Planning and Optimization
D7. M&S: Generic Algorithm and Heuristic Properties
Gabriele R¨oger and Thomas Keller
Universit¨at Basel
November 7, 2018
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 1 / 41
Planning and Optimization
November 7, 2018 — D7. M&S: Generic Algorithm and Heuristic Properties
D7.1 Generic Algorithm D7.2 Heuristic Properties D7.3 Summary
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 2 / 41
Content of this Course
Planning
Classical
Tasks Progression/
Regression Complexity Heuristics
Probabilistic
MDPs Uninformed Search
Heuristic Search Monte-Carlo
Methods
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 3 / 41
Content of this Course: Heuristics
Heuristics
Delete Relaxation
Abstraction
Abstractions in General
Pattern Databases
Merge &
Shrink Landmarks
Potential Heuristics Cost Partitioning
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 4 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
D7.1 Generic Algorithm
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 5 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Content of this Course: Merge & Shrink
Merge & Shrink
Synchronized Product Merge & Shrink Algorithm
Heuristic Properties Strategies Label Reduction
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 6 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Generic Merge-and-shrink Abstractions: Outline
Using the results from the previous chapter, we can develop the ideas of a generic abstraction computation procedurethat takes all state variables into account:
I Initialization step: Compute all abstract transition systems for atomic projections to form the initial abstraction collection.
I Merge steps: Combine two abstract systems in the collection by replacing them with their synchronized product. (Stop once only one transition system is left.)
I Shrink steps: If the abstractions in the collection are too large to compute their synchronized product, make them smaller by abstracting them further (applying an arbitrary abstraction to them).
We explain these steps with our running example.
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
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 variablepackage: {L,R,A,B}
I state variabletruck A:{L,R}
I state variabletruck B:{L,R}
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Initialization Step: Atomic Projection for Package
Tπ{package}:
L
A
B
R
M???
PAL DAL
M???
DAR PAR
M???
PBR DBR
M???
DBL PBL
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 9 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Initialization Step: Atomic Projection for Truck A
Tπ{truck A}:
L R
PAL,DAL,MB??, PB?,DB?
MALR
MARL
PAR,DAR,MB??, PB?,DB?
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 10 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Initialization Step: Atomic Projection for Truck B
Tπ{truck B}:
L R
PBL,DBL,MA??, PA?,DA?
MBLR
MBRL
PBR,DBR,MA??, PA?,DA?
current collection: {Tπ{package},Tπ{truck A},Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 11 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Merge Step
T1:=Tπ{package}⊗ Tπ{truck A}:
LL LR
AL AR
BL BR
RL RR
MALR MARL
MALR MARL
MALR MARL
MALR MARL PAL
DAL
PADARR
PBR DBL DBR
PBL
PBL DBL
DBR PBR
MB?? MB??
MB?? MB??
MB?? MB??
MB?? MB??
current collection: {T1,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 12 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Need to Simplify?
I If we have sufficient memory available, we can now compute T1⊗ Tπ{truck B}, which would recover the complete transition system of the task.
I However, to illustrate the general idea, let us assume that we do not have sufficient memory for this product.
I More specifically, we will assume that after each product operation we need to reduce the result transition system to four states to obey memory constraints.
I So we need to reduceT1 to four states. We have a lot of leeway in decidinghow exactly to abstractT1.
I In this example, we simply use an abstraction that leads to a good result in the end.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 13 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2:= some abstraction ofT1
LL LR
AL AR
BL BR
RL RR
MALR MARL
MALR MARL
MALR MARL
MALR MARL PAL
DAL
PADARR
PBR DBL DBR
PBL
PBL DBL
DBR PBR
MB?? MB??
MB?? MB??
MB?? MB??
MB?? MB??
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 14 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2 := some abstraction ofT1
LL LR
AL AR
BL BR
R
MALR MARL
MALR MARL
MALR MARL PAL
DAL
DAR PAR
DBRPBR DBL
PBL
PBL DBL
DBR PBR MB??
MB?? MB??
MB??
MB??
M???
MB??
current collection: {T2,Tπ{truck B}}
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2:= some abstraction ofT1
LL LR
AL AR
AL AR
BL BR
R
MALR MARL
MALR MARL
MALR MARL PAL
DAL
DAR PAR
DBRPBR DBL
PBL
PBL DBL
DBR PBR MB??
MB?? MB??
MB??
MB??
M???
MB??
current collection: {T2,Tπ{truck B}}
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2 := some abstraction ofT1
LL LR
A
BL BR
R
MALR MARL
MALR MARL PAL
DAL
DAR PAR
DBRPBR DBL
PBL
PBL DBL
DBR PBR MB??
M???
MB??
MB??
M???
MB??
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 17 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2:= some abstraction ofT1
LL LR
A
BL BR
BL BR
R
MALR MARL
MALR MARL PAL
DAL
DAR PAR
DBRPBR DBL
PBL
PBL DBL
DBR PBR MB??
M???
MB??
MB??
M???
MB??
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 18 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2 := some abstraction ofT1
LL LR
A
B
R
MALR MARL
PAL DAL
DAR PAR
PBR DBL DBR
PBL PBL DBL MB??
M???
MB??
M???
M???
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 19 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2:= some abstraction ofT1
LL LR
A A
B B
R
MALR MARL
PAL DAL
DAR PAR
PBR DBL DBR
PBL PBL DBL MB??
M???
MB??
M???
M???
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 20 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2 := some abstraction ofT1
LL LR I R
MALR MARL MB??
MB??
D?R M???
P?R M???
PBL DBL P?L D?L
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 21 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
First Shrink Step
T2:= some abstraction ofT1
LL LR I R
MALR MARL MB??
MB??
D?R M???
P?R M???
PBL DBL P?L D?L
current collection: {T2,Tπ{truck B}}
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 22 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Second Merge Step
T3 :=T2⊗ Tπ{truck B}: LRL
LRR
LLL
LLR
IL
IR
RL
RR
MBLR MBRL
MBLR MBRL
MBLR MBRL
MBLR MBRL DAR
PAR
D?R P?R P?L
D?L
PAL DAL MALR MARL MALR MARL
PBL DBL
MA??
MA?? MA??
MA??
current collection: {T3}
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Another Shrink Step?
I Normally we could stop now and use the distances in the final abstract transition system as our heuristic function.
I However, if there were further state variables to integrate, we would simplify further, e.g. leading to the following
abstraction (again with four states):
LRR LLLLRL
LLR I R
M??? M???
M???
M?RL M?LR
P?L D?L
D?R P?R
I We get a heuristic value of 3 for the initial state,better than any PDB heuristicthat is a proper abstraction.
I The example generalizes to more locations and trucks, even if we stick to the size limit of 4 (after merging).
D7. M&S: Generic Algorithm and Heuristic Properties Generic Algorithm
Generic Algorithm Template
Generic Merge & Shrink Algorithm abs := {Tπ{v} |v ∈V}
whileabs contains more than one abstract transition system:
selectA1,A2 from abs
shrinkA1 and/orA2 until size(A1)·size(A2)≤N abs :=abs \ {A1,A2} ∪ {A1⊗ A2}
return the remaining abstract transition system inabs N: parameter bounding number of abstract states Questions for practical implementation:
I Which abstractions to select? merging strategy
I How to shrink an abstraction? shrinking strategy
I How to chooseN? usually: as high as memory allows
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 25 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
D7.2 Heuristic Properties
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 26 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Content of this Course: Merge & Shrink
Merge & Shrink
Synchronized Product Merge & Shrink Algorithm
Heuristic Properties Strategies Label Reduction
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 27 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Heuristic Properties
I Each iteration of the algorithm corresponds to a
transformation of the collection absof transition systems.
I The exact transformation depends on the specific instantiation of the generic algorithm
(e.g. of the merging and the shrinking strategy).
I For analyzing the properties of the resulting heuristic, we analyze properties of the individual transformations.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 28 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Collections of Transition Systems
Definition (Collection of Transition Systems)
A set X of transition systems is a collection of transition systemsif allT ∈X have the same set of labels and the same cost function.
Thecombined systemis TX :=N
T ∈XT.
Remark: Strictly speaking, the combined system is not well-defined as the Cartesian product is neither commutative nor associative.
For our purpose it is sufficient that the results of all possible combination orders are isomorphic.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 29 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Safe Transformations
Definition (Safe Transformation)
LetX andX0 be collections of transition systems with label setsL andL0 and cost functionsc andc0, respectively.
The transformation fromX to X0 is safeif there exist functions σ andλ mapping the states and labels ofTX to the states and labels ofTX0 such that
I c0(λ(`))≤c(`) for all`∈L,
I ifhs, `,ti is a transition ofTX then hσ(s), λ(`), σ(t)i is a transition of TX0, and
I ifs is a goal state of TX thenσ(s) is a goal state ofTX0.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 30 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Examples
X: Collection of transition systems
Replacement with Synchronized Product is Safe
Let T1,T2 ∈X withT1 6=T2. The transformation fromX to X0 := (X\ {T1,T2})∪ {T1⊗ T2} is safe withσ = id andλ= id.
Abstraction is Safe
Let αbe an abstraction for Ti ∈X. The transformation fromX to X0 := (X\ {Ti})∪ {Tiα} is safe withλ= id and
σ(hs1, . . . ,sni) =hs1, . . . ,si−1, α(si),si+1, . . . ,sni.
(Proofs omitted.)
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Heuristic Properties (1)
Theorem
Let X and X0 be collections of transition systems. If the
transformation from X to X0 is safewith functionsσ andλthen h(s) =h∗T
X0(σ(s))is a safe, goal-aware, admissible, and consistent heuristic forTX.
Proof.
We prove goal-awareness and consistency, the other properties follow from these two.
Goal-awareness: For all goal states s? of TX, stateσ(s?) is a goal state of TX0 and therefore h(s?) =h∗T
X0(σ(s?)) = 0. . . .
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Heuristic Properties (2)
Proof (continued).
Consistency: Letc andc0 be the label cost functions of X andX0, respectively. Consider states of TX and transitionhs, `,ti.
As TX0 has a transitionhσ(s), λ(`), σ(t)i, it holds that h(s) =h∗T
X0(σ(s))
≤c0(λ(`)) +h∗T
X0(σ(t))
=c0(λ(`)) +h(t)
≤c(`) +h(t)
The second inequality holds due to the requirement on the label costs.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 33 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Exact Transformations
Definition (Exact Transformation)
LetX andX0 be collections of transition systems with label setsL andL0 and cost functionsc andc0, respectively.
The transformation fromX toX0 is exactif there exist functionsσ andλ mapping the states and labels ofTX to the states and labels ofTX0 such that
1 σ andλ satisfy the requirements of safe transformations,
2 ifhs0, `0,t0iis a transition ofTX0 thenhs, `,tiis a transition of TX for all s ∈σ−1(s0),t∈σ−1(t0) and some`∈λ−1(`0),
3 ifs0 is a goal state of TX0 then all states s ∈σ−1(s0) are goal states ofTX, and
4 c(`) =c0(λ(`)) for all `∈L.
no “new” transitions and goal states, no cheaper labels
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 34 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Examples
Replacement with Synchronized Product is Exact
Let T1,T2 ∈X withT1 6=T2. The transformation fromX to X0 := (X\ {T1,T2})∪ {T1⊗ T2} is exact withσ= id and λ= id.
(Proof omitted.)
More examples will follow.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 35 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Heuristic Properties with Exact Transformations (1)
Theorem
Let X and X0 be collections of transition systems. If the
transformation from X to X0 is exactwith functions σ andλ then h∗T
X(s) =h∗T
X0(σ(s)).
Proof.
As the transformation is safe,h∗T
X0(σ(s)) is admissible forTX and therefore h∗T
X(s)≥h∗T
X0(σ(s)).
For the other direction, we show that for every states0 ofTX0 and goal pathπ0 for s0, there is for each s ∈σ−1(s0) a goal path inTX
that has the same cost. . . .
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 36 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Heuristic Properties with Exact Transformations (2)
Proof (continued).
Proof via induction over the length of π0.
|π0|= 0: If s0 is a goal state ofTX0 then eachs ∈σ−1(s0) is a goal state of TX and the empty path is a goal path for s in TX.
|π0|=i+ 1: Letπ0 =hs0, `0,t0iπ0t0, where π0t0 is a goal path of length i from t0. Then there is for each t ∈σ−1(t0) a goal pathπt of the same cost in TX. Furthermore, for alls ∈σ−1(s0) there is a label`∈λ−1(`0) such that TX has a transitionhs, `,ti with t ∈σ−1(t0). The path π=hs, `,tiπt is a solution fors inT. As` and`0 must have the same cost andπt andπt00 have the same cost, π has the same cost asπ0.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 37 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Sequences of Transformations
Theorem (Sequences of Transformations)
Let X1, . . . ,Xn be collections of transition systems.
If for i ∈ {1, . . . ,n−1} the transformation from Xi to Xi+1 is safe (exact) then the transformation from X1 to Xn is safe (exact).
Proof idea: The composition of the σ andλ functions of each step satisfy the required conditions.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 38 / 41
D7. M&S: Generic Algorithm and Heuristic Properties Heuristic Properties
Consequences
Generic Merge & Shrink Algorithm abs := {Tπ{v} |v ∈V} =: X0
whileabs contains more than one abstract transition system:
selectA1,A2 from abs
shrinkA1 and/orA2 until size(A1)·size(A2)≤N abs :=abs \ {A1,A2} ∪ {A1⊗ A2}
return the remaining abstract transition system inabs
I Initially Tabs is the concrete transition system.
I Each iteration performs a safe transformation of abs.
→the corresponding abstraction heuristic is safe, goal-aware,
→consistent, and admissible.
I If shrinking is exact, the corresponding heuristic is perfect.
D7. M&S: Generic Algorithm and Heuristic Properties Summary
D7.3 Summary
D7. M&S: Generic Algorithm and Heuristic Properties Summary
Summary
I Projections perfectlyreflecta few state variables.
Merge-and-shrink abstractions are ageneralization that can reflectallstate variables, but in apotentially lossy way.
I Themerge stepscombine two abstract transition systems by replacing them with theirsynchronized product.
I Theshrink stepsmake an abstract system smaller by abstracting it further.
I As we only use safe transformations, the resulting heuristic is alwaysadmissible.
I If we use onlyexacttransformations, the resulting heuristic is perfect.
G. R¨oger, T. Keller (Universit¨at Basel) Planning and Optimization November 7, 2018 41 / 41