the rules of modelling automatic generation of constraint programs through refinement
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The Rules of Modelling Automatic Generation of Constraint Programs through Refinement. Artificial Intelligence Group Dept of Computer Science University of York. School of Computer Science University of St Andrews. Alan M. Frisch Ian Miguel - PowerPoint PPT PresentationTRANSCRIPT
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Alan M. Frisch Ian Miguel
Joint work with Matt Grum, Chris Jefferson & Bernadette Martinez Hernandez
The Rules of ModellingAutomatic Generation of Constraint
Programs through Refinement
Artificial Intelligence Group
Dept of Computer Science
University of York
School of Computer Science
University of St Andrews
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Allocating a Divisible Resource:An Application to Grid Computing
Alan Frisch
Artificial Intelligence Group
Dept. Computer Science
University of York
4pm, CS103
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Modelling Bottleneck
• Useful for solving a wide range of important, complex problems including scheduling, allocation, layout, configuration,…
• Modelling a problem as a constraint program requires moderate/great expertise.
• Major barrier to widespread use of CP.
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What are CP researchers doing?
• Model Generation:– Done in private.
• Model Selection:– Case studies– Some generalisations from analysis
& experiments
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Reducing the Modelling Bottleneck:
Automated Modelling• 2 to 3 years: We have been generating
models more and more systematically.• Embed systematic viewpoint in
automated system (CONJURE).– Takes specification in high-level language
(ESSENCE).– Reformulates specification into a set of
correct constraint programs.
• Ongoing: add heuristics for model selection.
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Outline
• Reduced constraint programming.• ESSENCE.• CONJURE.• The big picture.
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Part IReduced Constraint
Programming
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What is Constraint Programming?
• Broad view: programming in which constraints play a central role.
• Narrow view: solve combinatorial (optimisation) problem by • mapping it to a constraint satisfaction
problem• solving the constraint satisfaction
problem• mapping solution back to original
problem
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What is the (finite domain) CSP?
An instance comprises:• Finite set of variables• Each associated with a finite domain• Finite set of constraints on the
values taken by the variables• (Objective function)Solution is assignment of values to
variables that satisfies the constraints (and optimises objective function)
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Modelling: Just Do It!
Find 3 distinct non-zero digits that sum to 9.
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Just Do It!
Find 3 distinct non-zero digits that sum to 9.
• Variables: X, Y, Z
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Just Do It!
Find 3 distinct non-zero digits that sum to 9.
• Variables/Domains: X, Y, Z::{1,..9}
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Just Do It!
Find 3 distinct non-zero digits that sum to 9.
• Variables/Domains: X, Y, Z::{1,..9}• Constraints: X Y, Y Z, X Z, X+Y+Z=9
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Just Do It!
Find 3 distinct non-zero digits that sum to 9.
• Variables/Domains: X, Y, Z :: {1..9}• Constraints: X Y, Y Z, X Z, X+Y+Z=9
X Y Z
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Just Do It!
Find 3 non-zero digits that sum to 9.• Variables/Domains: X, Y, Z :: {1..9}• Constraints: X Y, Y Z, X Z, X+Y+Z=9
X Y Z X < Y < Z
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Just Do It for a Problem
Given n and s, find n distinct non-zero digits that sum to s.
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Model: Explicit Representation
Given n and s, find n distinct non-zero digits that sum to s.
givenfindsuch that
n:nat, s:natX: matrix (indexed by 1.. n) of 1..9?
1 nX
2 31..9 1..9 1..9…
…
1..9
AllDiff(X)
i1..n X[i] = s
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Model: Occurrence Representation
givenfindsuch that
n:nat, s:natD: matrix (indexed by 1..9) of 0..1?
Given n and s, find n distinct non-zero digits that sum to s.
10/1 …
9D 0/1 0/1
2 30/1
…
D[1] + 2D[2] + … + 9D[9] = sD[1] + D[2] + … + D[9] = n
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The SONET Problem
• Given nrings rings, nnodes nodes, a set of pairs of nodes (communication demand) and an integer capacity (of each ring). Install nodes on rings satisfying demand and capacity constraints. Minimise installations.
• nrings=2, nnodes=5, capacity = 4
• demand: n1 & n3, n1 & n4, n2 & n3, n2 & n4, n3 & n5
Specification
Instance
Solution
n2 n1
n3n4 n5
n3
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A Model of SONET Problem
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
Nodes
Ringsrings-nodes
rings-nodes[r,n] = 1iff
node n is installed on ring r.
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A Model of SONET Problem
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
0..1 0..1 0..1 0..1
Nodes
Ringsrings-nodes
c
ScalarProduct > 0
c…
Minimise:
rings-nodes[r,n]r Rings n Nodes
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Part II
Abstract Problem Specifications
The ESSENCE Language
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ESSENCE
• The language enables problems to be specified at a level of abstraction above that at which modelling decisions are made.
• This requires features not found in current constraint programming languages.
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SONET Specification
given
letting
given find
minimising
such that
nrings:nat, nnodes:nat, capacity:nat
Nodes be 1..nnodes
demand: set of set (size 2) of Nodes
rings: mset (size nrings) of set (maxsize capacity) of Nodes
r rings .|r|
pair demand . r rings . pair r
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ESSENCE Provides Abstract Types
Sets, Multisets, Partitions, Range Types, Unnamed TypesFunctions, Relations, Enumerated Types, Sequences…
given
letting
given find
minimising
such that
nrings:nat, nnodes:nat, capacity:nat
Nodes be 1..nnodes
demand: set of set (size 2) of Nodes
rings: mset (size nrings) of set (maxsize capacity) of Nodes
r rings .|r|
pair demand . r rings . pair r
Abstract types
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ESSENCE SupportsArbitrarily-Nested Types
given
letting
given find
minimising
such that
nrings:nat, nnodes:nat, capacity:nat
Nodes be 1..nnodes
demand: set of set (size 2) of Nodes
rings: mset (size nrings) of set (maxsize capacity) of Nodes
r rings .|r|
pair demand . r rings . pair r
Arbitrary nesting of types
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ESSENCE Supports Quantification over Decision
Variables
given
letting
given find
minimising
such that
nrings:nat, nnodes:nat, capacity:nat
Nodes be 1..nnodes
demand: set of set (size 2) of Nodes
rings: mset (size nrings) of set (maxsize capacity) of Nodes
r rings .|r|
pair demand . r rings . pair r
QuantifyingOver Decision
Variables
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How Usable is ESSENCE?
• Specifications of ~50 problems found in the CSP literature written by an undergraduate with no background in constraint programming.
• URL: http://www.cs.york.ac.uk/aig/constraints/
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ESSENCE’
• ESSENCE’ has a similar level of abstraction to existing constraint languages.
ESSENCE’ = ESSENCE -
Abstract types
Arbitrary nesting of types
QuantifyingOver Decision
Variables
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Model Generation: Informal
Given n and s, find n distinct non-zero digits that sum to s.
givenfindsuch that
n:nat, s:natX: matrix (indexed by 1.. n) of 1..9AllDiff(X)i1..n X[i] = s
givenfindsuch that
n:nat, s:natD: matrix (indexed by 1..9) of 0..1D[1] + 2D[2] + … + 9D[9] = sD[1] + D[2] + … + D[9] = n
ESSENCE’
ESSENCE’
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givenfind
n:nat, s:intX: set (size n) of 1..9x = sx X
such that
Formalisation of Model Generation
givenfindsuch that
n:nat, s:natX: matrix (indexed by 1.. n) of 1..9AllDiff(X)i1..n X[i] = s
givenfindsuch that
n:nat, s:natD: matrix (indexed by 1..9) of 0..1D[1] + 2D[2] + … + 9D[9] = sD[1] + D[2] + … + D[9] = n
ESSENCE’
ESSENCE’
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Part III
Generating Models:The CONJURE System
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A Theory of Model Generation Must Account
for
• Representing complex decision variables.• SONET involves finding a multiset of sets.
• Exploiting channelling.• Generate, maintain multiple representations.
• Identifying and breaking symmetries.• X+Y+Z = c X <= Y <= Z
• Performing transformations to improve efficiency.• X<=Y<=Z, AllDiff(X,Y,Z) X < Y < Z
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CONJURE
• CONJURE automatically generates a set of alternative ESSENCE’ models from ESSENCE specifications.
• Core: set of refinement rules.• Refine an ESSENCE expression into a set of
ESSENCE’ expressions.• Produce a set because often many
alternative refinements of an ESSENCE expression.
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Rules Must be Compositional
• Compositional refinement: • Refinement of an expression is composed
of the refinements of its sub-expressions.
• Necessary to handle:• Unbounded nesting of constraints (usual)
• |A (B C) | < X
• Unbounded nesting of types (unique)• set, set of sets, set of set of sets, …
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Challenge: Compositionality is Difficult
• Complex variables: can have multiple refinements. Not all support all operations.
• Nested variables: refinement of outer type must be determined without looking arbitrarily deep into the nesting.
• Operators: refinement depends on refinement of operands.
• Operators: recursion needed to handle arbitrary nesting of operands, e.g.
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Refinement via Operator
• Function mapping ESSENCE expression to a set of ESSENCE’ expressions.
• Defined by set of equalities.• One per syntactic construct.
• For easy presentation, each equation broken into clauses:• RuleName1 (e) e1’
• RuleName2 (e) e2’
• …
• RuleNamen (e) en’
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Refining a Specification
• Refine constraints & objective function in turn.• Compose results to form final model.
• Multiple refinements composed = multiple models.
givenlettingfindsuch that
nnodes:nat, capacity:natNodes be 1..nnodes
4 ring
MicroSonet:
4 ring : set (maxsize capacity) of Nodes
ring : set (size capacity) of Nodes
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Refining MicroSonet
• Refinement begins with (4 ring):• Relevant clause of Element rule refines
both sides, composes the results.
givenlettingfindsuch that
nnodes:nat, capacity:natNodes be 1..nnodesring : set (size capacity) of Nodes
4 ring
(ring)(4) = {4}
Compositionality
at work
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FixedSizeSet1: Occurrence Rep
FixedSizeSet1 (S: set (size n) of lb..ub )
= n
such that sum(S’ )= n
{
|}
• Generates a 1-d matrix, S’
Indexed by lb..ub0/1 0/1 0/1 0/1 0/1 …
S’ = genSym(S, matrix (indxd by lb..ub) of bool)
S’
true
• Associate an axiom schema with this rule:• i . i S S’ [i]
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ESSENCE’ Occ Model: MicroSonet
givenlettingfindsuch that
nnodes:nat, capacity:natNodes be 1..nnodes
ring’ : matrix (indexed by Nodes) of boolsum(ring’ ) = capacityring’ [4]=1
0/1 0/1 0/1 0/1…ring’1 2 3 nnodes
givenlettingfindsuch that
nnodes:nat, capacity:natNodes be 1..nnodesring :set (size capacity) of Nodes
4 ring
ESSENCE’
ESSENCE
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FixedSizeSet2: Explicit RepFixedSizeSet2 (S: set (size n) of )
{
}
|
• S’ refined to give fully refined matrix S’’
S’’
S’’ (S’ )
S’’ : matrix (indexed by 1..n) of _
such that
(AllDiff(S’ ))
• All-different constraint refined, added to local model.
AllDiff(S’ )
S’ = genSym(S, matrix (indexed by 1..n) of )
• Generates intermediate 1-d matrix, S’ :
1 nS’
2 3
…
…
• Axiom schema: i . i S j 1..n (S’ [j]=i )
NB could be arbitrarily complex
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ESSENCE’ Explicit Model: MicroSonet
nnodes:nat, capacity:natNodes be 1..nnodes
ring’:matrix (indexed by 1..capacity) of NodesAllDifferent(ring’ )ring’ [1] = 4 OR ring’ [2] = 4 OR…
1Nodes
capacity2ring’ Nodes Nodes…
givenlettingfindsuch that
nnodes:nat, capacity:natNodes be 1..nnodesring :set (size capacity) of Nodes
4 ring
ESSENCE
givenlettingfind
such that
ESSENCE’
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Channelling Models
• Good constraint models often contain multiple representations of same abstract variable.
• Different constraints can often be stated most effectively on different representations.
• We use channelling constraints to maintain consistency between the representations.
• CONJURE can generate these constraints automatically.
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Automatically GeneratingChannelling Models
S:set (size n) of
S1’ :matrix (indxd by ) of bool S2’ :matrix (indxd by 1..n) of
(1) i (i S S1’ [i])(2) i (i S j 1..n (S2’ [j]=i ))
(1) (2)
(3) i (S1’ [i] j 1..n (S2’ [j]=i ))
(3)
FixedSizeSet1 FixedSizeSet2
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(Some) Symmetry Detection is Free!
• Some refinements, whether manual or automatic, introduce symmetry.
• Some rule clauses introduce symmetry every time they are used.
• The clause can annotate the refined expression with a description of the symmetry.
• E.g. explicit representation of a set of 3 integers drawn from 1..n.
1..n1 2 3
• Have named the indices, where the set did not.
• Introduced index symmetry.
1..n 1..n
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Part IV
Conclusion: The Big Picture
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What we Hope to Achieve
• Much of modelling can be formalised as refining ESSENCE to ESSENCE’• Reduce modelling bottleneck with CONJURE.
• Ongoing work:• Develop heuristics to select among
competing models.
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E
Why Might we Fail to Build a Competent Automatic
Modeller?• Can’t get compositionality to work.• Some/much of modelling is involved in
writing ESSENCE specs.
M4M3
E2
M2M1
E1
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Why Might we Fail to Build a Competent Automatic
Modeller?• Can’t get compositionality to work.• Some/much of modelling is involved in
writing ESSENCE specs.• Failure to reach closure:
– Endless list of new modelling steps to encapsulate in rules.
• General mathematical reasoning needed to perform transformations:– Adding implied constraints.– Obtaining rule applicability.– Obtaining more effective models.
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Benefits other than Automation
• A more systematic approach to modelling.– We’ve noticed missing combinations.– Reveal gaps in understanding.
• Generalisation/Patterns.– See a technique used in generating one model
and recognise its general usefulness.• Pedagogical purposes.
– Modelling almost absent from existing textbooks.
• ESSENCE could prove useful.• We look at modelling differently.
– Eventually, so will others.
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• ESSENCE specifications and CONJURE paper:– http://www.cs.york.ac.uk/aig/constraints/