What Is a Good Domain Description? Evaluating & Revising Action Theories in Dynamic Logic

IntroductionMain Results

Concluding Remarks

What is a Good Domain Description?Evaluating & Revising Action Theories in Dynamic Logic

Ivan José Varzinczak

IRIT – Université Paul Sabatier

October 27th 2006

Ivan José Varzinczak What is a Good Domain Description?


Concluding Remarks

Reasoning About Actions

Problem: describing domains by logical formulasActions and their effectsExecutabilities of actionsInexecutabilities of actionsDomain constraints

ExampleA turkey that walks is aliveTeasing a turkey makes it walkIt is always possible to tease a turkeyA dead turkey remains dead after teasing it



Concluding Remarks


Problem: describing domains by logical formulasActions and their effectsExecutabilities of actionsInexecutabilities of actionsDomain constraints

ExampleA turkey that walks is aliveTeasing a turkey makes it walkIt is always possible to tease a turkeyA dead turkey remains dead after teasing it



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Goal: inference tasksPredictionExplanationPlanning



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Prediction: reasoning about the future

actionsInitialstate

Resultstate

?

After shooting, the turkey stops walking



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Explanation: reasoning about the past

stateCurrent

stateInitial actions

?

After shooting, the turkey is dead: the gun was loaded



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Planning: what to do to achieve a goal

?actionsstate

Currentstate

Desired

To have the turkey dead: load the gun, then shoot



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Other important tasksConsistency checkTest of executability/inexecutabilityTheory change. . .



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Outline

1 IntroductionDescribing Action TheoriesUnwanted Conclusions

2 Main ResultsDecomposing TheoriesLogical ModularityExploiting ModularityTheory Change

3 Concluding Remarks



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Describing Action TheoriesUnwanted Conclusions

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Formalizing Domains

Several base formalismsSituation calculus [McCarthy & Hayes, 1969]� s �� Holds � loaded � s �� Holds � alive � do � shoot � s ��Languages � , �� , etc. [Lifschitz et al., 90’s]

shoot causes � alive if loadedFluent calculus [Thielscher, 1995]

Poss � shoot � tk �� s ��State � do � shoot � tk �� s �� State � s �� dead � tk �� alive � tk �

. . .



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Formalizing Domains

In this work. . .we have chosen Modal Logic

Weak version of Propositional Dynamic Logic (PDL)Simple and decidableWith a tableaux-based theorem prover: Lotrec



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Logical PreliminariesOntology

Actions: �� a1 � a2 �! ! ! #"Atomic propositions: $&%!')(*�� p1 � p2 �! ! ! #"Literals: +-,.��/$&%!'0(�12�43 p 5 p 67$&%!'0( "Classical formulas: 8:9<;=�>�4? 1 � ? 2 �! ! ! #"

Action operatorsFor each a 6@�� , a modal operator A a BA a BC? : “after execution of a, ? is true”D

a EGF : “a is inexecutable”Ha I�?J� def 3KA a BC3L?M

a NPO : “a is executable”Complex formulas: Q 1 � Q 2 �! ! !



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Logical Preliminaries

ExampleActions: shoot, teasePropositions: loaded, alive, walkingFormulas: alive RS3 walking,

Htease I�T ,

loaded UVA shoot BC3 alive



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Semantics

Multimodal logic K [Popkorn 94,Blackburn et al. 2001].

DefinitionModels W � H W � R I where

W X 2 Y[ZC\] : set of possible worlds (states)R: ��_^`U 2W a W

Definitionb �)cw p (p is true at world w of model W ) iff p 6 wb �)cw A a BdQ iff for every w e such that wRaw e , b �)cw f Qthe usual truth conditions for the other connectives



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Semantics

Multimodal logic K [Popkorn 94,Blackburn et al. 2001].

DefinitionModels W � H W � R I where

W X 2 Y[ZC\] : set of possible worlds (states)R: ��_^`U 2W a W

Definitionb �)cw p (p is true at world w of model W ) iff p 6 wb �)cw A a BdQ iff for every w e such that wRaw e , b �)cw f Qthe usual truth conditions for the other connectives



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Semantics

ExampleIf ��>� a1 � a2 " , and $&%!'0(g�� p1 � p2 " , W � H W � R I , where

W �>�h� p1 � p2 "i� � p1 � 3 p2 "i� �43 p1 � p2 "h"i�R j a1 k � l jm� p1 � p2 "i� � p1 � 3 p2 "4kn� jm� p1 � p2 "i� �43 p1 � p2 "4kn�jm�43 p1 � p2 "i� �43 p1 � p2 "4kn� jm�43 p1 � p2 "i� � p1 � 3 p2 "4kJo

R j a2 k ��ijm� p1 � p2 "i� � p1 � 3 p2 "4kn� jm� p1 � 3 p2 "i� � p1 � 3 p2 "4kp"is a model



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SemanticsExample

W :

p1 q p2 r p1 s p2

p1 str p2

a1

a1

a2a1

a1

a2

u v)wp1 x p2u v w p1 y{z a2 |C} p2u v)w } p1 y{~ a1 ��u v w } p2 yVz a1 |��



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Semantics

Definitionb � c Q iff for all w 6 W,b � cw Qb �)c�� iff

b �)c�Q for every Q�62�DefinitionQ is a consequence of the set of global axioms � in allPDL-models (noted � b � PDL Q ) iff for every W , if

b � c � , thenb � c Q .



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Semantics

Definitionb � c Q iff for all w 6 W,b � cw Qb �)c�� iff

b �)c�Q for every Q�62�DefinitionQ is a consequence of the set of global axioms � in allPDL-models (noted � b � PDL Q ) iff for every W , if

b � c � , thenb � c Q .



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The Tale Again

ExampleA turkey that walks is alive: walking U aliveTeasing a turkey makes it to walk: A teaseB walkingIt is always possible to tease a turkey:

Htease I�T

A dead turkey remains dead after teasing it� alive � Dtease E.� alive

If the gun is loaded, shooting kills the turkeyloaded � D

shoot Ed� aliveTeasing does not unload the gun

loaded � Dtease E loaded



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The Tale Again

Example�� Htease I�T �

walking U alive �A teaseB walking �loaded U{A shoot BC3 alive �3 alive UVA tease BC3 alive �loaded UVA tease B loaded

� ��N.B.: Such a description is consistent

b � A tease B aliveb ��3 alive UVA teaseBPj alive RS3 alive kb ��3 alive U{A teaseB��b � alive

What is the problem?



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The Tale Again




b � A tease B aliveb ��3 alive UVA teaseBPj alive RS3 alive kb ��3 alive U{A teaseB��b � alive




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The Tale Again




b � A tease B aliveb � 3 alive UVA teaseBPj alive RS3 alive kb ��3 alive U{A teaseB��b � alive




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The Tale Again




b ��A tease B aliveb ��3 alive UVA teaseBPj alive RS3 alive kb ��3 alive U{A teaseB��b � alive




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The Tale Again








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Decomposing TheoriesLogical ModularityExploiting ModularityTheory Change

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Natural Modules in Action Theories

Types of domain lawsStatic laws : walking U aliveEffect laws : loaded UVA shoot B�3 alive

Executability laws : hasGun U Hshoot I�T

Inexecutability laws : 3 hasGun UVA shoot BC�! only formulas of these types



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Defining modules� : set of static lawsGiven a 6@�� a: effect laws for a� a: executability laws for a� a: inexecutability laws for aH � �� a �n� a �� a I : domain description for a� �� a �4��#� � a, � �� a �4��#� � a, and � �� a ��#� � aH � ��n� �� I : the action theory of a given domain



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What About the Frame Problem?

In our exampleIf we had an action wait¡ � � � � � �£¢¤ � PDL loaded � D

wait E loaded

DefinitionDependence relation [Castilho et al. 99]: ¥¦X§��_¨©+ª,.�Example

shoot ¥«3 alive, tease ¥ walking, tease ¬¥ aliveFrom wait ¬¥«3 loaded conclude loaded UVAwait B loaded



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wait E loaded





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wait E loaded





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wait E loaded





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Restriction on modelsFor all wRaw e :¬b � cw p implies ¬b � cw f p, if a ¬¥ pb �)cw p implies

b �0cw f p, if a ¬¥3 p.

New logical consequenceb ��® instead ofb � PDL

Example� ��n� �� b � ® loaded U¯AwaitB loaded



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b �0cw f p, if a ¬¥3 p.





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b �0cw f p, if a ¬¥3 p.





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The dependence-based approach. . .solves the frame problemsubsumes Reiter’s regression [Demolombe et al. 2003]does not entirely solve the ramification problem

e.g. shoot °±� walkingBut is the only approach that works for domains withactions with both indeterminate and indirect effects[Castilho et al. 2002], [Herzig & Varzinczak 2004]



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Consistency and MorePostulatesPC (Consistency): � �� a �n� a �� a ¬b � ® �PS (No implicit static laws): if � �� a �n� a �� a

b � ® ? , then � b ��?PI (No implicit inexecutability laws):

if � �� a �n� a �� a

b � ® ?²U{A aB�� ,then � �� a

b � PDL ?²UVA a BC�PX (No implicit executability laws):

if � �� a �n� a �� a

b ��®³?²U Ha I�T ,

then � �n� a

b � PDL ?JU Ha I�T

MotivationBetter control what is going on



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if � �� a �n� a �� a

b � ® ?²U{A aB�� ,then � �� a


if � �� a �n� a �� a

b ��®³?²U Ha I�T ,

then � �n� a





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if � �� a �n� a �� a

b � ® ?²U{A aB�� ,then � �� a


if � �� a �n� a �� a

b ��®³?²U Ha I�T ,

then � �n� a





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if � �� a �n� a �� a

b � ® ?²U{A aB�� ,then � �� a


if � �� a �n� a �� a

b ��®³?²U Ha I�T ,

then � �n� a





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if � �� a �n� a �� a

b � ® ?²U{A aB�� ,then � �� a


if � �� a �n� a �� a

b ��®³?²U Ha I�T ,

then � �n� a





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No Implicit Static Laws

Example� �� walking U alive "i�_� � l A tease B walking �loaded UVA shoot BC3 alive o �� H tease I�T "i�[� ��43 alive UVA tease BC� "

tease ¥ walking � shoot ¥«3 alive� �� tease �n� tease �� tease

b � ® aliveBut � ¬b � alive

! Postulate PS violated



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Idea of algorithmFor each ?JU H

a I�T1 find ?[e�UVA a BC� entailed by the theory2 if ?´R©?[e is consistent with ��Gµ&¶�µ�·¸� is possibly an implicit law

Result: the set of all implicit static laws



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Idea of algorithmFor each ?JU H

a I�T1 find ?[e�UVA a BC� entailed by the theory2 if ?´R©?[e is consistent with ��Gµ&¶�µ�·¸� is possibly an implicit law

Result: the set of all implicit static laws



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No Implicit Static LawsAlgorithm 1: Finding all implicit static laws induced by ainput: ¹¸º_»�¼ a »½ a ».¾ a ¿ and Àoutput: º imp*, the set of all implicit static laws of ¹¸º_»�¼ a »P½ a »#¾ a ¿calls: NewCons ÁÃÂ.Ä:ÅÇÆ PI ÂGÈÊÉËÄ:Å=Ì PI ÂGÈÇÅº imp*:= Í

repeatº imp:= Ífor all ÈÏÎ¦¹ a ¿�ÐÒÑ ½ a do

for all Ó a fÃÔ ¼ a Õ ¾ a doÈ×Ö a Ø := ÙKÚnÈ i Û È i Î¦Ü a Ý�Ä iÑ Ó a f¸ÞÄ Ö a Ø := ÙKÚßÄ i Û È i Î¦Ü a Ý�Ä iÑ Ó a f¸Þ

for all à Ñ NewConsá�Â.ÄÇÖ a Ø Å doif º Õ º imp*

Õ ÚpÈ`»PÈ×Ö a Ø�»âtà Þ�ãgä and å li Ñ à�» a æÀ li thenº imp:= º impÕ ÚnâçÂGÈÊÉèÈ Ö a Ø ÉéâtàÇÅ Þº imp*:= º imp*

Õ º impuntil º imp Æ7Í



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ExampleºSÆ©Ú walking Î alive Þ »`¼ÏÆ ê Ü tease Ý walking »loaded Î¦Ü shootÝ#â alive ë »½�Æ©Úì¹ tease ¿�Ð Þ »i¾@Æ©Úpâ alive Î¦Ü tease Ý ä_Þ

tease À walking » shoot Àíâ alive

ForHtease I�T and A tease B walking:

NewCons îLj walking k � alive: A tease B alivetease ¬¥ alive: 3 alive U{A teaseB�3 aliveHence 3 alive U{A tease BC�� 12�4T�RS3 alive " ¬b �� : � ¬b � alive�

imp �� alive "Ivan José Varzinczak What is a Good Domain Description?


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ExampleºSÆ©Ú walking Î alive Þ »`¼ÏÆ ê Ü tease Ý walking »loaded Î¦Ü shootÝ#â alive ë »½�Æ©Ú ¹ tease ¿�Ð Þ »i¾@Æ©Úpâ alive Î¦Ü tease Ý ä_Þ






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Example (cont.)Alternatives for repairing:�

:=� 1©� alive "

add tease ¥ aliveweaken A teaseB walking: alive UVA tease B walkingweaken

Htease I�T : alive U H

tease I�Tcontraction of action theories (addressed later)



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:=� 1©� alive "






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:=� 1©� alive "






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TheoremH � �� a �n� a �� a I and ¥ satisfy Postulate PS iff � imp* �ðï .TheoremLet � imp* be the output of Algorithm 1 on input

H � �� a �n� a �� a Iand ¥ . ThenH � 1 � imp* �� a �n� a �� a I has no implicit static law.� �� a �n� a �� a

b � ®³ñ � imp*.

CorollaryFor all ?ò6@8:9<; , � �� a �n� a �� a

b � ® ? iff � 1 � imp*

b ��? .



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b � ®³ñ � imp*.


b � ® ? iff � 1 � imp*

b ��? .



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b � ®³ñ � imp*.


b � ® ? iff � 1 � imp*

b ��? .



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No Implicit Inexecutability LawsExampleºSÆ©Ú walking Î alive Þ »`¼ÏÆ ê Ü tease Ý walking »

loaded Î¦Ü shootÝ#â alive ë »½óÆ*¾<ÆSÍì» tease À walking » shoot Àôâ alive� �� tease

b �ß®õA tease B alive� �� tease

b � ® 3 alive UVA teaseB�3 alive (from tease ¬¥ alive)Thus � �� tease �n� tease �� tease

b � ® 3 alive UVA tease BC�But � �� tease ¬b � PDL 3 alive UVA tease BC�

! Postulate PI violated



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No Implicit Inexecutability Laws

Idea of algorithmFor each combination of effect laws

1 find inconsistent consequents2 mark it as an implicit inexecutability

Result: the set of all implicit inexecutabilities



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Idea of algorithmFor each combination of effect laws

1 find inconsistent consequents2 mark it as an implicit inexecutability

Result: the set of all implicit inexecutabilities



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Algorithm 2: Finding implicit inexecutability laws for ainput:

H � �� a �� a I and ¥output: � a

imp, the set of implicit inexecutability laws for acalls: NewCons öçj�÷ k � PI jP?7R´÷ k�ø PI jP? k� a

imp:= ïfor all � a e X � a do?�ù a f := ñ �4? i 5=? i UVA a Bd÷ i 6 � a e "÷�ù a f := ñ �ì÷ i 5Ã? i UVA a Bd÷ i 6 � a e "

for all úò6 NewCons î j�÷ªù a f k doif û li 67ú � a ¬¥ li and � �� a ü jP?[ù a f RS3�ú k UVA a BC� then� a

imp:= � aimp 12�ijP?[ù a f R73ªú k UVA a BC� "Ivan José Varzinczak What is a Good Domain Description?


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ExampleºSÆ©Ú walking Î alive Þ »`¼ÏÆ ê Ü tease Ý walking »loaded Î¦Ü shootÝ#â alive ë »½�Æg¾@Æ7Íì» tease À walking » shoot Àôâ alive

For action tease:NewCons îLj walking k � alive: A tease B alivetease ¬¥ alive: 3 alive U{A teaseB�3 aliveThen 3 alive UVA tease BC�� tease ¬b � PDL 3 alive UVA tease BC�� tease

imp ��43 alive UVA tease B�� "Ivan José Varzinczak What is a Good Domain Description?


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TheoremIfH � �� a �n� a �� a I and ¥ satisfy Postulate PS, thenH � �� a �n� a �� a I and ¥ satisfy Postulate PI iff � imp �ðï .



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Generalizing the Postulates

PostulatePS* (No implicit static laws):

if � ��n�ý�� b ��®³? , then � b � PDL ?TheoremH � ��n� �� I and ¥ satisfy PS* iff

H � �� a �n� a �� a I and ¥satisfy PS for all a 67�� .



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PostulatePS* (No implicit static laws):

if � ��n�ý�� b ��®³? , then � b � PDL ?TheoremH � ��n� �� I and ¥ satisfy PS* iff

H � �� a �n� a �� a I and ¥satisfy PS for all a 67�� .



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PostulatePC* (Logical consistency): � ��n�ý�� ¬b � ® �

TheoremIfH � ��n� �� I and ¥ satisfy PS*, then

H � ��n�J�� I and ¥satisfy PC* iff

H � �� a �n� a �� a I and ¥ satisfies PC for all a 67�� .



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PostulatePC* (Logical consistency): � ��n�ý�� ¬b � ® �

TheoremIfH � ��n� �� I and ¥ satisfy PS*, then

H � ��n�J�� I and ¥satisfy PC* iff

H � �� a �n� a �� a I and ¥ satisfies PC for all a 67�� .



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PostulatePI* (No implicit inexecutability laws):

if � ��n� �� b ��®þ?JUVA a B�� , then � �� b � PDL ?JUVA a BC�TheoremLet

H � ��n� �� I and ¥ satisfy PS*.H � ��n� �� I and ¥

satisfy PI* iffH � �� a �n� a �� a I and ¥ satisfy PI for all a 6@�� .



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PostulatePI* (No implicit inexecutability laws):

if � ��n� �� b ��®þ?JUVA a B�� , then � �� b � PDL ?JUVA a BC�TheoremLet

H � ��n� �� I and ¥ satisfy PS*.H � ��n� �� I and ¥

satisfy PI* iffH � �� a �n� a �� a I and ¥ satisfy PI for all a 6@�� .



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Reasoning Modularly

IfH � ��n� �� I and ¥ satisfy Postulate PS*, then

Theorem� ��n� �� b � ® � iff � b �� .

Theorem� ��n� �� b � ® ?JUVA a Bd÷ iff � �� a �� a

b � ® ?JUVA a BG÷ .



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Reasoning Modularly


Theorem� ��n� �� b � ® � iff � b �� .

Theorem� ��n� �� b � ® ?JUVA a Bd÷ iff � �� a �� a

b � ® ?JUVA a BG÷ .



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Reasoning Modularly


Theorem� ��n� �� b � ® ?JU Ha I�T iff � �n� a

b � ® ?JU Ha I�T .

CorollaryPX is a consequence of PS.

TheoremIfH � ��n� �� I and ¥ satisfy Postulates PS* and PI*, then� ��n� �� b � ® ?JUVA a BC� iff � �� a

b � ® ?JUVA aB�� .



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Reasoning Modularly



b � ® ?JU Ha I�T .



b � ® ?JUVA aB�� .



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Reasoning Modularly



b � ® ?JU Ha I�T .



b � ® ?JUVA aB�� .



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Theorem� ��n� �� b � ® ?JUVA a1 �! ! ! ÿ� an Bd÷ iff� �� a1 �� an �� a1 �� an

b � ® ?JUVA a1 �! ! ! ÿ� an Bd÷ .

Theorem� ��n� �� b � ® ?JU Ha1 �! ! ! ÿ� an IP÷ iff� �� a1 �� an �n� a1 �� an �� a1 �� an

b � ® ?JU Ha1 �! ! ! ÿ� an IP÷ .



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Reasoning Modularly


Theorem� ��n� �� b � ® ?JUVA a1 �! ! ! ÿ� an Bd÷ iff� �� a1 �� an �� a1 �� an

b � ® ?JUVA a1 �! ! ! ÿ� an Bd÷ .

Theorem� ��n� �� b � ® ?JU Ha1 �! ! ! ÿ� an IP÷ iff� �� a1 �� an �n� a1 �� an �� a1 �� an

b � ® ?JU Ha1 �! ! ! ÿ� an IP÷ .



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Another Tale

ExampleIf the switch is up, the room is lit up

up � lightToggling the switch changes its position� up � D

toggle E upup � D

toggle Ed� upIt is always possible to toggle the switchM

toggle N�OIvan José Varzinczak What is a Good Domain Description?


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The Need for Theory Change

You observe that. . .even if the switch is up the light is off.in a blackout, you do not succeed to switch the light on.despite your efforts you do not manage to toggle theswitch.



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Contraction: Motivation

Contracting by a static lawYou observe that even if the switch is up the light is offStatic law up U light must be given upCan we just contract the static laws of � ?

May not be enough: side effects!Conflict with ½The contracted law may be an implicit one



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Contracting by a static lawYou observe that even if the switch is up the light is offStatic law up U light must be given upCan we just contract the static laws of � ?

May not be enough: side effects!Conflict with ½The contracted law may be an implicit one



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Contracting by an effect lawDuring a blackout you do not succeed to switch the light onEffect law 3 up UVA toggleB light must be given upImportant issue: give up as few as possible



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Contracting by an effect lawDuring a blackout you do not succeed to switch the light onEffect law 3 up UVA toggleB light must be given upImportant issue: give up as few as possible



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Contracting by an executability lawDespite your efforts you do not manage to toggle the switchExecutability law up U H

toggle I�T must be given upSide effects?



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Contraction: Semantics

Playing with models

� up q � light up s lighttoggle

toggle

Semantical contraction produces a set of models:~ W � R � �� v�� ~ W � R �� Ivan José Varzinczak What is a Good Domain Description?


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Playing with models


toggle

Semantical contraction produces a set of models:~ W � R � �� v�� ~ W � R �� Ivan José Varzinczak What is a Good Domain Description?


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Contraction: SemanticsContracting static lawsH

W � R I��up � light ��Intuition: add some up R73 light-worlds to W

� up q � light up s light

up s×r light

toggle

toggle

Don’t add new arrows to R!Ivan José Varzinczak What is a Good Domain Description?


Concluding Remarks


Contraction: SemanticsContracting static lawsH

W � R I��up � light ��Intuition: add some up R73 light-worlds to W

� up q � light up s light

up s×r light

toggle

toggle

Don’t add new arrows to R!Ivan José Varzinczak What is a Good Domain Description?


Concluding Remarks



Contracting static lawsRely on any belief change operator for classical logic

Say PMA, . . .HW � R I �up � light �� H W e � R e I " , where

W · � W � PMAup � lightR · � R

N.B.:HW e � R e Ié¬b � H toggle I�T

Executability laws to be weakened!



Concluding Remarks










Concluding Remarks










Concluding Remarks



Contracting effect lawsHW � R I �� up �� toggle � light ��

Intuition: add some arrows from 3 up-worlds to3 light-worlds


toggle

toggle



Concluding Remarks



Contracting effect lawsHW � R I �� up �� toggle � light �� H W � R 1 R ea I§5 R ea X �ij w � w e k 5 w

b ��3 up "h"Problems:

Don’t link light-worldsDon’t link all � light-worlds



Concluding Remarks



Contracting effect lawsHW � R I �� up �� toggle � light �� H W � R 1 R ea I§5 R ea X �ij w � w e k 5 w

b ��3 up "h"Problems:

Don’t link light-worldsDon’t link all � light-worlds



Concluding Remarks



Contracting executability lawsHW � R I �up �� toggle ��

Intuition: delete some arrows leaving up-worlds




Concluding Remarks



Contracting executability lawsHW � R I �up �� toggle �� H W � R ø R ea I§5 R ea Xð�ij w � w e k 5 wRaw e and w

b � up "h"N.B.: if there is no up-world, then contraction is notsuccessful!



Concluding Remarks



Contracting executability lawsHW � R I �up �� toggle �� H W � R ø R ea I§5 R ea Xð�ij w � w e k 5 wRaw e and w

b � up "h"N.B.: if there is no up-world, then contraction is notsuccessful!



Concluding Remarks


Contraction: Syntax

Domain descriptionsSimplification:

H � ��n� IResulting action theoryM ¡ � � � � N! " � M ¡ · � � · � � · N



Concluding Remarks


Contraction: Syntax

Contracting static lawsH � ��n� I �up � light � H � e �� e I , where¡ · � ¡ � PMAup � light �$# light � up %� · �#��P� � up & light � ¶Ïµ:�[� Mtoggle NPO('Kµ@� M

toggle NPO*) � %



Concluding Remarks


Contraction: Syntax

Contracting static lawsH � ��n� I �up � light � H � e �� e I , where¡ · � ¡ � PMAup � light �$# light � up %� · �#��P� � up & light � ¶Ïµ:�[� Mtoggle NPO('Kµ@� M

toggle NPO*) � %



Concluding Remarks


Contraction: Syntax

Contracting effect lawsH � ��n� I �� up �� toggle� light � H � � � e �n� I , where� · �+#�� up ¶�µ�� Dtoggle E-,.'Kµ´� D

toggle E-,/) � %



Concluding Remarks


Contraction: Syntax

Contracting executability lawsH � ��n� I �up �� toggle �0� � H � �� e I , where� · �+#��C� up ¶�µ�� Mtoggle N�O('Kµ@� M

toggle NPO*) � %



Concluding Remarks


Soundness

TheoremIfHW � R I b � � R � R � , then

HW � R I �1 b � H � ��n� I �1 .



Concluding Remarks


Incompleteness

Example� �ðï , � �� p UVA a BC� " , � �� H a I�T "Unique model:

HW � R I-� H �h�43 p "h"i� �ijm�43 p "i� �43 p "4kp" IH � ��n� I �p �� a �0� � H ï � � p UVA a B�� "i� �43 p U H

a I�T " I .Syntactically: successful, as � ��n� e ¬b � PDL p U H

a I�T .Semantically: contraction is unsuccessful!



Concluding Remarks


Incompleteness







Concluding Remarks


Incompleteness


HW � R I-� H �h�43 p "h"i� �ijm�43 p "i� �43 p "4kp" IH � ��n� I �p �� a �0� � H ï � � p UVA a B�� "i� � 3 p U H





Concluding Remarks


Incompleteness







Concluding Remarks


Incompleteness







Concluding Remarks


Completeness: Modularity

TheoremIfH � ��n� I and ¥ satisfy Postulate PS*, then

H � ��n� I �1 b �32 iffHW � R I �1 b �32 , for every

HW � R I such that

HW � R I b � � R � R � .



Concluding Remarks


Outlook: Semantics of Revision

Levi identityRevise by glued U{A toggleBC� amounts to

1 Contract by �� glued � Dtoggle EGFK�

2 Expand by glued � Dtoggle EGF

Problem: we can contract by domain laws only�� glued � Dtoggle EGFK�54 � glued ¶ M toggle NPOK�M ¡ � � � � N! glued 687 toggle 90: not defined

What is the negation ofan effect law?an executability law?



Concluding Remarks


Outlook: Semantics of Revision

Levi identityRevise by glued U{A toggleBC� amounts to

1 Contract by �� glued � Dtoggle EGFK�

2 Expand by glued � Dtoggle EGF

Problem: we can contract by domain laws only�� glued � Dtoggle EGFK�54 � glued ¶ M toggle NPOK�M ¡ � � � � N! glued 687 toggle 90: not defined

What is the negation ofan effect law?an executability law?



Concluding Remarks

Related Work

Modularity[Pirri & Reiter 1999]: deterministic actions withoutramifications in Situation Calculus[Amir 2000]: object-oriented concepts in Situation Calculus[Zhang et al. 2002]: EPDL approach/normal form[Lang et al. 2003]: computational complexity[Kakas et al. 2005]: elaboration tolerance, concurrentactions[Ghilardi, Lutz & Wolter, KR’06]: uniform interpolation andconservative extensions in ��;=<



Concluding Remarks

Related Work

Theory change[Li& Pereira 1996]: motivations[Liberatore 2000]: meta-results[Eiter et al. 2005/06]: update in action languages



Concluding Remarks

SummaryClaim

Consistency is not enough to evaluate a domaindescriptionThe dynamic part of an action theory should not influencethe non-dynamic one (otherwise: problems)

ContributionFine-grained postulates of modularityAlgorithms to check/give hints on modularitySatisfaction of modularity

More efficient reasoningImportant for updating theories [Herzig et al. ECAI’06]

Our results apply to every approach allowing for � , � , �and �



Concluding Remarks

SummaryClaim

Consistency is not enough to evaluate a domaindescriptionThe dynamic part of an action theory should not influencethe non-dynamic one (otherwise: problems)

ContributionFine-grained postulates of modularityAlgorithms to check/give hints on modularitySatisfaction of modularity

More efficient reasoningImportant for updating theories [Herzig et al. ECAI’06]

Our results apply to every approach allowing for � , � , �and �



Concluding Remarks

Summary

Contribution (cont.)Semantics of action theory contraction

Domain-independentDoes not require extra information (preferences/epistemicentrenchment relation/. . . )Fully automatic

Completeness result: highlights importance of modularity



Concluding Remarks

Summary

Modularity is also fruitful. . .for theories in general [Herzig & Varzinczak AiML’04]in the Situation Calculus [Herzig & Varzinczak IJCAI’05]in Description Logics [Herzig & Varzinczak JELIA’06]

(See next slide)

Future workFine tune contraction of effect lawsContract by any formulas (not just laws)Postulates about effect laws? about causation?



Concluding Remarks

Summary

Modularity is also fruitful. . .for theories in general [Herzig & Varzinczak AiML’04]in the Situation Calculus [Herzig & Varzinczak IJCAI’05]in Description Logics [Herzig & Varzinczak JELIA’06]

(See next slide)

Future workFine tune contraction of effect lawsContract by any formulas (not just laws)Postulates about effect laws? about causation?



Concluding Remarks

Outlook: Modularity in Description Logics

ExampleSuppose a passport control system in an airportSuch a system is composed of many software componentsOne of them an ontology (knowledge base) aboutpassengersAll passengers must be controlled



Concluding Remarks


Example (Ontology)A passenger has a passportEuropean citizens have European passportsForeigners have non-European passportsSomeone with double citizenship is a foreigner and aEuropean



Concluding Remarks


Example (The ontology in DL)Terminology:

Passenger >@? passport � OEUcitizen > � passport �EUForeigner > � passport � � EU2Citizen > Foreigner A EUcitizen

Assertions:EU � POLAND �EUcitizen � JAN �passport � JAN � POLAND �



Concluding Remarks

Outlook: Modularity in Description LogicsNevertheless

BCCD CCEPassenger FHG passport I Ð »EUcitizen Fgå passport IEU »

Foreigner F*å passport I â EU »2Citizen F Foreigner J EUcitizen

K CCLCCMN Æ 2Citizen Fgå passport I Â EU JÊâ EU ÅN Æ 2Citizen Fgå passport I äN Æ 2Citizen F´â Passenger

Someone with double citizenship is not a passenger

Hence. . .

! if we have 2Citizen j BINLADEN k ,this individual is not obliged to be controlled!



Concluding Remarks






Hence. . .




Concluding Remarks






Hence. . .




Concluding Remarks






Hence. . .




Concluding Remarks






Hence. . .




Concluding Remarks






Hence. . .




Concluding Remarks


Our results. . .can be applied in DL, too



Concluding Remarks

Thank you!Merci beaucoup !

Danke schön!Choukran!

¡Muchas gracias!Muito obrigado!


Can We Ask for More?

Postulate about effectsPE (No implicit effect laws):

if � ��n�ý�� b � ® ?²U{A aBd÷ and � ��n� �� ¬b � ® ?JUVA a BC� ,then � �� b � ® ?JUVA aBd÷



Example� �ðï �L� � l loaded U{A shoot BC3 alive �jP3 loaded R alive k U{A shootB alive o� �� hasGun U Hshoot I�T "i�[� ��43 hasGun U{A shoot BC� "

shoot ¥3 alive� ��n�J�� b � ® 3 hasGun O loaded UVA shoot B�3 alive� ��n�J�� ¬b � ® 3 hasGun O loaded UVA shoot B��but � �� ¬b � ® 3 hasGun O loaded UVA shoot BC3 alive











Postulate about effectsP � (No unattainable effects):

if � �� b � ® ?JUVA a BG÷ , then � ��n�J�� ¬b � ® ?²UVA a BC�




shoot ¥3 alive� b � ® jP3 hasGun R loaded k UVA shoot BC3 alivebut � ��n�ý�� b � ® jP3 hasGun R loaded k UVA shoot BC�




shoot ¥3 alive� b � ® jP3 hasGun R loaded k UVA shoot BC3 alivebut � ��n�ý�� b � ® jP3 hasGun R loaded k UVA shoot BC�


Technology

What Is a Good Domain Description? Evaluating & Revising Action Theories in Dynamic Logic