Planning and Optimization D6. Merge-and-Shrink Abstractions: Synchronized Product Gabriele R¨oger and Thomas Keller

(1)

Planning and Optimization

D6. Merge-and-Shrink Abstractions: Synchronized Product

Gabriele R¨ oger and Thomas Keller

Universit¨ at Basel

November 5, 2018

G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 5, 2018 1 / 46

Planning and Optimization

November 5, 2018 — D6. Merge-and-Shrink Abstractions: Synchronized Product

D6.1 Motivation

D6.2 Synchronized Product

D6.3 Synchronized Products and Abstractions D6.4 Summary

Content of this Course

Planning

Classical

Tasks Progression/

Regression Complexity Heuristics

Probabilistic

MDPs Uninformed Search

Heuristic Search Monte-Carlo

Methods

Content of this Course: Heuristics

Heuristics

Delete Relaxation

Abstraction

Abstractions in General

Pattern Databases

Merge &

Shrink Landmarks

Potential

Heuristics

Cost Partitioning

(2)

D6. Merge-and-Shrink Abstractions: Synchronized Product Motivation

D6.1 Motivation

Beyond Pattern Databases

I Despite their popularity, pattern databases have some fundamental limitations ( example on next slides).

I For the rest of this week, we study a class of abstractions called merge-and-shrink abstractions.

I Merge-and-shrink abstractions can be seen as a proper generalization of pattern databases.

I

They can do everything that pattern databases can do (modulo polynomial extra effort).

I

They can do some things that pattern databases cannot.

Back to the Running Example

LRR LLL

LLR

LRL

ALR

ALL

BLL

BRL

ARL

ARR

BRR

BLR

RRR RRL

RLR

RLL

Logistics problem with one package, two trucks, two locations:

I state variable package: {L, R, A, B}

I state variable truck A: {L, R }

I state variable truck B: {L, R }

Example: Projection

T ^π

^{package}

:

LRR LLL

LLR

LRL LRR

LLR

LRL LLL

ALR ARL

ALL ARR

ALR ARL

ARR ALL

BLL

BRL

BRR

BLR BLL BRR

BLR BRL

RRR RRL

RLR

RLL RLL RRL

RLR

RRR

(3)

Example: Projection (2)

T ^π

{package,truck A}

:

LRR

LRL LRR

LRL LLL LLR LLR

LLL

ALR

ALL ALR

ALL

ARL

ARR ARL

ARR

BRR

BLL BLR

BRL

BLL BLR

BRL BRR

RRR RRL RRL

RRR

RLR

RLL RLL

RLR

Limitations of Projections

How accurate is the PDB heuristic?

I consider generalization of the example:

N trucks, M locations (fully connected), still one package

I consider any pattern that is a proper subset of variable set V .

I h(s ₀ ) ≤ 2 no better than atomic projection to package These values cannot be improved by maximizing over several patterns or using additive patterns.

Merge-and-shrink abstractions can represent heuristics with h(s ₀ ) ≥ 3 for tasks of this kind of any size.

Time and space requirements are polynomial in N and M .

Merge-and-Shrink Abstractions: Main Idea

Main Idea of Merge-and-shrink Abstractions (due to Dr¨ ager, Finkbeiner & Podelski, 2006):

Instead of perfectly reflecting a few state variables, reflect all state variables, but in a potentially lossy way.

Merge-and-Shrink Abstractions: Idea

Start from projections to single state variables

(4)

Merge-and-Shrink Abstractions: Idea

Successively replace two transition systems with their product.

T

M

B

L R

TL TR

ML MR

BL BR

Merge-and-Shrink Abstractions: Idea

If too large, replace a transition system with an abstract system.

Merge-and-Shrink Abstractions: Idea

I Given two abstract transition systems, we can merge them into a new abstract product transition system.

I The product transition system captures all information of both transition systems and can be better informed than either.

I It can even be better informed than their sum.

I If merging with another abstract transition system exceeded memory limitations, we can shrink an intermediate result using any abstraction and then continue the merging process.

D6. Merge-and-Shrink Abstractions: Synchronized Product Synchronized Product

D6.2 Synchronized Product

(5)

Content of this Course: Merge & Shrink

Merge & Shrink

Synchronized Product Merge & Shrink Algorithm

Heuristic Properties Strategies Label Reduction

Running Example: Explanations

I Atomic projections – projections to a single state variable – play an important role for merge-and-shrink abstractions.

I Unlike previous chapters, transition labels are critically important for this topic.

I Hence we now look at the transition systems for atomic projections of our example task, including transition labels.

I We abbreviate operator names as in these examples:

I

MALR: move truck A from left to right

I

DAR: drop package from truck A at right location

I

PBL: pick up package with truck B at left location

I We abbreviate parallel arcs with commas and wildcards (?) in the labels as in these examples:

I

PAL, DAL: two parallel arcs labeled PAL and DAL

I

MA??: two parallel arcs labeled MALR and MARL

Running Example: Atomic Projection for Package

T ^π

^{package}

:

L

A

B

R

M???

PAL DAL

M???

DAR PA R

M???

PBR DBR

M???

DBL PBL

Running Example: Atomic Projection for Truck A

T ^π

^{{truck A}}

:

L R

PAL,DAL,MB??, PB?,DB?

MALR

MARL

PAR,DAR,MB??,

PB?,DB?

(6)

Running Example: Atomic Projection for Truck B

T ^π

^{{truck B}}

:

L R

PBL,DBL,MA??, PA?,DA?

MBLR

MBRL

PBR,DBR,MA??, PA?,DA?

Synchronized Product of Transition Systems

Definition (Synchronized Product of Transition Systems)

For i ∈ {1, 2}, let T _i = hS _i , L, c , T _i , s _0i , S _?i i be transition systems with identical label set and identical label cost function.

The synchronized product of T ₁ and T ₂ , in symbols T ₁ ⊗ T ₂ , is the transition system T ⊗ = hS ⊗ , L, c , T ⊗ , s _0⊗ , S _?⊗ i with

I S _⊗ := S ₁ × S ₂

I T ⊗ := {hhs ₁ , s ₂ i, l, ht ₁ , t ₂ ii | hs ₁ , l , t ₁ i ∈ T ₁ and hs ₂ , l , t ₂ i ∈ T ₂ }

I s _0⊗ := hs ₀₁ , s ₀₂ i

I S _?⊗ := S _?1 × S _?2

Example: Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

:

LL LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PAR DAR

PBR DBR DBL

PBL

PBL DBL

DBR PBR

MB?? MB??

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

:

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL

PBL

⊗

^L ^R

PAL,DAL,MB??, PB?,DB?

MALR MARL

PAR,DAR,MB??, PB?,DB?

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBR DBL

PBL PBL DBL

DBR PBR

MB?? MB??

(7)

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: S _⊗ = S ₁ × S ₂

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL

A

⊗

^L ^R

MALR MARL

L

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBL DBR

PBL PBL DBL

DBR PBR

MB?? MB??

AL

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: s _0⊗ = hs ₀₁ , s ₀₂ i

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL

L

⊗

^L ^R

MALR MARL

R

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBL DBR

PBL PBL DBL

DBR PBR

MB?? MB??

LR

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: S _?⊗ = S _?1 × S _?2

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL

PBL R

⊗

^L ^R

MALR MARL

L R

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBR DBL

PBL PBL DBL

DBR PBR

MB?? MB??

RL RR

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: T _⊗ := {hhs ₁ , s ₂ i, l, ht ₁ , t ₂ ii | . . . }

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL PAL

⊗

^L ^R

MALR MARL

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBR DBL

PBL PBL DBL

DBR PBR

MB?? MB??

PAL

(8)

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: T _⊗ := {hhs ₁ , s ₂ i, l , ht ₁ , t ₂ ii | . . . }

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL M???

⊗

^L ^R

MALR MARL

MALR

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL DAR

PAR

PBR DBL DBR

PBL PBL DBL

DBR PBR

MB?? MB??

MALR

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: T _⊗ := {hhs ₁ , s ₂ i, l, ht ₁ , t ₂ ii | . . . }

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL

PBL

⊗

^L ^R

MALR MARL

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBL DBR

PBL PBL DBL

DBR PBR

MB?? MB??

PBL

Example: Computation of Synchronized Product

T ^π

^{package}

⊗ T ^π

^{{truck A}}

: T _⊗ := {hhs ₁ , s ₂ i, l , ht ₁ , t ₂ ii | . . . }

L

A

B

R M???

PAL DAL

M???

DAR PAR

M???

PBR DBR

M???

DBL PBL

M???

⊗

^L ^R

MALR MARL

=

^LL ^LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PARDAR

PBR DBR DBL

PBL PBL DBL

DBR PBR

MB?? MB??

MB??

D6. Merge-and-Shrink Abstractions: Synchronized Product Synchronized Products and Abstractions

D6.3 Synchronized Products and

Abstractions

(9)

Synchronized Product of Functions

Definition (Synchronized Product of Functions)

Let α ₁ : S → S ₁ and α ₂ : S → S ₂ be functions with identical domain.

The synchronized product of α ₁ and α ₂ , in symbols α ₁ ⊗ α ₂ , is the function α _⊗ : S → S ₁ × S ₂ defined as α _⊗ (s ) = hα ₁ (s), α ₂ (s )i.

Synchronized Product of Abstractions

Theorem

Let α ₁ and α ₂ be abstractions of transition system T such that α _⊗ := α ₁ ⊗ α ₂ is surjective.

Then α _⊗ is an abstraction of T and a refinement of α ₁ and α ₂ .

Proof.

Abstraction: suitable domain as α ₁ , α ₂ are abstractions of T , Abstraction: surjective by premise

Refinement: For i ∈ {1, 2}, α _i = β _i ◦ α _⊗ with β _i (hx ₁ , x ₂ i) = x _i .

Preserving Abstractions

I It would be very nice if we could prove that if α ₁ and α ₂ are abstractions of T then there is an abstraction of T inducing T ^α

¹

⊗ T ^α

²

.

I However, this is not true in general.

I It is not even true for SAS ⁺ tasks.

I But there is an important sufficient condition for preserving the abstraction property.

Synchronized Products and Abstractions

Theorem (Synchronized Products and Abstractions) Let Π be a SAS ⁺ planning task with variable set V , and let V ₁ and V ₂ be disjoint subsets of V .

For i ∈ {1, 2}, let α _i be an abstraction of T (Π) such that α _i is a coarsening of π _V

_i

.

Then α _⊗ := α ₁ ⊗ α ₂ is surjective and T ^α

¹

^⊗α

²

= T ^α

¹

⊗ T ^α

²

.

(10)

Synchronized Products and Abstractions

Proof.

Let T = hS , L, c , T , s ₀ , S _? i and

for i ∈ {1, 2} let T ^α

ⁱ

= hS _i , L, c , T _i , s _0i , S _?i i (with α _i : S → S _i ).

α ₁ ⊗ α ₂ is surjective:

Since α _i is a coarsening of π _V

_i

there is a β _i such that α _i = β _i ◦ π _V

_i

with β _i : S| _V

_i

→ S _i .

Consider an arbitrary hs ₁ , s ₂ i ∈ S ₁ × S ₂ .

As α ₁ , α ₂ are surjective (because they are abstractions), there are s ₁ ⁰ , s ₂ ⁰ ∈ S such that α _i (s _i ⁰ ) = s _i .

As S consists of all valuations of V , also state s with s | _V

₁

= s ₁ ⁰ | _V

₁

and s | _V _\V

₁

= s ₂ ⁰ | _V _\V

₁

is in S .

Then α _i (s ) = β _i ◦ π _V

_i

(s) = β _i ◦ π _V

_i

(s _i ⁰ ) = α _i (s _i ⁰ ) = s _i and hence α ₁ ⊗ α ₂ (s ) = hα ₁ (s), α ₂ (s )i = hs ₁ , s ₂ i. . . .

Synchronized Products and Abstractions

Proof (continued).

T ^α

¹

^⊗α

²

= T ^α

¹

⊗ T ^α

²

: S _α

₁

⊗α

₂

= S ₁ × S ₂ = S ⊗

s _0α

₁

_⊗α

₂

= α ₁ ⊗ α ₂ (s ₀ ) = hα ₁ (s ₀ ), α ₂ (s ₀ )i = hs ₀₁ , s ₀₂ i = s _0⊗

S _?α

₁

_⊗α

₂

= {α ₁ ⊗ α ₂ (s) | s ∈ S _? }

= {hα ₁ (s), α ₂ (s )i | s ∈ S _? }

⊆ {hα ₁ (s), α ₂ (s ⁰ )i | s , s ⁰ ∈ S _? }

= {hs ₁ , s ₂ i | s ₁ ∈ S _?1 , s ₂ ∈ S _?2 }

= S _?1 × S _?2

= S _?⊗

. . .

Synchronized Products and Abstractions

Proof (continued).

For equality, we also need to establish that

{hα ₁ (s ), α ₂ (s ⁰ )i | s, s ⁰ ∈ S _? } ⊆ {hα ₁ (s), α ₂ (s )i | s ∈ S _? }.

Consider arbitrary s, s ⁰ ∈ S _? .

Define s ⁰⁰ as s ⁰⁰ | _V

₁

= s | _V

₁

and s ⁰⁰ | _V _\V

₁

= s ⁰ | _V _\V

₁

.

It holds that α ₁ (s ⁰⁰ ) = α ₁ (s ) and α ₂ (s ⁰⁰ ) = α ₂ (s ⁰ ) because α _i is a coarsening of π _V

_i

.

Furthermore, s ⁰⁰ ∈ S _? : the goal formula γ of a SAS ⁺ task is a conjunction of atoms v = d . If v ∈ V ₁ , then s ⁰⁰ (v ) = d because s ∈ S _? , otherwise s ⁰⁰ (v ) = d because s ⁰ ∈ S _? . Overall, s ⁰⁰ | = γ.

. . .

Synchronized Products and Abstractions

Proof (continued).

We still need to show the equality of the sets of transitions.

T _α

₁

⊗α

₂

= {hα ₁ ⊗ α ₂ (s), o, α ₁ ⊗ α ₂ (t)i | hs, o, ti ∈ T }

= {hhα ₁ (s ), α ₂ (s)i, o, hα ₁ (t), α ₂ (t)ii | hs , o, ti ∈ T }

⊆ {hhα ₁ (s ), α ₂ (s ⁰ )i, o, hα ₁ (t ), α ₂ (t ⁰ )ii

| hs, o, ti, hs ⁰ , o , t ⁰ i ∈ T }

= {hhs ₁ , s ₂ i, o, ht ₁ , t ₂ ii | hs ₁ , o, t ₁ i ∈ T ₁ , hs ₂ , o , t ₂ i ∈ T ₂ }

= T _⊗

For equality, we need to show that for hs, o, ti, hs ⁰ , o, t ⁰ i ∈ T there is a transition hs ⁰⁰ , o, t ⁰⁰ i ∈ T with

α ₁ (s ) = α ₁ (s ⁰⁰ ), α ₁ (t) = α ₁ (t ⁰⁰ ), α ₂ (s ⁰ ) = α ₂ (s ⁰⁰ ), α ₂ (t ⁰ ) = α ₂ (t ⁰⁰ ).

. . .

(11)

Synchronized Products and Abstractions

Proof (continued).

Consider s ⁰⁰ ∈ S with s ⁰⁰ | _V

₁

= s| _V

₁

and s ⁰⁰ | _V _\V

₁

= s ⁰ | _V _\V

₁

and t ⁰⁰ ∈ S with t ⁰⁰ | _V

₁

= t| _V

₁

and t ⁰⁰ | _V _\V

₁

= t ⁰ | _V _\V

₁

.

Since pre(o ) is a conjunction of atoms and consist(eff(o)) ≡ >, o is applicable in s ⁰⁰ by an analogous argument as for the goal.

As t = s J o K , we have t| _V \vars(eff(o)) = s | _V \vars(eff(o)) , analogously for t ⁰ and s ⁰ . Hence t ⁰⁰ | _V \vars(eff(o)) = s ⁰⁰ | _V \vars(eff(o)) .

As eff(o) contains no conditional effect, it holds for all atomic effects v := d in eff(o) that t(v ) = t ⁰ (v ) = d and hence t ⁰⁰ (v ) = d . Overall, t ⁰⁰ = s ⁰⁰ J o K and hs ⁰⁰ , `, t ⁰⁰ i ∈ T .

The requirements on the abstractions are again satisfied by the construction of s ⁰⁰ and t ⁰⁰ and α _i being coarsenings of π _V

_i

.

Example: Product for Disjoint Projections

T ^π

^{package}

⊗ T ^π

^{{truck A}}

∼ T ^π

{package,truck A}

:

LL LR

AL AR

BL BR

RL RR

MALR MARL

MALR MARL PAL

DAL

PAR DAR

PBR DBL DBR

PBL

PBL DBL

DBR PBR

MB?? MB??

Synchronized Products of Projections

Corollary (Synchronized Products of Projections)

Let Π be a SAS ⁺ planning task with variable set V , and let V ₁ and V ₂ be disjoint subsets of V .

Then T ^π

^V¹

⊗ T ^π

^V²

∼ T ^π

^V¹^∪V²

. (Proof omitted.)

By repeated application of the corollary, we can recover all pattern database heuristics of a SAS ⁺ planning task from the abstract transition systems induced by atomic projections.

Recovering T (Π) from the Atomic Projections

Moreover, by computing the product of all atomic projections, we can recover the identity abstraction id = π _V .

Corollary (Recovering T (Π) from the Atomic Projections) Let Π be a SAS ⁺ planning task with variable set V . Then T (Π) ∼ N

v∈V T ^π

^{v}

.

This is an important result because it shows that the transition

systems induced by atomic projections contain all information of a

SAS ⁺ task.

(12)

D6. Merge-and-Shrink Abstractions: Synchronized Product Summary

D6.4 Summary

D6. Merge-and-Shrink Abstractions: Synchronized Product Summary

Planning and Optimization D6. Merge-and-Shrink Abstractions: Synchronized Product Gabriele R¨oger and Thomas Keller

Planning and Optimization

D6. Merge-and-Shrink Abstractions: Synchronized Product

Gabriele R¨ oger and Thomas Keller

Universit¨ at Basel

November 5, 2018

Planning and Optimization

November 5, 2018 — D6. Merge-and-Shrink Abstractions: Synchronized Product

D6.1 Motivation

D6.2 Synchronized Product

D6.3 Synchronized Products and Abstractions D6.4 Summary

Content of this Course

Planning

Classical

Tasks Progression/

Regression Complexity Heuristics

Probabilistic

MDPs Uninformed Search

Heuristic Search Monte-Carlo

Methods

Content of this Course: Heuristics

Heuristics

Delete Relaxation

Abstraction

Abstractions in General

Pattern Databases

Merge &

Shrink Landmarks

Potential

Heuristics

Cost Partitioning

D6.1 Motivation

Beyond Pattern Databases

I Despite their popularity, pattern databases have some fundamental limitations ( example on next slides).

I For the rest of this week, we study a class of abstractions called merge-and-shrink abstractions.

I Merge-and-shrink abstractions can be seen as a proper generalization of pattern databases.

They can do everything that pattern databases can do (modulo polynomial extra effort).

They can do some things that pattern databases cannot.

Back to the Running Example

LRR LLL

LLR

LRL

ALR

ALL

BLL

BRL

ARL

ARR

BRR

BLR

RRR RRL

RLR

RLL

Logistics problem with one package, two trucks, two locations:

I state variable package: {L, R, A, B}

I state variable truck A: {L, R }

I state variable truck B: {L, R }

Example: Projection

T π

:

LRR LLL

LLR

LRL LRR

LLR

LRL LLL

ALR ARL

ALL ARR

ALR ARL

ARR ALL

BLL

BRL

BRR

BLR BLL BRR

BLR BRL

RRR RRL

RLR

RLL RLL RRL

RLR

RRR

Example: Projection (2)

T ^π

T ^π

I h(s ₀ ) ≤ 2 no better than atomic projection to package These values cannot be improved by maximizing over several patterns or using additive patterns.

Merge-and-shrink abstractions can represent heuristics with h(s ₀ ) ≥ 3 for tasks of this kind of any size.

T ^π