Answer Set Programming vs CSP: Power of Constraint Propagation Compared Jia-Huai You ( 犹嘉槐 ) University of Alberta Canada

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

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)

Blocks World Planning AB C TABLE

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.

Example: 8-Queens Generate-and-test, 8 8 combinations

Example: Map Coloring WA NT SA Q NSW V T WA NT SA Q NSW V T

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

Approaches to Constraint Programming Constraint Satisfaction Problem (CSP) Systems: Constraint Logic Programming Propositional Satisfiability (SAT) Systems: SAT solvers Answer Set Programming (ASP) Systems: Smodels, …

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.

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).

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

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

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

Constraint Satisfaction Problem(CSP) CSP Variable set Domain set Constraint set Instantiation

A CSP x y z support consistent

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. i=2 arc-consistency (AC) i=3 path-consistency (PC)

Maintaining AC x y z \ \ \ \ \ \

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

Lookahead in Smodels  Answer set program  Stable model (answer set) of

Lookahead Expand ( ) conflict

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.

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.

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

Direct Encoding Uniqueness rules Denial rules

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

Example xy z  Enforcing AC cannot remove any value.  How about lookahead?

Unique value propagation(UP) Generate an superset of which is

Theorem 1 lookahead=AC+UP. Propagation Arc- Consistency(PAC)

Theorem 2 (i-1)-lookahead i-consistency

Support Encoding Uniqueness rules The same as direct encoding Support rules

Under support encoding Does the pruning power of lookahead equal to that of PAC?

Singleton arc-consistency(SAC)

Theorem 3 and 4 Lookahead=SAC SAC > PAC

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

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

Relationships Niemela ’ s encoding Uniquness rules The same as direct encoding Allowed rules Lookahead has the same pruning power under the direct encoding as under Niemela ’ s encoding

Relationships Performance of the direct and support encodings. Problem set 10 variables Domain size of constraints

Experiments

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

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

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

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

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