On Action Theory Change: Semantics for Contraction and its Properties

On Action Theory Change:Semantics for Contraction and its Properties

Ivan Jose Varzinczak

Knowledge Representation and Reasoning

Meraka Institute, CSIR

Pretoria, South Africa

Ivan Jose Varzinczak (KRR–Meraka) On Action Theory Change 1 / 29

Motivation


Motivation

Knowledge Base

A coffee is a hot drink

With a token I can buy coffee

After buying I have a hot drink

. . .


Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b


Motivation

Observations

I have got a cold coffee

I cannot buy

I bought and I got no hot drink


Motivation

Observations

I have got a cold coffee

I cannot buy

I bought and I got no hot drink

Need for changing the laws about the behavior of actions


Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b



Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

c,¬h

b b

b



Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b



Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b

b



Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b



Motivation

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b



Outline

1 PreliminariesAction Theories

2 Contracting Action LawsSemanticsAlgorithmsProperties

3 ConclusionContributionsFuture Work


Outline





Outline





Outline





Action Theories

Knowledge bases about the dynamics of the world

Actions

Effects

Preconditions

Usually 3 types of laws

Static laws : ‘a coffee is a hot drink’

Effect laws : ‘after buying I get a coffee’

Executability laws : ‘if I have a token, I can buy’

Reasoning tasks

Projection : ‘do I have a hot drink after I buy?’

Explanation : ‘I hold a coffee. I bought. Did I have a token?’

Planning : ‘how to get a hot drink?’

. . .


Action Theories


Actions

Effects

Preconditions





Reasoning tasks




. . .


Action Theories


Actions

Effects

Preconditions





Reasoning tasks




. . .


Action Theories in Multimodal Logic

Multimodal Logic

Propositional logic + modal operators

I [a] : every a-arrow

I 〈a〉 : some a-arrow

Well defined semantics

I Possible worlds models

Expressive

I Actions, state constraints, nondeterminism

Decidable

I EXPTIME-complete, though

More elegant than FOL



Multimodal Logic






Expressive


Decidable





Multimodal Logic






Expressive


Decidable





Multimodal Logic






Expressive


Decidable





Multimodal Logic






Expressive


Decidable





Multimodal Logic






Expressive


Decidable





Possible worlds semantics: transition systems M = 〈W ,R 〉

W : possible worlds

R : accessibility relations

M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1

Satisfaction in a modelq → p

p → [a1]¬q

p → 〈a1〉>

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


p → [a1]¬q

p → 〈a1〉>

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


p → [a1]¬q

p → 〈a1〉>

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1

Satisfaction in a model

q → p "

p → [a1]¬q

p → 〈a1〉>

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


q → p "

p → [a1]¬q "

p → 〈a1〉>

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


q → p "

p → [a1]¬q "

p → 〈a1〉> "

p → 〈a2〉>

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


q → p "

p → [a1]¬q "

p → 〈a1〉> "

p → 〈a2〉> %

(p ∧ ¬q)→ [a2]⊥




W : possible worlds


M :

p, q p,¬q

¬p,¬q

a1

a1

a1

a2

a2

a1


q → p "

p → [a1]¬q "

p → 〈a1〉> "

p → 〈a2〉> %

(p ∧ ¬q)→ [a2]⊥ "



Example

Static Law: coffee→ hot

Executability Law: token→ 〈buy〉>

Effect Law: ¬coffee→ [buy]coffee, ¬token→ [buy]⊥, hot→ [buy]hot

Definition

Action Theory T = S ∪ E ∪ X



Example

Static Law: coffee→ hot

Executability Law: token→ 〈buy〉>

Effect Law: ¬coffee→ [buy]coffee, ¬token→ [buy]⊥, hot→ [buy]hot

Definition

Action Theory T = S ∪ E ∪ X



Example

T = S ∪ E ∪ X =

coffee→ hot, token→ 〈buy〉>,

¬coffee→ [buy]coffee, token→ [buy]¬token,¬token→ [buy]⊥,

coffee→ [buy]coffee, hot→ [buy]hot

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b



Example

T = S ∪ E ∪ X =

coffee→ hot, token→ 〈buy〉>,

¬coffee→ [buy]coffee, token→ [buy]¬token,¬token→ [buy]⊥,

coffee→ [buy]coffee, hot→ [buy]hot

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b


Outline





Intuitions about Model Contraction

Contracting a law

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

Make the law false in the model



Contracting coffee→ hot

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

Make coffee ∧ ¬hot true in one world




M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

t, c,¬h

b b

b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

¬t, c,¬h

b b

b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

t, c,¬h¬t, c,¬h

b b

b




Contracting a law

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b




Contracting token→ 〈buy〉>

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

Make token ∧ [buy]⊥ true in one world




M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b

b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h




Contracting a law

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b




Contracting token→ [buy]hot

M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

Make token ∧ 〈buy〉¬hot true in one world




M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b





M :

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b

b b



Action Theory Change

Principles (Dalal, 1988)

Maintenance of Consistency

Primacy of New Information

Persistence of Prior Knowledge

Fairness

Irrelevance of Syntax

"

"

"

"

+−

Assumptions in Reasoning about Actions (Shanahan, 1997)

Status of static laws

Focus on the effect laws

Executability laws: very difficult

"

"

"







Fairness


"

"

"

"

+−





"

"

"







Fairness


"

"

"

"

+−





"

"

"







Fairness


"

"

"

"

+−





"

"

"


Choosing Models

Distance between models

Prefer models closest to the original one

Hamming/Dalal distance, etc

Distance dependent on the type of law retracted

Static law: look at the set of worlds

Action laws: look at the set of arrows

Definition

M ′ is as close to M as M ′′ iff

either W −W ′ ⊆W −W ′′

or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′

Notation: M ′ �M M ′′


Choosing Models







Definition



or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′



Choosing Models







Definition



or W −W ′ = W −W ′′ and R −R ′ ⊆ R −R ′′



Choosing Models

Contracting ϕ

Definition

M ′ is a candidate iff

W ⊆W ′

R = R ′

There is w ∈W ′ falsifying ϕ

Take the models that are minimal w.r.t. �M


Choosing Models

Contracting ϕ

Definition


W ⊆W ′

R = R ′

There is w ∈W ′ falsifying ϕ



Choosing Models


¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

t, c,¬h

b b

b �M

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

t, c,¬h¬t, c,¬h

b b

b


Choosing Models


¬t, c, h

t, c, h t,¬c, h

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

¬t, c,¬h

b b

b

Incomparable


Choosing Models

Contracting ϕ→ 〈a〉>

Definition


W ′ = W

R ′ ⊆ R

There is w ∈W ′ falsifying ϕ→ 〈a〉>



Choosing Models

Contracting ϕ→ 〈a〉>

Definition


W ′ = W

R ′ ⊆ R

There is w ∈W ′ falsifying ϕ→ 〈a〉>



Choosing Models


¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b

b �M

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h


Choosing Models


¬t, c, h

t, c, h 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

b b

Incomparable


Choosing Models

Contracting ϕ→ [a]ψ

Definition


W ′ = W

R ⊆ R ′

If (w ,w ′) ∈ R ′ \ R , then w ′ is a target (details in the JAIR paper)

There is w ∈W ′ falsifying ϕ→ [a]ψ



Choosing Models

Contracting ϕ→ [a]ψ

Definition


W ′ = W

R ⊆ R ′

If (w ,w ′) ∈ R ′ \ R , then w ′ is a target (details in the JAIR paper)

There is w ∈W ′ falsifying ϕ→ [a]ψ



Choosing Models


¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b

�M

¬t, c, h

t, c, h t,¬c, h

t,¬c,¬h¬t,¬c,¬h ¬t,¬c, h

b b

b

b

b


Choosing Models


¬t, c, h

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¬t,¬c,¬h ¬t,¬c, h

b b

b

b

Incomparable


Outline





Quick look: Algorithms

We have defined algorithms that contract T giving T ′

Theorem

The algorithms are correct w.r.t. our semantics (details in the JAIR paper)

Theorem

Complexity is exponential, though

Nevertheless

Theorem

The algorithms always terminate

Theorem

Size of T ′ is linear in that of T




Theorem


Theorem


Nevertheless

Theorem


Theorem





Theorem


Theorem


Nevertheless

Theorem


Theorem





Theorem


Theorem


Nevertheless

Theorem


Theorem



Outline





Properties

Monotonicity

T |= T ′

Preservation

If T 6|= α, then T ≡ T ′

Success

If T 6|= ⊥ and 6|= α, then T ′ 6|= α


Properties

Monotonicity

T |= T ′

Preservation


Success

If T 6|= ⊥ and 6|= α, then T ′ 6|= α


Properties

Monotonicity

T |= T ′

Preservation


Success

If T 6|= ⊥ and 6|= α, then T ′ 6|= α


Properties

Equivalences

Contracting with equivalent formulas give the same result

Recovery

T ′ ∪ {α} |= T


Properties

Equivalences

Contracting with equivalent formulas give the same result

Recovery

T ′ ∪ {α} |= T


Outline





Contributions

Approach for action theory change

I Contraction: falsifying a law

I Revision: making a law valid (details in the NRAC’2009 paper)

Intuitive semantics

I Simple operations: add and remove

I Distance between models

I Minimal change

Syntactic operators (algorithms)

I Correct w.r.t. the semantics

Investigation on postulates for action theory change


Contributions




Intuitive semantics



I Minimal change





Contributions




Intuitive semantics



I Minimal change





Contributions




Intuitive semantics



I Minimal change





Outline





Future (rather outstanding) Work

More ‘orthodox’ approach to non-classical revision

I Other distances

I Representation result

Revision of general formulas

I Not only ϕ, ϕ→ 〈a〉>, ϕ→ [a]ψ

I More expressive logics: PDL

I Less expressive logics: Causal Theories of Action

Applications in Description Logics

I Ontology repair




I Other distances







I Ontology repair




I Other distances







I Ontology repair


Reference

I.J. Varzinczak. On Action Theory Change. Journal of ArtificialIntelligence Research (JAIR) vol. 37, 2010.

Thank you!


Reference

I.J. Varzinczak. On Action Theory Change. Journal of ArtificialIntelligence Research (JAIR) vol. 37, 2010.

Thank you!


Technology

On Action Theory Change: Semantics for Contraction and its Properties