Intention Interleaving Via Classical Replanning Mengwei Xu , Kim - - PowerPoint PPT Presentation

Intention Interleaving Via Classical Replanning

Mengwei Xu, Kim Bauters, Kevin McAreavey, Weiru Liu

Extending Belief-Desire-Intention (BDI) Agents to Managing Intention Interleaving

Intention Merging: to facilitate positive interference Intention Resolution: to avoid negative interference

Guarantee the achievability of intentions when interleaving the steps in different intentions Perform one task once for at least two goals, i.e. “kill two birds with one stone”

Intention Resolution

Careless interleaving could result in that neither of its intention can be completed.

Motivation to Manage Intention Interleaving

Motivation to Manage Intention Interleaving

Intention Merging

TransmitSoilResults TransmitImageResults EstablishConnection SendSoilResults SendImageResults BreakConnection

execute them once for both intentions

𝐻! 𝑏! 𝑏" 𝑏# 𝐻" 𝑏$

EstablishConnection

𝑏!

BreakConnection

𝑏$

Belief-Desire-Intention: Literature

AgentSpeak [Rao, 1996] CAN [Winikoff et al., 2002] CANPLAN [Sardina et al., 2011] Jason [Bordini et al., 2007] Jack [Winikoff, 2005] Jadex [Pokahr et al., 2013]

Programming Languages Software Platforms Logics

[Cohen & Levesque, 1990] [Rao & Georgeff, 1991] [Shoham, 2009]

BDI Agent ℬ, Λ, Π

Initial belief base

Belief base specifying agent’s initial beliefs

Action library

Set of STRIPS-style action descriptions

Plan library

Set of plan rules

BDI Agent ℬ, Λ, Π

Belief base ℬ ⊆ ℒ

Set of formulas from logical language ℒ ℬ must support:

• ℬ ⊨ 𝜒

(Entailment)

• ℬ ∪ 𝜒

(Addition)

• ℬ ∖ 𝜒

(Deletion) Assume ℬ is a set of atoms

Initial belief base

Belief base specifying agent’s initial beliefs

CAN: Agent ℬ, Λ, Π

Action description act ∶ 𝜒 ← ℬ# ; ℬ$

Primitive action symbol Precondition 𝜒 ∈ ℒ Set of “delete” atoms ℬ! ⊆ ℒ Set of “add” atoms ℬ" ⊆ ℒ

Action library

Set of STRIPS-style action descriptions

BDI Agent ℬ, Λ, Π

Plan rule P = 𝐻: 𝜒 ← ℎ%; ⋯ ; ℎ&

𝑰𝒇𝒃𝒆 𝑸 : 𝐻

e.g. new goal

𝑫𝒑𝒐𝒖𝒇𝒚𝒖 𝑸 : 𝜒 ∈ ℒ

Formula from ℒ

Plan library

Set of plan rules

body 𝑸 : ℎ!; ⋯ ; ℎ%

e.g. a sequence of actions or goals

BDI Operational Mechanism Sketch

where 𝓒 ⊨ φ𝒌𝟐, 𝑘 ∈ 1, ⋯ , 𝑜 Relevant Plans Goal 𝐻 𝐻 ∶ 𝜒! ← 𝑄

!

𝐻 ∶ 𝜒" ← 𝑄

"

𝐻 ∶ 𝜒# ← 𝑄

#

⋮ 𝐻 ∶ 𝜒% ← 𝑄

%

Applicable Plans 𝐻 ∶ 𝜒!! ← 𝑄

!!

𝐻 ∶ 𝜒"! ← 𝑄

"!

𝐻 ∶ 𝜒#! ← 𝑄

#!

⋮ 𝐻 ∶ 𝜒%! ← 𝑄

%!

⊆ Π

select select

repeat for the subgoals A tree structure representing all possible ways of achieving a goal 𝐻

Our Intention Interleaving Framework in BDI

• 1. Intention Formalisation
• Model an intention as an AND/OR graph
• Define the execution trace for multiple intentions
• Define the conflict-free and maximal-merged execution trace for multiple intentions
• 2. Intention Interleaving Planning Preparation
• Indexing nodes
• Defined terminal, initial node sets, and progression links of intentions
• Computing overlapping programs between multiple intentions
• 3. Intention Interleaving Planning Formalism
• Formalise FPP problem of interleaving intentions
• Correctness Proof
• 4. Implementation
• 5. Evaluation

AND/OR Graphs for Intentions

𝑄

! = 𝐻!: 𝜒! ← 𝑏!; 𝑏"; 𝑏$

𝑄

" = 𝐻": 𝜒" ← 𝑏!; 𝑏#; 𝑏$

OR-nodes OR-edges AND-nodes AND-edges OR-nodes

Execution Trace for An Intention

𝜐! 𝑈" = 𝐻"; 𝑄"; 𝑏#; 𝑏$ 𝜐% 𝑈" = 𝐻"; 𝑄

#; 𝑐#; 𝑐$; 𝑐&

Execution trace for 𝑈#: Execution trace for 𝑈

!:

𝜐 𝑈

! = 𝐻!; 𝑄 !; 𝑏! ; 𝑏" ; 𝑏$

To identifies every unique way in which a given intention can be achieved

Execution Trace for Multiple Intentions

Potential execution trace for 𝑈

% and 𝑈H: 𝜏 = 𝑯𝟐; 𝑸𝟐; 𝐻H; 𝑄H; 𝒃𝟐; 𝑏J ; 𝒃𝟑 ; 𝒃𝟓 ; 𝑏M by interleaving 𝜐 𝑈

! = 𝑯𝟐; 𝑸𝟐; 𝒃𝟐 ; 𝒃𝟑 ; 𝒃𝟓 and 𝜐! 𝑈# = 𝐻#; 𝑄 #; 𝑏$; 𝑏*

The construction of an execution trace of a set of intentions is to interleave elements in the execution traces of different intentions

Execution Trace for Intentions (Cont.)

Conflict-free Execution Trace:

𝜏 1 𝜏 2 ⋯ 𝜏 𝑘 − 1 𝜏 𝑘 𝜏 𝑘 + 1 ⋯ 𝜏 𝑜 ℬ! ℬ" ℬ

+,!

ℬ

+

ℬ

+-!

ℬ% where ℬ

+ is the belief base before the execution of the 𝑘./ element of an execution trace (i.e. 𝜏 𝑘 )

An execution trace 𝜏 is conflict-free if and only if the following hold:

• 1. if 𝜏 𝑘 = 𝑄 ∈ Π, then ℬ' ⊨ 𝑑𝑝𝑜𝑢𝑓𝑦𝑢(𝑄), i.e. the context of plan 𝑄 must be met before selection
• 2. if 𝜏 𝑘 = 𝑏 ∈ Λ, then ℬ' ⊨ 𝜔(𝑏), i.e. the pre-condition of action `𝑏′ must be met before selection

To model the successful interleaving which achieves all intentions

Execution Trace for Intentions (Cont.)

Mergeable Execution Trace of 𝑼𝟐, ⋯ , 𝑼𝒏

𝜏 1 𝜏 2 ⋯ 𝜏 𝑘 ⋯ 𝜏 𝑜 ⋯ 𝜏 𝑘 + 1 𝜏 𝑘 + 𝑙 − 1 𝜏 𝑘 + 𝑙

An execution trace 𝜏 is a mergeable execution trace if and only if the following hold: 1. ∃𝑘 ∈ 1, ⋯ , 𝑜 such that 𝜏 𝑘 = 𝜏 𝑘 + 1 = ⋯ 𝜏 𝑘 + 𝑙 where 2 ≤ 𝑙 ≤ 𝑜 − 𝑘; 2. ∀𝑚 ∈ 1, ⋯ , 𝑛 , ∄𝑡, 𝑢 ∈ 𝑘, ⋯ , 𝑘 + 𝑙 where 𝑡 ≠ 𝑢 such that 𝜏 𝑡 ⊆ 𝜐 𝑈* ⊆ 𝜏 and 𝜏 𝑢 ⊆ 𝜐 𝑈* ⊆ 𝜏; 3. 𝜏+ is a conflict-free execution trace where 𝜏+ is the merged execution trace of 𝜏 by reducing each subsequence consisting of consecutive identical elements characterized by 1 and 2 in 𝜏 to only one element left.

𝑙 consecutive same element from all difference intentions in 𝜏

𝝉: 𝝉𝒏: 𝜏 1 𝜏 2 ⋯ 𝜏 𝑘 ⋯ 𝜏 𝑜 𝜏 𝑘 + 𝑙 + 1

To capture the overlapping programs of different intentions

Execution Trace for Intentions (Cont.)

Maximal-merged Trace of 𝑼𝟐, ⋯ , 𝑼𝒏 The merged execution trace 𝜏+ of a mergeable execution trace 𝜏 of 𝑈

!, ⋯ , 𝑈 + is maximal-merged

if there is no another mergeable execution trace 𝜏, of 𝑈

!, ⋯ , 𝑈 + such that 𝜏,+ < 𝜏+ where

𝜏 stands for the length of 𝜏.

To capture the most merged execution trace of multiple intentions

𝜏1 = 𝐻!; 𝑄

!; 𝐻"; 𝑄 "; 𝑏! ; 𝑏" ; 𝑏# ; 𝑏$

the potential maximal-merged trace of 𝑈

!, 𝑈"

Perform action a! and a$ once for both two goals 𝑈

! and 𝑈"

Indexing Nodes

A node 𝑜 is a top-level goal of intention 𝑈: 𝑈 W 𝑜 The nodes of actions and subgoals of intention 𝑈: 𝑜2,+,4to denote the 𝑘./ member of 𝑐𝑝𝑒𝑧(𝑄) in 𝑈 A plan node in intention 𝑈: 𝑜4 Terminal node set for a goal node: a collection of the last element of each execution trace of a goal Initial node set for intentions 𝑈

!, ⋯ , 𝑈 1 :

𝑨5 = 𝑈

! W

𝑜 , ⋯ , 𝑈

1 W

𝑜 Terminal node set for intentions I = 𝑈

!, ⋯ , 𝑈 1

𝑨6 = 𝑢𝑜!, ⋯ , 𝑢𝑜1 where 𝑢𝑜7 is a terminal node of 𝑈7 W 𝑜 𝑨6 ⊳.% 𝐽 if 𝑨6 is a terminal node set of 𝐽

To ensure that e.g. the same actions in distinct plans is seen as different

Progression Links

To visualise the progression order of execution elements in the context of indexes

The progression links of execution trace 𝜐(𝑈

!)

The progression links of execution trace 𝜐(𝑈") (𝑈

!(W

𝑜) → 𝑄

! 4

!)

(𝑄

! 4

! → 𝑏!2!,!,4 !)

(𝑏!2!,!,4

! → 𝑏"2!,",4 !)

(𝑏"2!,",4

! → 𝑏$2!,#,4 !)

(𝑈"(W 𝑜) → 𝑄

" 4

")

(𝑄

" 4

" → 𝑏!2",!,4 ")

(𝑏!2",!,4

" → 𝑏#2",",4 ")

(𝑏#2",",4

" → 𝑏$2",#,4 ")

They are also called primitive progression links

Overlap Set of Multiple Intentions

The progression links of execution trace 𝜐(𝑈

!)

The progression links of execution trace 𝜐(𝑈") (𝑈

!(W

𝑜) → 𝑄

! 4

!)

(𝑄

! 4

! → 𝒃𝟐𝑸𝟐,𝟐,𝑼𝟐)

(𝑏!2!,!,4

! → 𝑏"2!,",4 !)

(𝑏"2!,",4

! → 𝑏$2!,#,4 !)

(𝑈"(W 𝑜) → 𝑄

" 4

")

(𝑄

" 4

" → 𝒃𝟐𝑸𝟑,𝟐,𝑼𝟑)

(𝑏!2",!,4

" → 𝑏#2",",4 ")

(𝑏#2",",4

" → 𝑏$2",#,4 ")

The overlap set of 𝑈

!, ⋯ , 𝑈 + is a set of tuples of the form

𝑗𝑒𝑦-

! → 𝑗𝑒𝑦. ! , ⋯ , 𝑗𝑒𝑦- / → 𝑗𝑒𝑦. /

if: 1. J 𝑗𝑒𝑦.

! = ⋯ = J 𝑗𝑒𝑦. / where J 𝑗𝑒𝑦.

represents the actual node of the ending index 𝑗𝑒𝑦.

0;

2. ∀𝑚 ∈ 1, ⋯ , 𝑛 , ∄𝑡, 𝑢 ∈ 𝑘, ⋯ , 𝑘 + 𝑙 where 𝑡 ≠ 𝑢 s.t. 𝑗𝑒𝑦-

1 → 𝑗𝑒𝑦. 1 ∈ 𝜐 𝑈* and 𝑗𝑒𝑦- 2 → 𝑗𝑒𝑦. 2 ∈ 𝜐 𝑈* ;

The overlap set of intention 𝑈

!, 𝑈% has two elements as follows:

1. 𝑄

! 3

& → 𝒃𝟐𝑸𝟐,𝟐,𝑼𝟐 , (𝑄%

3

( → 𝒃𝟐𝑸𝟑,𝟐,𝑼𝟑)

where J 𝑏!7&,!,3

& = J 𝑏!7(,!,3 ( = 𝑏!;

2. 𝑏%7&,%,3

& → 𝒃𝟓𝑸𝟐,𝟒,𝑼𝟐 , (𝑏"7(,%,3 ( → 𝒃𝟓𝑸𝟑,𝟒,𝑼𝟑)

where J 𝑏#7&,",3

& = J 𝑏#7(,",3 ( = 𝑏#

To compute all potential overlapping programs among a set of intentions

Overlap Progression Links

Let an element of overlap set of 𝑈

!, ⋯ , 𝑈 + be

𝑗𝑒𝑦-

! → 𝑗𝑒𝑦. ! , ⋯ , 𝑗𝑒𝑦- / → 𝑗𝑒𝑦. /

. The overlap set of intention 𝑈

!, 𝑈% has two elements as follows:

1. 𝑄

! 3

& → 𝑏!7&,!,3 & , (𝑄%

3

( → 𝑏!7(,!,3 ()

2. 𝑏%7&,%,3

& → 𝑏#7&,",3 & , (𝑏"7(,%,3 ( → 𝑏#7(,",3 ()

𝛽: = 𝑗𝑒𝑦-

!, ⋯ , 𝑗𝑒𝑦- / → 𝑗𝑒𝑦. !, ⋯ , 𝑗𝑒𝑦. /

Then we have a corresponding overlap progression link

𝑄

! 4

!, 𝑄

" 4

" → 𝑏!2!,!,4 !, 𝑏!2",!,4 "

𝑏"2!,",4

!, 𝑏#2",",4 " → 𝑏$2!,#,4 !, 𝑏$2",#,4 "

such that the side of 𝛽: is 𝑡𝑗𝑨𝑓 𝛽: = k − 1, i.e. merging 𝑙 − 1 extra primitive progression links. by fault, the size of a primitive progression link 𝛽; is 𝑡𝑗𝑨𝑓 𝛽: = 0, i.e. no merging at all.

Intention Interleaving Planning Formalism

A First-principles Planning (FPP) problem of interleaving intentions 𝐽 = 𝑈

!, ⋯ , 𝑈 1 is a tuple

Ω = Σ, 𝑌, 𝑃, 𝑡!, 𝑇"

a finite set of (propositional) atoms 𝑌 = ⋃#$%

& 𝑈 #(𝑂∨ ∪ 𝑂∧) is the set of node indexes of 𝐽

𝑃 = 𝑃) ∪ 𝑃* is a set of progression links where 𝑃) (resp. 𝑃*) is the collection of primitive (resp. overlap) progression links 𝑡+ = ℬ+ ∪ 𝑨+ is the initial state where ℬ+ is the initial belief base and 𝑨+ is the initial node set of 𝐽 𝑇, = 𝑨-|𝑨- ⊳./ 𝐽 is the goal state where 𝑨- is the terminal node set of 𝐽

A FPP problem of interleaving intentions 𝐽 = 𝑈

!, ⋯ , 𝑈 1 is a tuple

Ω = Σ, 𝑌, 𝑃, 𝑡!, 𝑇"

𝑃 = 𝑃X ∪ 𝑃Y

Intention Interleaving Planning Formalism (Cont.)

𝛽: = 𝑗𝑒𝑦;

!, ⋯ , 𝑗𝑒𝑦; < → 𝑗𝑒𝑦= !, ⋯ , 𝑗𝑒𝑦= <

∈ 𝑃: in which 𝛽7

> = 𝑗𝑒𝑦; 7 → 𝑗𝑒𝑦= 7

∈ 𝑃>

• 𝑞𝑠𝑓 𝛽: = 𝑞𝑠𝑓(𝛽!

>) ∪ ⋯ ∪ 𝑞𝑠𝑓(𝛽< >)

• 𝑒𝑓𝑚 𝛽: = 𝑒𝑓𝑚(𝛽!

>) ∪ ⋯ ∪ 𝑒𝑓𝑚(𝛽< >)

• 𝑏𝑒𝑒 𝛽: = 𝑏𝑒𝑒(𝛽!

>) ∪ ⋯ ∪ 𝑏𝑒𝑒(𝛽< >)

A FPP problem of interleaving intentions 𝐽 = 𝑈

!, ⋯ , 𝑈 1 is a tuple Ω = Σ, 𝑌, 𝑃, 𝑡5, 𝑇?

Intention Interleaving Planning Formalism (Cont.)

Definition 1: The result of applying a progression link 𝛽 ∈ 𝑃 to a state 𝑡 = ℬ ∪ 𝑨 is described by the transition function 𝑔: 2@ ∪ 2A×𝑃 → 2@ ∪ 2A defined as follows: 𝑔 𝑡, 𝛽 = t(𝑡\del 𝛽 ) ∪ 𝑏𝑒𝑒(𝛽) 𝑗𝑔 𝑡 ⊨ 𝑞𝑠𝑓(𝛽) 𝑣𝑜𝑒𝑓𝑔𝑗𝑜𝑓𝑒 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 Definition 2: The result of applying a sequence of progression links to a state specification 𝑡 is defined inductively: Res(s, = 𝑡 Res(s, 𝛽5; ⋯ ; 𝛽% = 𝑆𝑓𝑡(𝑔 𝑡, 𝛽5 , 𝛽!; ⋯ ; 𝛽% ) Definition 3: A sequence of progression links ∆= 𝛽5; 𝛽!; ⋯ ; 𝛽% is a solution to a FPP problem Ω = Σ, 𝑌, 𝑃, 𝑡5, 𝑇? , denoted as ∆= 𝑡𝑝𝑚 Ω , iff Res s, ∆ ⊨ 𝑇?. We also say that ∆ is optimal if the sum of the size of the progression link 𝑡𝑗𝑨𝑓(𝛽7) is maximum where 𝑗 = 0, ⋯ , 𝑜. Theorem: we have a maximal-merged trace 𝜏1 of intention 𝐽 = 𝑈

!, ⋯ , 𝑈 1 if and only if there

exists an optimal solution Δ to Ω.

A FPP problem of interleaving intentions 𝐽 = 𝑈

!, ⋯ , 𝑈 1 is a tuple Ω = Σ, 𝑌, 𝑃, 𝑡5, 𝑇?

Intention Interleaving Planning Formalism (Cont.)

line 5-7 instruct the procedures for failure backtracking and initial node state modification

To adapt to the dynamic environment

Implementation

Operator Files

(containing progression links)

primitive progression links:

• verlap progression links:

Fact Files

(containing initial/goal state description) declare all objects in the plan problem instance initial belief base and the top-level goals of intentions reach any terminal node

• f each intention

Planning Domain Definition Language (PDDL)

Evaluation: A Manufacturing Scenario

block 1 block 2 block 3 block 4

twisting-drilling 10cm reaming boring

Operation 1 Operation 2 Operation 3

twisting-drilling 15cm reaming boring twisting-drilling 20cm reaming boring twisting-drilling 25cm reaming boring Details can be found in my github

https://github.com/Mengwei-Xu/manufacturing-evaluation

Summary:

• 1. Formalise an intention as AND/OR graph
• 2. Formalise the conflict-free execution trace of multiple intentions
• 3. Formalise the maximal-merged execution trace of multiple intentions
• 4. Define the concept of overlapping programs between different intentions
• 5. Both formally and practically compile the intention interleaving problem into a planning problem
• 6. Provide a preliminary evaluation of a planning-centric intention interleaving problem

Future Work:

• 1. A complete algorithm of computing overlap set of intentions
• 2. Further test the costs and benefits of our approach empirically in a wider range of applications
• 3. Investigate the collaboration between multi-BDI agents, e.g. how to discover and exploit collaboration opportunities