Dynamic Csp Thesis

8/13/2019 Dynamic Csp Thesis

1/50

Dynamic Flexible Constraint

Satisfaction and Its Application toAI Planning

Ian Miguel

AI GroupDepartment of Computer Science

University of York


2/50

Outline

Part I Constraint Satisfaction.

Weaknesses/Remedies.

Dynamic Flexible Constraint Satisfaction. Part II

AI Planning.

Flexible Planning. Plan Synthesis via dynamic flexible CSP.


3/50


4/50

The Constraint Satisfaction Problem

(CSP)

Given:

1. A set of variables.2. Each variable has an associated finite domainof

potential values.

3. A set of constraintsover these variables.

Find:

A complete assignmentof values to variables that

satisfies all constraints.

Specify allowed combinations of assignments

of values to variables.


5/50

Applications

Combinatorial Mathematics.

Fault diagnosis.

Machine vision.

Planning.

Scheduling. Systems Simulation.

Csplib.org


6/50

Example CSPCourse Scheduling

Decide the number of lecture, exercise and

training sessions.

So we need 3 variables.

Constraints:

There must be a total of 8 sessions.

Professor A will give 4 or 5 lectures.

Dr B will give 3 or 4 exercise sessions.

There must be 1 or 2 training sessions.

These unary constraints

determine the domains

in this simple example.


7/50

Solving CSPs

Most often depth-first search (I.e. Backtrack). Select an unassigned variable.

Select a value compatible with all previously

assigned variables.

If no such value, backtrack and find a new value for

previous variableRoot

1stvariable

2ndvariable


8/50

Course Scheduling: A Solution

Lectures = 4

Exercise = 3

Training = 1

Constraints:

There must be a total of 8 sessions.

Professor A will give 4 or 5 lectures. Dr B will give 3 or 4 exercise sessions.

There must be 1 or 2 training sessions.


9/50

The Problem Changes

Constraints: There must be a total of 8 sessions.

Professor A will now give 3 or 4lectures.

Dr B agrees to give 4 or 5exercise sessions. There must be 1 or 2 training sessions.

The solution to the old problem:

Lectures = 4 Exercise = 3

Training = 1


10/50

Weakness 1: Static Formulation

Classical CSP has no way of dealing with this

change gracefully.

Naively we can solve the new problem from scratch.

This wastes all work on the old problem!

So?

We use an extension called dynamic CSP.


11/50

Dynamic CSP

A dynamic environment is viewed as a sequenceof static CSPs.

Linked by:

Restriction(constraints added)

Relaxation(constraints removed)


12/50

Solving Dynamic CSPs

Oracles:

Start from scratch.

Previous solution guides value assignment.

Local Repair:

Start from previous solution.

Modify individual assignments until find a solution.

Constraint Recording:

Record new constraints during search.

Carry new constraints over to future problems to

restrict search there.


13/50

Course Scheduling

In problem 2:

Dr B agrees to give 4 or 5exercise sessions.

Problem 1 Problem 2Solution:

Lectures = 4

Exercise = 3

Training = 1

Solution:

Lectures = 3

Exercise = 4

Training = 1Incompatible with

solution to problem 1

Repair method:

Start with violated exercise constraint: Exercise = 4

Then repair violated sum constraint: Lectures = 3


14/50

The Problem Changes Again

Constraints: There must be a total of 7sessions.

Professor A will give 3 or 4 lectures.

Dr B will give 4 or 5 exercise sessions. There must be 1 or 2 training sessions.

This problem has no solution.


15/50

Weakness 2: Hard Constraints

Classical CSP has no way of finding acompromise.

Constraints are hard:

Imperative: Valid solution must satisfy all of them. Inflexible: Constraints wholly satisfied or wholly

violated.

So? We use an extension called flexible CSP.


16/50

Flexible CSP

An umbrella term for a variety of methods.

Max-CSP: Maximise the number of

satisfied constraints/Minimise violations.

WeightedMax-CSP.

Weighted Preference.

FuzzyCSP.

General frameworks:

Partial CSP, Valued CSP, Semi-ring CSP.


17/50

Solving Flexible CSPs

Branch and boundis most common technique. Search for a first solution (minimise violations).

Use this as a bound on future solutions.

As soon as current search branch can be shown to be

equal or exceed the bound, prune it.

When a new solution is found, use as new bound,

etc

New bound

E.g. 3 violatedconstraints.

Equals bound:

3 violated

already. Prune.


18/50

Course Scheduling

In problem 3:

There must be a total of 7sessions.

Problem 2 Problem 3Solution:

Lectures = 3

Exercise = 4

Training = 1

Incompatible with

solution to problem 2

Max-CSP: Pretty good solutiononly 1 violated

constraint.

Or attach priorities/preferences to each constraint.


19/50

The Gap in the Market

Dynamic CSP research is founded almost

exclusively on hard constraints.

Flexible CSP research is founded almostexclusively on static problems.

We want to combine the two to bring to

bear the benefits of both.


20/50

Dynamic Flexible CSP

Also an umbrella term for a variety of methods.

FlexibleCSP

Techniques

Dynamic CSP Techniques

DFCSP Instance

Solution methods

Max

Weighted

Max

WeightedPreference

Fuzzy


21/50

Fuzzy rrDFCSP

Restriction/Relaxation DCSP + Fuzzy CSP.

Fuzzy CSP:

A totally ordered satisfaction scale.

Endpoints signify total violation/satisfaction.

Constraints modelled by fuzzy relations: map

variable assignments to points in the scale. Overall satisfaction determined by fuzzy

conjunction: the minoperator.


22/50

Two Algorithms

Dynamic version of a flexible CSP algorithm.

Using oraclesmethod.

Flexible version of a dynamic CSP algorithm. Flexible Local Changes.

Each algorithm has several variants, which do

an increasing amount of work to spot deadends early.


23/50

Flexible Local Changes

Complete Local repair algorithm.

Divides Variables into three sets:

1. Assigned & Fixed.

2. Assigned & Not Fixed.

3. Unassigned.

Uses these sets to control search

procedure.


24/50

Flexible Local Changes: Operation

Maintain aboundon best

solution found for each

sub-problem.

1. Assigned & Fixed.

2. Assigned & Not Fixed.

3. Unassigned.

1 2 3 1 2 3

1 2 3 1 2 3

1 2 3

1 2 3

Sub-sub problem

solved.

Sub-problemsolved.


25/50

Experiments

We wanted to investigate:

The structure of fuzzy rrDFCSPs.

The relative performance of the algorithms.

We varied:

Problem size.

Connectivity.

Proportion of variables related by a constraint.

Constraint tightness. Proportion of disallowed value combinations.

Satisfaction scale size.

The amount of changebetween instances.

Each sequence contains 10 problems.


26/50

Experiments: Measurements

Constraint checks.

Every time we query a constraint for the

satisfaction degree of an assignment.

Search nodes.

Every time we assign a value to a variable.

Size of search tree.

Solution stability.

Proportion of assignments that remain the same

between instances.


27/50

Results: Search Effort Mean over 3 sequences of 10 instances.

Scale size: 3, Change 1 constraint between instances.

Instance size: 20 variables, connectivity: 0.25

0

500

1000

1500

2000

2500

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Tightness

Checks

FLC

BB


28/50

Search Effort: Trends

Branch and bound finds solutions moreefficiently than FLC.

BB has a more rigid search structure.

This allows stronger inferences to be made at eachnode in the search tree.

Result: BB spots dead ends earlier.

Algorithm variants: Doing more work to spot dead ends guarantees a

smaller or equal-sized search tree.

But often costs more in terms of constraint checks.


29/50

Search Effort: Trends

Phase transition behaviour.

Increasing some parameters generally increases

difficulty:

Problem size.

Connectivity.

Size of satisfaction degree scale.

Amount of change between instances.


30/50

Results: Stability

Scale size: 3, Change 1 constraint between instances Problem size: 20 variables, connectivity: 0.25

90

91

92

93

94

95

96

97

98

99

100

0.1 0.2 0.3 0.4 0.5 0.6

Tightness

Stability

FLC

BB


31/50

Stability: Trends

FLC produces more stable solutions than BB. Both algorithms prefer stable assignments where

possible.

FLC actively seeks to leave areas of the previous

solution undisturbed.

Variants: More work to spot dead ends means

more stability.

Extra work gives us a better idea of the utility of

potential assignments.

Can reveal that they are no better than our

preferred stable assignment.


32/50

Utility of Dynamic Information

We tested `crippled versions of the algorithms.

Branch and bound loses its oracle.

FLC starts from an empty assignment. Crippled versions:

Explore significantly larger search trees.

Give significantly lower solution stability. Extent varies according to the particular dynamic

sequence.


33/50

Part II

Application to AI Planning


34/50

Plan: Course of action to achieve pre-specified

goals.

Components of a planning problem:

Plan objects.

Initial state.

Goal state.

Operators.

AI Planning

c1 c2 c4

c3

r

r2 r3

m1 m2

pkg1 pkg2

guard1


35/50

Characteristics of AI Planning

Inflexible operators.

Imperative goals.

Suffers from similar problems to classical CSP.

If no perfect solution, no plan returned.

We want to give the planner the ability to

compromise.


36/50

Flexible Planning

Incorporate preferences into operators and goals.

Model both via fuzzy relations:

Map from preconditions onto a totally ordered

satisfaction scale.

Load-truck operator has preconditions:

Truck and package in same place.

Guard must be present.

Can relax, with associated

damage to resultant plan

Imperative

Preference


37/50

Example

Goals:

Both packages to c4

.

pkg2is worth less, dont deliver: l1

Guard to c3.

Can also leave guard at c2or c4: l2

c1 c2 c4

c3

r1

r2 r3

m1 m2

pkg1 pkg2

guard1

L={l , l1, l2, lT}


38/50

Example

Operators:

Drive-truck.

Avoid mountains or: l1 Load/Unload-truck.

For valuable package, guard present or: l2 Guard-boards/leaves-truck.

c1 c2 c4

c3

r1

r2 r3

m1 m2

pkg1 pkg2

guard1

L={l , l1, l2, lT}


39/50

Graphplan

Solution procedure for classical planning problems. Constructs/analyses aplanning graph.

Forward phase extends planning graph until goals found.

Add binary mutexconstraints betweenactions/propositions that conflict.

Backward phase extracts valid plan.Inconsistent:

Preconditions

Effects

Initial

Conditions

Actions1 Propositions1

. . .

. . .

. . .


40/50

The Flexible Planning Graph

Actions annotated with their satisfaction degrees.

Actions1 Propositions1. . .

. . .

. . .

l2

l3

l1


41/50

The CSP Viewpoint

Variables: proposition nodes. Domains: actions who assert these propositions

as effects.

Constraints: Binary mutex + Unary fuzzy.Actions1 Propositions1

. . .

. . .

. . .

l2

l3

l1


42/50

Plan Synthesis via Fuzzy rrDFCSP

Goals, and their domains form a first sub-problem. Action pre-conditions specify new sub-problems

GoalSub-problem

Solutions at level nare likely to intersect.

So, problems at level n-1form a related sequence:

Each level is a DFCSP.


43/50

Guiding Overall Search

Goal: Solve as few sub-problems as possible. Generate memosetsfrom unsolvable sub-problems.

Propagate memosets up the level hierarchy.

Goal

Sub-problem


44/50

4-step Solution

1. Load-truckpkg1

truckl2

.

2. Drive-trucktruckc1to c2via r1 lT.

3. Drive-trucktruckc2to c4via m2 l1.

4. Unload-truckpkg1truckl2.

c1 c2 c4

c3

r1

r2 r3

m1 m2

pkg1 pkg2guard1

L={l , l1, l2, lT}

Satisfaction: l1


45/50

6-step Solution

1. Load-truckpkg1truckl2.


3. Load-truckpkg2trucklT.



6. Unload-truckpkg1truckl2,pkg2trucklT.

c1 c2 c4

c3

r1

r2 r3

m1 m2

pkg1 pkg2guard1

L={l , l1, l2, lT}

Satisfaction: l2


46/50

Compromise-free Solution

10 steps.

First collect the guard. Return to loadpkg1.

Use the main roads to deliver both packages

and guard.

c1 c2 c4

c3

r1

r2 r3

m1 m2

pkg1 pkg2

guard1L={l , l1, l2, lT}


47/50

Flexible Graphplan: Observations

It is more expensive to search for a range of

plans than for one compromise-free plan.

But it is often possible to find short,compromise plans quickly.

Supports anytimebehaviour.

Range of plans trade length versus numberand severity of the compromises made.


48/50

Conclusions

Dynamic flexible constraint satisfactioncombines two extensions to classical CSP.

Multiple instances of DFCSP. Considered fuzzy rrDFCSP in detail.

Two solution procedures (with variants).

Flexible planning.

Plan synthesis via hierarchical decompositionand fuzzy rrDFCSP.


49/50

Future Work

We have examined one instance of DFCSP

in detail.

Examine other instances. Modify existing solution techniques/consider

new ones (e.g. local search).

Other applications. Compositional Modelling [Keppens 2002].

Dynamic Scheduling.

Interactive configuration


50/50

Acknowledgements

Qiang Shen, University of Edinburgh.

Peter Jarvis, SRI International.

Dynamic Csp Thesis

Documents

Transcript of Dynamic Csp Thesis