Answer Set Programming vs CSP: Power of Constraint Propagation Compared

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Answer Set Programming vs CSP Power of
Constraint Propagation Compared

• Jia-Huai You (???)
• University of Alberta
• Canada

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Constraint Programming

• Studies computational models for solving problems
  that can be expressed as constraints
• Applications
• -NP-hard problems
• -Optimization
• -Planning (PSPACE-complete), logistics,
  scheduling
• -Combinatorial problems

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Approaches to Constraint Programming

• Constraint Satisfaction Problem (CSP)
• (Maybe embedded in a programming language
  such as Constraint Logic Programming)
• Propositional satisfiability (SAT)
• Answer set programming (ASP)

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Blocks World Planning
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Planning Example Logistics World

• There are two types of vehicles trucks and
  airplanes. Trucks are used to transport packages
  within a city, and airplanes to transport
  packages between airports.
• The problem in this domain starts with a
  collection of packages at various locations in
  various cities, and the goal is to redistribute
  these packages to their destinations.

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Example 8-Queens
Generate-and-test, 88 combinations
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Example Map Coloring
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Cryptarithmetic puzzles

• The problem is to assign distinct (decimal)
  digits to letters so that adding two words yields
  the third. E.g.
• S E N D M O R E M O N E Y
• D O N A L D G E R A L D R O B E R T

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Approaches to Constraint Programming

• Constraint Satisfaction Problem (CSP)
• Systems Constraint Logic Programming
• Propositional Satisfiability (SAT)
• Systems SAT solvers
• Answer Set Programming (ASP)
• Systems Smodels,

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Graph Coloring in SAT (or ASP)

• Write clauses (or rules in ASP) that express the
  constraints
• Any region must be colored with exactly one (but
  any) color.
• No model (or answer set in case of ASP) may have
  two (distinct) adjacent regions colored with the
  same color.

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Graph Coloring in ASP

• coloring(N,C) ? node(N),color(C),
• not other_color(N,C).
• other_color(N,C) ? node(N),color(C),
• color(C), C\C,
  coloring(N,C).
• hasColor(N) ? coloring(N,C), color(C).
• ? node(N),node(N),color(C), edge(N,N),
• coloring(N,C), coloring(N,C).

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Contents

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in ASP
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Motivation

• Constraint Satisfaction Problem(CSP)
• Maintaining local consistency
• SAT solver or answer set programming
• Unit propagation, lookahead
• Pruning power comparison

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Structure

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in ASP
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Constraint Satisfaction Problem(CSP)

• CSP
• Variable set
• Domain set
• Constraint set
• Instantiation

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A CSP
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i-consistency

• A CSP is i-consistent iff given any consistent
  assignment of any i-1 variables, there exists an
  assignment of any ith variable such that the i
  values taken together satisfy all of the
  constraints among the i variables.
• i2 arc-consistency (AC)
• i3 path-consistency (PC)

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Maintaining AC
x
y
\
\
z
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Structure

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in Smodels
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Lookahead in Smodels

• Answer set program

• Stable model (answer set) of

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Lookahead
conflict
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Propagation rules

• 1. Adds the head of a rule to A if the
  body is true in A.
• 2. If there is no rule with h as the head whose
  body is not false w.r.t A, then add not h to A.

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Propagation rules

• 3. If h belongs to A, the only rule with
  h as the head must have its body true.
• 4. If the head of a rule is false in A, the body
  must be false.

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Structure

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in Smodels
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Direct Encoding

• Uniqueness rules

• Denial rules

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Lookahead vs AC

• The solution of a CSP corresponds to the stable
  model of the ASP.
• Does Lookahead AC?

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Example

• Enforcing AC cannot remove any value.

• How about lookahead?

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Unique value propagation(UP)
which is

• Generate an superset of

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Theorem 1

• lookaheadACUP.

Propagation Arc-Consistency(PAC)
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Theorem 2

• (i-1)-lookahead i-consistency

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Support Encoding

• Uniqueness rules
• The same as direct encoding

• Support rules

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Under support encoding

• Does the pruning power of lookahead equal to that
  of PAC?

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Singleton arc-consistency(SAC)
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Theorem 3 and 4

• LookaheadSAC
• SAC gt PAC

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Theorem 5

• 2-lookahead SRPC
• Restricted path consistency(RPC)
• Check the unique support pair
• Singleton RPC(SRPC)
• Restrict the domain to be a single value

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Structure

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in Smodels
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Relationships

• Niemelas encoding
• Uniquness rules
• The same as direct encoding
• Allowed rules

• Lookahead has the same pruning power under the
  direct encoding as under Niemelas encoding

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Relationships

• Performance of the direct and support encodings.
• Problem set
• 10 variables
• Domain size of 30
• 20 constraints

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Experiments
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Structure

• Motivation
• Maintaining local consistencies in CSP
• Lookahead in Smodels
• Pruning power comparison
• Relationships between encodings
• Conclusions

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Conclusions

• When testing single literal, lookahead has the
  same prunning power under the direct encoding as
  well as Niemelas encoding.
• When Testing (i-1)-tuples, lookahead captures
  i-consistency under direct encoding.

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Conclusions

• Under the support encoding, lookahead has the
  same pruning power as SAC.
• Under support encoding, when testing pair of
  literals, lookahead captures SRPC.

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Conclusions

• Lookahead performs more efficiently under the
  support encoding than under the direct encoding.

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• Thank You!

Comments and Questions?