implausibility of w given B is:

r(w|B) = −rmin(B) + r(w).

Intuition: rmin(B) is the “height” of the most plausible member of B. r(w|B) is the “height” of w above the most plausible member of B.

Nina Gierasimczuk Day 1: Belief Revision Epistemic State and Plausibility State

Definition An epistemic space, B = (S, Prop), consists of a state space S and a family of propositions Prop ⊆ P(S).

The set of true propositions in a given state s:

Props := {P ∈ Prop : s ∈ P}.

Definition A plausibility space( S, Prop, ≤) is an epistemic space (S, Prop) together with a preorder ≤ on S, called plausibility order.

Nina Gierasimczuk Day 1: Belief Revision Belief operator in plausibility spaces

Let us take (S, Prop, ≤), s ∈ S, and p ∈ Prop,

if ≤ is well-founded: p is believed in s iff ∀u ∈ min≤S u ∈ p; in general: p in believed in s iff ∃w ≤ s ∀u ≤ w u ∈ p.

Nina Gierasimczuk Day 1: Belief Revision DEL belief revision methods

Conditioning of (S, Prop, ≤) with a proposition p, eliminates all worlds of S that do not satisfy p. Lexicographic upgrade of (S, Prop, ≤) with a proposition p, rearranges the preorder by putting all worlds satisfying p to be more plausible than others. Minimal upgrade of (S, Prop, ≤) with a proposition p, rearranges the preorder by making only the most plausible states satisfying p more plausible than all others, leaving the rest of the preorder the same.

Nina Gierasimczuk Day 1: Belief Revision Motivation for Plausibility States: Doxastic Logic

Modal logics of epistemic and doxastic change. Used to analyze the information flow in multi-agent systems. The approach of dynamic epistemic logic.

Nina Gierasimczuk Day 1: Belief Revision Doxastic Epistemic Logic

Definition An epistemic-plausibility model M is a triple

(W , (∼i )i∈A, (≤i )i∈A, V ),

where W 6= ∅ is a set of states, for each i ∈ A, ≤i is a total well-founded preorder on W , and V : Prop → P(W ) is a valuation. For each i ∈ A we will assume that ≤i ⊆ ∼i .

Nina Gierasimczuk Day 1: Belief Revision Doxastic Epistemic Logic

Definition (Syntax of LDOX)

The syntax of doxastic-epistemic language LDOX is defined as follows:

ψ ϕ := p | ¬ϕ | ϕ ∨ ϕ | Ki ϕ | Bi ϕ where p ∈ Prop, i ∈ A.

Definition (Semantics of LDOX)

We interpret LDOX in the states of doxastic-epistemic models in the following way.

M, w |= p iff w ∈ V (p) M, w |= ¬ϕ iff it is not the case that M, w |= ϕ M, w |= ϕ ∨ ψ iff M, w |= ϕ or M, w |= ψ M, w |= Ki ϕ iff for all v such that w∼i v we have M, v |= ϕ ψ M, w |= Bi ϕ iff for all v ∈ Ki [w] if v ∈ min≤i (Ki [w] ∩ kψk) then v |= ϕ We define kϕk such that kϕk = {w ∈ W | w |= ϕ}.

Nina Gierasimczuk Day 1: Belief Revision END OF DAY 1

Nina Gierasimczuk Day 1: Belief Revision