dissertation: advances in answer set planning · dissertation: advances in answer set planning...
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![Page 1: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/1.jpg)
DISSERTATION:
Advancesin Answer SetPlanningDipl.-Ing. Axel Polleres
supervised by
O.Univ.Prof. Dipl.-Ing. Dr.techn. Thomas Eiter
Institut fur Informationssysteme, Abteilung fur Wissensbasierte Systeme
– p.1/21
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Overview – Contrib utions
Preliminaries
Novel Declarative Planning LanguageSyntax,Semantics
Complexity of Planning in
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in to ASP
Ready-to-Use Planning System DLV
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
![Page 3: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/3.jpg)
Overview – Contrib utions
Preliminaries
Novel Declarative Planning Language
��Syntax,Semantics
Complexity of Planning in
��
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in to ASP
Ready-to-Use Planning System DLV
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
![Page 4: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/4.jpg)
Overview – Contrib utions
Preliminaries
Novel Declarative Planning Language
��Syntax,Semantics
Complexity of Planning in
��
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in to ASP
Ready-to-Use Planning System DLV
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
![Page 5: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/5.jpg)
Overview – Contrib utions
Preliminaries
Novel Declarative Planning Language
��Syntax,Semantics
Complexity of Planning in
��
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in��
to ASP
Ready-to-Use Planning System DLV
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
![Page 6: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/6.jpg)
Overview – Contrib utions
Preliminaries
Novel Declarative Planning Language
��Syntax,Semantics
Complexity of Planning in
��
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in��
to ASP
Ready-to-Use Planning System DLV
�
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
![Page 7: DISSERTATION: Advances in Answer Set Planning · DISSERTATION: Advances in Answer Set Planning Dipl.-Ing.Axel Polleres axel@kr.tuwien.ac.at supervised by O.Univ.Prof. Dipl.-Ing.Dr.techn](https://reader034.vdocuments.net/reader034/viewer/2022042411/5f2a3e4b4baa8c40f13b270c/html5/thumbnails/7.jpg)
Overview – Contrib utions
Preliminaries
Novel Declarative Planning Language
��Syntax,Semantics
Complexity of Planning in
��
Problem Solving in Answer Set Programming (ASP) by“Guess and Check”
Translation of Planning in��
to ASP
Ready-to-Use Planning System DLV
�
Implementation
Experiments
Application: Planning for MAS-Monitoring
– p.2/21
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Preliminaries:
Planning Problem: Find a sequence of actions to bring an agent froman initial state to a goal state
Input: A set of actions (preconditions, effects);
Fluents (state variables) and their
initial values and goal values
Output: Sequence of action sets (discrete notion of time)
Classical Planning (complete knowledge, deterministic actions)
Actions: move(B,L)
Fluents: on(b,table), on(c,a), on(a,table),
occupied(a), ...
initial:
bca
cba
goal:
Non-Classical Planning(Conformant Plans, Conditional Plans, . . . )
b
d
d cainitial: goal:
ca
?
db
– p.3/21
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Preliminaries:
Planning Problem: Find a sequence of actions to bring an agent froman initial state to a goal state
Input: A set of actions (preconditions, effects);
Fluents (state variables) and their
initial values and goal values
Output: Sequence of action sets (discrete notion of time)
Classical Planning (complete knowledge, deterministic actions)
Actions: move(B,L)
Fluents: on(b,table), on(c,a), on(a,table),
occupied(a), ...
initial:
bca
cba
goal:
Non-Classical Planning(Conformant Plans, Conditional Plans, . . . )
b
d
d cainitial: goal:
ca
?
db
– p.3/21
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Preliminaries:
Planning Problem: Find a sequence of actions to bring an agent froman initial state to a goal state
Input: A set of actions (preconditions, effects);
Fluents (state variables) and their
initial values and goal values
Output: Sequence of action sets (discrete notion of time)
Classical Planning (complete knowledge, deterministic actions)
Actions: move(B,L)
Fluents: on(b,table), on(c,a), on(a,table),
occupied(a), ...
initial:
bca
cba
goal:
Non-Classical Planning(Conformant Plans, Conditional Plans, . . . )
b
d
d cainitial: goal:
ca
?
db
– p.3/21
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Planning and Action Languages
Existing formal Languages: STRIPS, ADL, PDDL, ,�
, . . .
Here: Novel planning language
��
:
��
– Features:
A relative of action languages
�
(Gelfond & Lifschitz, 1993) and
�
(Giunchiglia & Lifschitz, 1998)
Incomplete states (“knowledge states”)
Default (nonmonotonic) negation and strong (classical) negation
Nondeterministic action effects
Action costs
– p.4/21
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Planning and Action Languages
Existing formal Languages: STRIPS, ADL, PDDL, ,�
, . . .
Here: Novel planning language
��
:
��
– Features:
A relative of action languages
�
(Gelfond & Lifschitz, 1993) and
�
(Giunchiglia & Lifschitz, 1998)
Incomplete states (“knowledge states”)
Default (nonmonotonic) negation and strong (classical) negation
Nondeterministic action effects
Action costs
– p.4/21
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Planning and Action Languages
Existing formal Languages: STRIPS, ADL, PDDL, ,�
, . . .
Here: Novel planning language
��
:
��
– Features:
A relative of action languages
�
(Gelfond & Lifschitz, 1993) and
�
(Giunchiglia & Lifschitz, 1998)
Incomplete states (“knowledge states”)
Default (nonmonotonic) negation and strong (classical) negation
Nondeterministic action effects
Action costs
– p.4/21
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Planning and Action Languages
Existing formal Languages: STRIPS, ADL, PDDL, ,�
, . . .
Here: Novel planning language
��
:
��
– Features:
A relative of action languages
�
(Gelfond & Lifschitz, 1993) and
�
(Giunchiglia & Lifschitz, 1998)
Incomplete states (“knowledge states”)
Default (nonmonotonic) negation and strong (classical) negation
Nondeterministic action effects
Action costs
– p.4/21
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Planning and Action Languages
Existing formal Languages: STRIPS, ADL, PDDL, ,�
, . . .
Here: Novel planning language
��
:
��
– Features:
A relative of action languages
�
(Gelfond & Lifschitz, 1993) and
�
(Giunchiglia & Lifschitz, 1998)
Incomplete states (“knowledge states”)
Default (nonmonotonic) negation and strong (classical) negation
Nondeterministic action effects
Action costs
– p.4/21
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�
Planning Domainsand Problems
Background Knowledge
: A logic program
with a single model,
(answer set) defining type information and static knowledge.
Action Description
��
:
fluents:
� % fluent declarations
actions:
� � % action type declarations
always:
��� % causation rules + exec. cond’s
initially:
��� % initial state constraints
Planning Domain:
� �� �� �
Planning Problem: additional goal
goal:
�� �� �
ground literal(s)
�
; plan length
� � �
.
– p.5/21
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Blocks world in
�
initial:
bca
cba
goal:
Background knowledge
�
=
block(a). block(b). block(c).
location(table).
location(L) :- block(L).
!
(Logic Program which has a single model - set of “invariant” facts)
– p.6/21
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Blocksworld:
�
Problemdescriptionfluents: on(B,L) requires block(B), location(L).
occupied(B) requires location(B).
actions: move(B,L) requires block(B), location(L) costs 1.
always: executable move(B,L) if not occupied(B), not occupied(L), B<>L.
caused on(B,L) after move(B,L).
caused on(B,L1) after move(B,L), on(B,L1), L<>L1.
caused occupied(B) if on(B1,B), block(B).
inertial on(B,L). % Explicit frame axioms!
noConcurrency. % Optionally, parallel actions!
initially: on(a,table). on(b,table). on(c,a).
goal: on(c,b),on(b,a),on(a,table)? (3)
Intuitively: Feasible plan is
initial:
bca
cba
goal:
move(c,table); move(b,a); move(c,b) COSTS: 3
– p.7/21
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Blocksworld:
�
Problemdescriptionfluents: on(B,L) requires block(B), location(L).
occupied(B) requires location(B).
actions: move(B,L) requires block(B), location(L) costs 1.
always: executable move(B,L) if not occupied(B), not occupied(L), B<>L.
caused on(B,L) after move(B,L).
caused "on(B,L1) after move(B,L), on(B,L1), L<>L1.
caused occupied(B) if on(B1,B), block(B).
inertial on(B,L). % Explicit frame axioms!
noConcurrency. % Optionally, parallel actions!
initially: on(a,table). on(b,table). on(c,a).
goal: on(c,b),on(b,a),on(a,table)? (3)
Intuitively: Feasible plan is
initial:
bca
cba
goal:
move(c,table); move(b,a); move(c,b) COSTS: 3
– p.7/21
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Blocksworld:
�
Problemdescriptionfluents: on(B,L) requires block(B), location(L).
occupied(B) requires location(B).
actions: move(B,L) requires block(B), location(L) costs 1.
always: executable move(B,L) if not occupied(B), not occupied(L), B<>L.
caused on(B,L) after move(B,L).
caused "on(B,L1) after move(B,L), on(B,L1), L<>L1.
caused occupied(B) if on(B1,B), block(B).
inertial on(B,L). % Explicit frame axioms!
noConcurrency. % Optionally, parallel actions!
initially: on(a,table). on(b,table). on(c,a).
goal: on(c,b),on(b,a),on(a,table)? (3)
Intuitively: Feasible plan is
initial:
bca
cba
goal:
move(c,table); move(b,a); move(c,b) COSTS: 3
– p.7/21
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Blocksworld:
�
Problemdescriptionfluents: on(B,L) requires block(B), location(L).
occupied(B) requires location(B).
actions: move(B,L) requires block(B), location(L) costs 1.
always: executable move(B,L) if not occupied(B), not occupied(L), B<>L.
caused on(B,L) after move(B,L).
caused "on(B,L1) after move(B,L), on(B,L1), L<>L1.
caused occupied(B) if on(B1,B), block(B).
inertial on(B,L). % Explicit frame axioms!
noConcurrency. % Optionally, parallel actions!
initially: on(a,table). on(b,table). on(c,a).
goal: on(c,b),on(b,a),on(a,table)? (3)
Intuitively: Feasible plan is
initial:
bca
cba
goal:
move(c,table); move(b,a); move(c,b) COSTS: 3
– p.7/21
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Blocks World in
�
(cont’d)
Incomplete Knowldege: b
d
d cainitial: goal:
ca
?
db
initially: total on(d,L).
caused "on(d,c).% State Axioms:
caused false if on(B,L), on(B,L1), L<>L1.
caused false if on(B1,B), on(B2,B), block(B), B1<>B2....
goal: on(a,c),on(c,d),on(d,b),on(b,table)? (4)
Feasible plans:move(c,d); move(a,c); (no action); (no action) COSTS: 2
move(d,c); move(d,b); move(c,d); move(a,c) COSTS: 4
– p.8/21
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Blocks World in
�
(cont’d)
Incomplete Knowldege: b
d
d cainitial: goal:
ca
?
db
initially: total on(d,L).
caused "on(d,c).% State Axioms:
caused false if on(B,L), on(B,L1), L<>L1.
caused false if on(B1,B), on(B2,B), block(B), B1<>B2....
goal: on(a,c),on(c,d),on(d,b),on(b,table)? (4)
Feasible plans:move(c,d); move(a,c); (no action); (no action) COSTS: 2
move(d,c); move(d,b); move(c,d); move(a,c) COSTS: 4
– p.8/21
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Semanticsof
�
– PlansMulti-valued transition function
# $&%(' ) *
, LP-based (Answer Sets!)
Optimistic plans: Secure plans:A1 A nA2
S1
... ... ...
S0 Sn......
...
...
Goal
A1 A nA2
S1
... ... ...
S0 Sn
Goal
Goal
Goal
...
...
...
...
...
Optimal plans: plans with lowest cost
Admissible plans: plans which stay within fixed cost limit
– p.9/21
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�
Complexity
plan length
+
in query ,- .0/1 2 3 4 + 5
67
fixed (=constant) arbitrary
general
89
/
�:<; /
= :<> -complete
9 ? 9 @A B
/
�:<; / NEXPTIME -complete
proper
89
/ co-
89
/
= :C; -complete
9 ? 9 @A B
/ co-
89/ NEXPTIME -complete
Complexity Results for Optimistic Planning / Security Checking / Secure Planning in
D
(Propositional Case)
Planning with Action costs:Computing optimal optimistic/secure plans is
E F :C; -complete/
E F :HG -complete.
Answer Set Programming (with weak constraints) can be used to solve some of these tasks!
– p.10/21
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Answer SetProgramming (ASP)
ASP with weak constraints is capable of solving problems beyond NP!(conformant planning for proper domains, general secure checking)
2
P
2
P
SAT, normal LP, HCF−LP UnSAT, ...
co−NPNP
= NPNP
NP=
P
3=P
co−NP
Solvers for
IJ
problems: Smodels, SAT-Solvers, . . .
Solvers for
K LNM problems: DLV, GnT, . . .
Solvers forO LNP problems: DLV with weak constraints . . .
– p.11/21
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Answer SetProgramming (ASP)
function-free, disjunctive Logic Programs, set of rules:
QSR TVU U U T QXW :- YR' U U U ' YCZ ' [ \ ] YZ ^ R' U U U [ \ ] YC_U
Semantics: Answer Sets Semantics for nonmonotonic logic programs(Gelfond & Lifschitz, 1991), minimal “stable” models
Extension: weak constraints (Buccafurri et.al., 1999):
`a YR' U U U ' YHZ ' [ \ ] YZ ^ R' U U U [ \ ] YH_U bc d
Semantics: Optimal Answer Sets (with minimal violation costs)
– p.12/21
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ProblemSolving in ASP:
“Guess and Check” Paradigm: a Simple Example:
ef g hjilk m n o ef g h ik p n o ef g h ik q n
:- rf sut h i nv wGuess
:- t s p t h ik x nk ef g h ik y nk e f g h xk y nv wSolution Check
Input: A graph represented by [ \ zj{ $_
*and { zj| { $
_' _
*
.
Problem: Assign a color to all nodes such that
adjacent nodes always have different colors.
NP-complete problem!
ASP is well suited for solving search problems with a finite
search space! Efficient solvers (DLV, smodels, . . . ) exist!– p.13/21
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Beyond NP: 2QBFs
} ~ ��� RU U U ��� Z ��� RU U U ��� _ �
� ~ �R ��� � � � �X���� ~ � ��� R �� � � � � �� W�� and
� � ��� � �u� � � R' U U U ' � Z ' � R' U U U ' � _ �Compute an assignment to � R' U U U ' � Z such that
}is true
... Guess
...
sat :- ....
sat :- .
:- sat. ... :- sat.
:- sat. ... :- sat.
:- sat.
Check
Check part uses “saturation” technique!
– p.14/21
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Beyond NP: 2QBFs
} ~ ��� RU U U ��� Z ��� RU U U ��� _ �
� ~ �R ��� � � � �X���� ~ � ��� R �� � � � � �� W�� and
� � ��� � �u� � � R' U U U ' � Z ' � R' U U U ' � _ �Compute an assignment to � R' U U U ' � Z such that
}is true
�� � � � ��� ... ��� � � ��� � �Guess�� � � �� � ... �¢¡ � � �¢¡ �
sat :- 1 �V£ � ¤ � � � ¤ 1 �£ ¥u¦ ....
sat :- 1 §£ � ¤ � � � ¤ 1 §£ ¥©¨ .�� :- sat. ... ��¡ :- sat.� �� :- sat. ... � �¢¡ :- sat.
:- ª « ¬ sat.
¯®°®±®±®²®±®±®²®±®±®°®°³®±®°®°®±®±®²®±®±®²®±®±´
Check
Check part uses “saturation” technique!
– p.14/21
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Integrate “Guess” and “Check”
Problems:
Integrated
µ 6¶ programs often hard to find,
“Guess” and “Check” structure hard to see.
µ 6¶ encodings use unintuitive “saturation” techniques
Solution: Interleaved Computation!
Separate programs and
Compute solutions interleaved, i.e. compute allsolutions to and only accept those where
has no answer set.
Thesis provides automatic method for combiningthese programs!
– p.15/21
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Integrate “Guess” and “Check”
Problems:
Integrated
µ 6¶ programs often hard to find,
“Guess” and “Check” structure hard to see.
µ 6¶ encodings use unintuitive “saturation” techniques
Solution: Interleaved Computation!
Separate programs
·¹¸º »¼ ¼ and
· � ½ »� ¾
Compute solutions interleaved, i.e. compute allsolutions
¿
to
·À¸º »¼ ¼ and only accept those where·Á� ½ »� ¾ h ¿ n
has no answer set.
Thesis provides automatic method for combiningthese programs!
– p.15/21
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Integrate “Guess” and “Check”
Problems:
Integrated
µ 6¶ programs often hard to find,
“Guess” and “Check” structure hard to see.
µ 6¶ encodings use unintuitive “saturation” techniques
Solution: Interleaved Computation!
Separate programs
·¹¸º »¼ ¼ and
· � ½ »� ¾
Compute solutions interleaved, i.e. compute allsolutions
¿
to
·À¸º »¼ ¼ and only accept those where·Á� ½ »� ¾ h ¿ n
has no answer set.
Thesis provides automatic method for combiningthese programs!
– p.15/21
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ASPTranslation for Planning Problems
Based on this method, we define ASP Translations for:
optimistic planning
general/proper secure checking
proper secure planning
Action costs (optimal/admissible planning):
Extend these translations by weak constraints feature
– p.16/21
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ASPTranslation for Planning Problems
Based on this method, we define ASP Translations for:
optimistic planning
general/proper secure checking
proper secure planning
Action costs (optimal/admissible planning):
Extend these translations by weak constraints feature
– p.16/21
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Implementation – DLV
Control FlowData Flow
Plan Printer
Controller
Datalog Parser Plan Generator
�
Parser
DLV Core
DLV
Â
Core
Plan CheckerKnowledgeBackground
�
input
Plan Generator: Computes Optimistic Plans
Plan Checker: Checks Optimistic Plans for Security
– p.17/21
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Planning for Multi-Agent Monitoring
Event
Agent
r/wr/w
r/wr/w
r
Management
DBEvent
DBManager
StatisticDB
PackageDB
PackageAgent
NotificationAgent
EventDispatcher
Agent
AgentZip
Idea:
Model collaborative Behavior in an MAS in
��
, and
derive valid messaging protocols from plans.
Further interesting applications:
Optimal Route planning with exceptional, time-dependent costs
Cheapest among the shortest plans, Shortest among the cheapest plans
Conformant planning examples from the literature (SQUARE, Bomb in Toilet)
– p.18/21
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Planning for Multi-Agent Monitoring
Event
Agent
r/wr/w
r/wr/w
r
Management
DBEvent
DBManager
StatisticDB
PackageDB
PackageAgent
NotificationAgent
EventDispatcher
Agent
AgentZip
Idea:
Model collaborative Behavior in an MAS in
��
, and
derive valid messaging protocols from plans.
Further interesting applications:
Optimal Route planning with exceptional, time-dependent costs
Cheapest among the shortest plans, Shortest among the cheapest plans
Conformant planning examples from the literature (SQUARE, Bomb in Toilet)
– p.18/21
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Conclusions& Outlook
Expressive planning language based on LogicProgramming (Knowledge Representation!)
Efficient planning system based on Answer SetTechniques is feasible: DLV
Encouraging Results (Experiments!)
Exploiting the full potential of ASP power( encodings), problems beyond SAT Solvers!
Further Work: Reactive Planning!
– p.19/21
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Conclusions& Outlook
Expressive planning language based on LogicProgramming (Knowledge Representation!)
Efficient planning system based on Answer SetTechniques is feasible: DLV
�
Encouraging Results (Experiments!)
Exploiting the full potential of ASP power( encodings), problems beyond SAT Solvers!
Further Work: Reactive Planning!
– p.19/21
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Conclusions& Outlook
Expressive planning language based on LogicProgramming (Knowledge Representation!)
Efficient planning system based on Answer SetTechniques is feasible: DLV
�Encouraging Results (Experiments!)
Exploiting the full potential of ASP power( encodings), problems beyond SAT Solvers!
Further Work: Reactive Planning!
– p.19/21
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Conclusions& Outlook
Expressive planning language based on LogicProgramming (Knowledge Representation!)
Efficient planning system based on Answer SetTechniques is feasible: DLV
�Encouraging Results (Experiments!)
Exploiting the full potential of ASP power(
µ ÃÄ encodings), problems beyond SAT Solvers!
Further Work: Reactive Planning!
– p.19/21
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Conclusions& Outlook
Expressive planning language based on LogicProgramming (Knowledge Representation!)
Efficient planning system based on Answer SetTechniques is feasible: DLV
�Encouraging Results (Experiments!)
Exploiting the full potential of ASP power(
µ ÃÄ encodings), problems beyond SAT Solvers!
Further Work: Reactive Planning!
– p.19/21
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SelectedPublications
T. Eiter, W. Faber, N. Leone, G. Pfeifer, and A. Polleres.A Logic Programming Approach to Knowledge-State Planning: Semantics andComplexity. ACM Transactions on Computational Logic, 2003. To appear.
T. Eiter, W. Faber, N. Leone, G. Pfeifer, and A. Polleres.A Logic Programming Approach to Knowledge-State Planning, II: the DLV
Å
System.Artificial Intelligence, 144(1–2):157–211, March 2003.
T. Eiter, W. Faber, N. Leone, G. Pfeifer, and A. Polleres.Answer Set Planning under Action Costs. Journal of Artificial Intelligence Research,19:25–71, 2003.
J. Dix, T. Eiter, M. Fink, A. Polleres, and Y. Zhang. Monitoring agents using planning.German Conference on Artificial Intelligence (KI2003), 2003.
T. Eiter and A. Polleres. Towards Automated Integration of Guess and CheckPrograms in Answer Set Programming. Accepted for 7th International Conference onLogic Programming and Nonmonotonic Reasoning (LPNMR-7).
– p.20/21
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Semanticsof
�
– TransitionsTransition-based semantics: “legal” transitions
Æ&%ÈÇ ' )' % R Écausedfl if Cond1after Cond2
A 1s s0
A . . . set of actions (executable)
Cond2 is evaluated in %ÈÇ
fl and Cond1 are evaluated in % RDefine new state % R by a non-monotonic logic program of rules
fl :- Cond1
Remark:e.g., transitive closure easily expressed (LP-flavored semantics of
�
)
– p.21/21