Conjure Revisited: Towards Automated Constraint Modelling · Introduction Approach Conclusion...

IntroductionApproach

Conclusion

Conjure Revisited: Towards AutomatedConstraint Modelling

Ozgur Akgun1,Alan M. Frisch2, Brahim Hnich3, Chris Jefferson1, Ian Miguel1

1 School of Computer Science, University of St Andrews, UK2 Artificial Intelligence Group, Dept. of Computer Science, University of York, UK

3 Department of Computer Engineering, Izmir University of Economics, Turkey

ModRef 2010The 9th International Workshop on

Constraint Modelling and Reformulation

Ozgur Akgun, et al. Conjure Revisited: Towards Automated Constraint Modelling

Conclusion

1 IntroductionWhat is?EssenceEssence by exampleEssence′

1 ApproachThe taskThe pipelineNon-deterministic RewritingSome rulesMatching expressions, not constraints

1 ConclusionCoverage and limitationsConclusion and future work

Conclusion

What is?EssenceEssence by exampleEssence′

What is?

Essence: a high level problem specification language

Conjure: a tool to generate multiple CSP models given aproblem specification

Essence′: a solver independent, problem class level CSPmodelling language

Conclusion

What is?

Conclusion

What is?

Conclusion

What is?

Conclusion

What is?

Conclusion

Essence

A high level problem specification language

Supports many type constructors that allow problems to bespecified in natural ways

boolean, integer, enumeration, unnamed types,set, multi-set, function, relation, tuple,and arbitrary nestings of these type constructors

No CSP modelling decisions involved

Conclusion

Essence

Conclusion

Essence

Conclusion

Essence by example

Problem

given n distinct items, with associated weights and valuesselect a set out of these items maximising total valuesuch that the total weight is not more than that of you cancarry

Conclusion

Essence by example

given i t em : enumgiven w : funct ion i t em −> i n t (0 . . )given v : funct ion i t em −> i n t (0 . . )given cap : i n t (0 . . )

f i nd x : set of i t em

maximising sum i : x . v ( i )such that sum i : x . w( i ) <= cap

Conclusion

Essence′

Almost a subset of Essence

Operates on problem class level

Supports boolean and integer decision variables, andmulti-dimensional matrices

Supports several global constraints, in addition to commonarithmetic and logical ones

Tailor compiles efficient CSP models to multiple targetsolvers

MinionGecodeFlatZinc

Conclusion

Essence′

Conclusion

Essence′

Conclusion

Essence′

Conclusion

Essence′

Conclusion

Essence′

Conclusion

The taskThe pipelineNon-deterministic RewritingSome rulesMatching expressions, not constraints

The task

Compile Essence specifications to multiple Essence′ models

Compilation process needs to be easily modifiable

A term rewriting infrastructure supported by a set of rewriterules

Conclusion

The task

Conclusion

The task

Conclusion

The pipeline

Parsing

Type checking

Validating the input

Representations phase

Auto-Channelling phase

Adding structural constraints

Expression rewriting

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline

Parsing

Type checking

Presentation

Conclusion

The pipeline: Representations phase

One of the two kinds of modelling decisions

Selecting the viewpoint

Select a representation for every parameter and decisionvariable

Possible to represent a variable in multiple ways

if it appears in more than one constraint

Conclusion

2 representations for sets

Given: set (size 2) of int(1..3)

Explicit representation:matrix indexed by [int(1..2)] of int(1..3)Occurrence representation:matrix indexed by [int(1..3)] of bool

Conclusion

Given: set (size 2) of int(1..3)Explicit representation:matrix indexed by [int(1..2)] of int(1..3)

Occurrence representation:matrix indexed by [int(1..3)] of bool

Conclusion

Given: set (size 2) of int(1..3)Explicit representation:matrix indexed by [int(1..2)] of int(1..3)Occurrence representation:matrix indexed by [int(1..3)] of bool

Conclusion

Example

given lb , ub , n : i n tgiven s : set of i n t ( l b . . ub )f i nd x : set ( s i z e n ) of i n t ( l b . . ub )

such thatx subseteq s ,f o r a l l i : x . k ( i )

Conclusion

Example

given lb , ub , n : i n tgiven s : set of i n t ( l b . . ub )f i nd x o c c r : set ( s i z e n ) of i n t ( l b . . ub )

such thatx o c c r subseteq s ,f o r a l l i : x o c c r . k ( i )

Conclusion

Example

given lb , ub , n : i n tgiven s : set of i n t ( l b . . ub )f i nd x e x p l : set ( s i z e n ) of i n t ( l b . . ub )

such thatx e x p l subseteq s ,f o r a l l i : x e x p l . k ( i )

Conclusion

Example

given lb , ub , n : i n tgiven s : set of i n t ( l b . . ub )f i nd x ? : set ( s i z e n ) of i n t ( l b . . ub )

such thatx o c c r subseteq s ,f o r a l l i : x e x p l . k ( i )

Conclusion

Example

given lb , ub , n : i n tgiven s : set of i n t ( l b . . ub )f i nd x e x p l : set ( s i z e n ) of i n t ( l b . . ub )f i nd x o c c r : set ( s i z e n ) of i n t ( l b . . ub )

such thatx o c c r subseteq s ,f o r a l l i : x e x p l . k ( i )

Conclusion

Example

such thatx o c c r subseteq s ,f o r a l l i : x e x p l . k ( i ) ,x o c c r = x e x p l

Conclusion

The pipeline: Auto-Channelling phase

More than one representation for a decision variable=>

pairwise equality constraints!

Conclusion

The pipeline: Adding structural constraints

Now, representations for decision variables are known

“Structural constraints” are added to the model

an alldiff constraint for x_expla cardinality constraint for x_occr

Conclusion

Example, structural constraints added

such thatx o c c r subseteq s ,f o r a l l i : x e x p l . k ( i ) ,x o c c r = x e xp l ,{ a l l d i f f on x e xp l ’ s r e f i n emen t } ,{ c a r d i n a l i t y on x occ r ’ s r e f i n emen t }

Conclusion

Example, final

g iven lb , ub , n : i n tg i ven s o c c r : mat r i x i ndexed by [ i n t ( l b . . ub ) ] of boo lf i n d x e x p l : mat r i x i ndexed by [ i n t (1 . . n ) ] of i n t ( l b . . ub )f i n d x o c c r : mat r i x i ndexed by [ i n t ( l b . . ub ) ] of boo l

such thatf o r a l l i : i n t ( l b . . ub ) . x o c c r [ i ] <= s o c c r [ i ] ,f o r a l l i : i n t (1 . . n ) . k ( x e x p l [ i ] ) ,f o r a l l i : i n t (1 . . n ) . x o c c r [ x e x p l [ i ] ] = 1 ,f o r a l l i : i n t ( l b . . ub ) . (

x o c c r [ i ] => e x i s t s j : i n t (1 . . n ) . x e x p l [ j ] = i) ,a l l d i f f ( x e x p l ) ,sum i : i n t ( l b . . ub ) . x o c c r [ i ] = n

Conclusion

Non-deterministic Rewriting

Given a set of rewrite rules and a starting term, apply therules repeatedly.

normal form

More than one rule can match a term.

Select one at random?Apply all matching rules? (produces a list of terms)

Conclusion

normal form

Conclusion

normal form

Conclusion

normal form

Select one at random?

Apply all matching rules? (produces a list of terms)

Conclusion

normal form

Conclusion

Rule representation

Rules are represented as directed equations with guards.

A * B -> A + A, if B is 2A / B -> A , if B is 1

This is a partial mapping.

Can combine multiple such rules, into a one-to-manymapping.

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Handle just one rule.

Conclusion

Rule representation

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Conclusion

Rule representation

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Conclusion

Rule representation

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Conclusion

Rule representation

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Conclusion

Rule representation

Subterms: {A,B,C,D}

rule1: A to B

rule2: A to C

rule3: B to D

allRules

{ (A, {B,C}) , (B, { D }) , (C, { C }) , (D, { D }) }

Conclusion

Some example rules

essence_expression equivalent_expression

guards: properties that essence_expression must satisfy

declarations: newly created variables and

local aliases for expressions

Conclusion

Example rules: ruleSetEq

a = b a subseteq b /\ b subseteq a

guards: a ∼ set of τ,b ∼ set of τ

Conclusion

Example rules: ruleSetSubsetEq

a subseteq b forall i : a . i elem b

guards: a ∼ set of τ,b ∼ set of τ

Conclusion

Example rules: ruleSetElem

e elem s exists i : s . i = e

guards: e ∼ τ,s ∼ set of τ

Conclusion

Example rules: ruleSetQuan

forall i : (a union b) . k forall i : a . k /\ forall i : b . k

exists i : (a union b) . k exists i : a . k \/ exists i : b . k

forall i : (a intersect b) . k forall i : a ( i elem b => k )

forall i : (a intersect b) . k forall i : b ( i elem a => k )

exists i : (a intersect b) . k exists i : a ( i elem b /\ k )

exists i : (a intersect b) . k exists i : b ( i elem a /\ k )

Conclusion

Matching expressions, not constraints

Some other tools reason about complete constraints

including previous implementations of Conjure

Now, can match with any subexpression, not necessarilycomplete constraints

Can work on a non-flat modelAble to reason about structureDo more with fewer rules

Conclusion

Can work on a non-flat model

Able to reason about structureDo more with fewer rules

Conclusion

Can work on a non-flat modelAble to reason about structure

Do more with fewer rules

Conclusion

Consider: (a union b) subseteq c

Flattened: aux subseteq c /\ aux = a union b

We could have: a subseteq c /\ b subseteq c

Conclusion

Flattening = lost information

We can still flatten things, but only if we want, using ourpowerful rewrite rules!

A small problem, where to put helper constraints?

Conclusion

given lb , ub , n ,m, k : i n tf i nd t : set ( s i z e n ) of i n t ( l b . . ub )f i nd A : set ( s i z e n ) set ( s i z e m) of i n t ( l b . . ub )

such thatf o r a l l s : A .

(max( s ) − max( t ) = k ) => ( k elem s )

Conclusion

@: the bubble attaching operator

max(s) max_s @ bubble

guards: s ∼ set of int

declarations: max_s : int

bubble = (max_s elem s) /\ (forall i : s . i <= max_s)

Conclusion

forall s: A .

(max(s) - max(t) = k) => (k elem s)

forall s: A .

((max_s@bubble_s) - max(t) = k) => (k elem s)

forall s: A .

((max_s@bubble_s) - (max_t@bubble_t) = k) => (k elem s)

forall s: A .

(((max_s-max_t) @ (bubble_s /\ bubble_t))=k) => (k elem s)

forall s: A .

(((max_s-max_t=k) @ (bubble_s /\ bubble_t))) => (k elem s)

forall s: A .

(((max_s-max_t=k) => (k elem s)) @ (bubble_s /\ bubble_t))

forall s: A .

(((max_s-max_t=k) => (k elem s)) /\ bubble_s /\ bubble_t)

bubble_t /\ forall s: A .

(((max_s-max_t=k) => (k elem s)) /\ bubble_s)

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

forall s: A .

Conclusion

Coverage and limitationsConclusion and future work

Coverage of Essence

Essence has 7 core type constructors

matrix, set, mset, partition, tuple, function, relation

Type constructors supported: all except partition

Also we haven’t yet implemented support for enumerated andunnamed types

There are nearly 30 operators defined on these typeconstructors

Almost all of them implemented

Conclusion

Coverage of Essence

Conclusion

Coverage of Essence

Conclusion

Coverage of Essence

Conclusion

Coverage of Essence

Conclusion

Differences to the prototype implementation

Broader coverage of Essence

Representation decisions

Auto-channelling becomes very easy

No flattening

Easier rule authoring

Conclusion

No flattening

Conclusion

No flattening

Conclusion

No flattening

Conclusion

No flattening

Conclusion

Conclusion and future work

New version of Conjure with far greater coverage of theEssence language

Immediate future work, covering all of the types andoperations in Essence

Capture best modelling practices

Model selection

Investigate multi-model search techniques

Conclusion

Model selection

Conclusion

Model selection

Conclusion

Model selection

Conclusion

Model selection

Conjure Revisited: Towards Automated Constraint Modelling · Introduction Approach Conclusion...

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