[PPT] - Constraint Satisf action CS 486/ 686 May 17, 2005 Universit y of PowerPoint Presentation

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Constraint Satisf action

CS 486/ 686 May 17, 2005 Universit y of Wat erloo

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Outline

What are CSPs?
St andard search and CSPs
I mprovement s

– Backt r acking – Backt r acking + heurist ics – Forward checking

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I ntroduction

I n t he last couple of lect ures we have

been solving problems by searching in a space of st at es

– Treat ing st at es as black boxes, ignoring any st ruct ure inside t hem – Using pr oblem-specif ic rout ines

Today we st udy problems where t he

st at e st ruct ure is import ant

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States: all arrangement s
f 0,1,…

, or 8 queens on t he board

I nitial state: 0 queens on

t he boar d

Successor f unction: Add a

queen t o t he board

Goal test: 8 queens on t he

board wit h no t wo of t hem at t acking each ot her

64x63x… 57 ≈ 3x1014 st at es

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States: all arrangement s k

queens (0 ≤ k ≤ 8), one per column in t he lef t most k columns, wit h no queen at t acking anot her

I nitial state: 0 queens on

t he boar d

Successor f unction: Add a

queen t o t he lef t most empt y column such t hat it is not at t acked

Goal test: 8 queens on t he

board 2057 St at es

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I ntroduction

Earlier search met hods st udied of t en

make choices in an arbit rary order

I n many problems t he same st at e can be

reached independent of t he order in which t he moves are chosen (commut at ive act ions)

Can we solve problems ef f icient ly by

being smart in t he order in which we t ake act ions?

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4- queens Constraint Propagation

Place a queen in a squar e Remove conf lict ing squares f rom consider at ion

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4- queens Constraint Propagation

Place a queen in a squar e Remove conf lict ing squares f rom consider at ion

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4- queens Constraint Propagation

Place a queen in a squar e Remove conf lict ing squares f rom consider at ion

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4- queens Constraint Propagation

Place a queen in a squar e Remove conf lict ing squares f rom consider at ion

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CSP Def inition

A const raint sat isf act ion problem (CSP)

is def ined by {V,D,C} where

– V ={V1,V2,… ,Vn} is a set of variables – D={D1,… ,Dn} is t he set of domains, Di is t he domain of possible values f or variable Vi – C={C

1,…

,C

m} is t he set of const raint s

Each const raint involves some subset of t he

variables and specif ies t he allowable combinat ions of values f or t hat subset

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CSP Def inition

A st at e is an assignment of values t o some or

all of t he variables

– {Vi=xi, Vj=xj,… }

An assignment is consist ent if it does not

violat e any const r aint s

A solut ion is a complet e, consist ent

assignment (“hard const raint s”)

– Some CSPs also require an obj ect ive f unct ion t o be

pt imized (“sof t const raint s”)

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Example 1: 8- Queens

64 variables Vij, i=1 t o 8, j =1 t o 8
Domain of each variable is {0,1}
Const raint s

– Vij=1 Vik=0 f or all k ≠ j – Vij=1 Vkj=0 f or all k ≠ i – Similar const raint f or diagonals – ∑i,j Vij=8 Binary const raint s relat e t wo variables

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Example 2 – 8 queens

8 variables Vi, i=1 t o 8
Domain of each variable is {1,2,…

,8}

Const raint s

– Vi=k Vj≠ k f or all j ≠ i – Similar const raint s f or diagonals Binary const raint s relat e t wo variables

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Example 3 - Map Coloring

7 var iables {WA,NT,SA,Q,NSW,V,T} Each variable has t he same domain: {red, green, blue} No t wo adj acent variables have t he same value:

WA≠ ≠ ≠ ≠NT, WA≠ ≠ ≠ ≠SA, NT≠ ≠ ≠ ≠SA, NT≠ ≠ ≠ ≠Q, SA≠ ≠ ≠ ≠Q, SA≠ ≠ ≠ ≠NSW, SA≠ ≠ ≠ ≠V,Q≠ ≠ ≠ ≠NSW, NSW≠ ≠ ≠ ≠V WA NT SA Q NSW V T WA NT SA Q NSW V T

T WA NT SA Q NSW V

Const raint graph

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Example 4 - St reet Puzzle

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Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

Who owns t he Zebra? Who drinks Wat er? Who owns t he Zebra? Who drinks Wat er?

Example from R and N, Annotations from Stanford CS121

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St reet Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

(Ni = English) ⇔ (C

i = Red)

(Ni = J apanese) ⇔ (J i = Paint er) (N1 = Norwegian) lef t as an exercise (C

i = Whit e) ⇔ (C i+1 = Green)

(C

5 ≠ Whit e)

(C

1 ≠ Green)

Example from R and N, Annotations from Stanford CS121

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St reet Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

(Ni = English) ⇔ (C

i = Red)

(Ni = J apanese) ⇔ (J i = Paint er) (N1 = Norwegian) (C

i = Whit e) ⇔ (C i+1 = Green)

(C

5 ≠ Whit e)

(C

1 ≠ Green)

unary const raint s Example from R and N, Annotations from Stanford CS121

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St reet Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

∀ ∀ ∀ ∀i,j ∈[1,5], i≠j , Ni ≠ Nj ∀ ∀ ∀ ∀i,j ∈[1,5], i≠j , C

i ≠ C j

...

Example from R and N, Annotations from Stanford CS121

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St reet Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t

N1 = Norwegian

The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk

D3 = Milk

The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

Example from R and N, Annotations from Stanford CS121

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St reet Puzzle

1 2 3 4 5

Ni = {English, Spaniard, Japanese, I talian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Cof f ee, Milk, Fruit- juice, Water} J i = {Painter, Sculptor, Diplomat, Violinist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house

C

1 ≠

≠ ≠ ≠ Red The Spaniard has a Dog

A1 ≠

≠ ≠ ≠ Dog The Japanese is a Painter The I talian drinks Tea The Norwegian lives in the f irst house on the lef t

N1 = Norwegian

The owner of the Green house drinks Cof f ee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk

D3 = Milk

The Norwegian lives next door to the Blue house The Violinist drinks Fruit juice

J 3 ≠

≠ ≠ ≠ Violinist The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s

Example from R and N, Annotations from Stanford CS121

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Example 5 - Scheduling

Four t asks T1, T2, T3, and T4 are relat ed by t ime const raint s:

T1 must be done during T3
T2 must be achieved bef or e T1 st ar t s
T2 must overlap wit h T3
T4 must st art af t er T1 is complet e

Are t he const raint s compat ible? What ar e t he possible t ime relat ions bet ween t wo t asks? What if t he t asks use resources in limit ed supply?

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Example 6 - 3- Sat

n Boolean variables, V1,…

,Vn

K const raint s of t he f orm Vi v Vj v Vk

where Vi is eit her t rue or f alse

NP-complet e

– Recall GSAT and WALKSAT

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Properties of CSPs

Types of variables

– Discret e and f init e

Map colouring, 8-queens, boolean CSPs

– Discret e variables wit h inf init e domains

Scheduling j obs in a calendar
Require a const raint language (J ob1+3 ≤ J ob2)

– Cont inuous domains

Scheduling on t he Hubble t elescope
Linear programming

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Properties of CSPs

Types of cont raint s

– Unary const raint rest rict s a variable t o a single value

Queensland=Blue, SA≠Green

– Binary const raint s relat es t wo variables

SA≠NSW
Can use a const raint graph t o repr esent CSPs wit h only

binary const raint s

– Higher order const raint s involve t hree of more variables

Alldif f (V1,…

,Vn)

Can use a const raint hypergraph t o represent t he problem

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CSPs and search

N var iables V1,…

,Vn

Valid assignment : {V1=x1,…

,Vk=xk} f or 0 ≤ k ≤ n such t hat values sat isf y const raint s on t he variables

St at es: valid assignment s
I nit ial st at e: empt y assignment
Successor:

– {V1=x1,… ,VK=xk} {V1=x1,… ,Vk=xk, Vk+1=xk+1}

Goal t est : complet e assignment
I f all domains all have size d, t hen t here are

O(dn) complet e assignment s

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CSPs and commutativity

CSPs are commut at ive!

– The order of applicat ion of any given set of act ions has no ef f ect on t he out come – When assigning values t o variables we reach t he same part ial assignment , no mat t er t he order – All CSP search algorit hms generat e successors by considering possible assignment s f or only a single variable at each node in t he search t ree

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CSPs and commutativity

3 var iables V1, V2, V3
Let t he current assignment be

– A={V1=x1}

Pick var iable 3
Let domain of V3 be {a,b,c}
The successors of A are

Backt racking search is an uninf ormed

search met hod

– Not very ef f icient

We can do bet t er by t hinking about t he

f ollowing quest ions

– Which variable should be assigned next ? – I n which order should it s values be t r ied? – Can we det ect inevit able f ailure early (and avoid t he same f ailure in ot her pat hs)?

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Most constrained variable

Choose t he variable which has t he

f ewest “legal” moves

– AKA minimum remaining values (MRV) heurist ic

DNT={green, blue} DSA={green, blue} Dothers={red, green, blue} DSA={blue} DQ={blue, red} Dothers={red,green,blue}

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Most constraining variable

Most const raining variable:

– choose t he variable wit h t he most const r aint s on remaining var iables

Tie-breaker among most const rained

variables

SA is involved in 5 constraints

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Least- constraining value

Given a variable, choose t he least

const raining value:

– t he one t hat rules out t he f ewest values in t he remaining variables

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Forward checking

The t hird quest ions was

– I s t here a way t o det ect f ailure early?

Forward checking

– Keep t rack of remaining legal values f or unassigned var iables – Terminat e search when any variable has no legal values

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Forward Checking in Map Coloring

RGB RGB RGB RGB RGB RGB RGB T SA V NSW Q NT WA

T WA NT SA Q NSW V

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RGB RGB RGB RGB RGB RGB R RGB RGB RGB RGB RGB RGB RGB T SA V NSW Q NT WA

T WA NT SA Q NSW V

Forward Checking in Map Coloring

Forward checking removes t he value Red of NT and of SA

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RGB GB RGB RGB G GB R RGB GB RGB RGB RGB GB R RGB RGB RGB RGB RGB RGB RGB T SA V NSW Q NT WA

Forward Checking in Map Coloring

T WA NT SA Q NSW V

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RGB B B RB G B R RGB B RGB RB G B R RGB GB RGB RGB RGB GB R RGB RGB RGB RGB RGB RGB RGB T SA V NSW Q NT WA

Forward Checking in Map Coloring

T WA NT SA Q NSW V

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RGB B B RB G B R RGB B RGB RB G B R RGB GB RGB RGB RGB GB R RGB RGB RGB RGB RGB RGB RGB T SA V NSW Q NT WA

What you should know

– How t o f ormalize problems as CSPs – Backt r acking sear ch – Heurist ics

Variable ordering
Value ordering

– Forward checking

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Next class

Adversarial search