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The Use and Implementation of Global Constraints in B-Prolog

The Use and Implementation of Global Constraints in B-Prolog. Neng-Fa Zhou CUNY Brooklyn College and Graduate Center. Outline. Global constraints element, all_distinct, circuit, cumulative Implementation language Action rules Implementation of propagators with AR

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The Use and Implementation of Global Constraints in B-Prolog

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  1. The Use and Implementation of Global Constraints in B-Prolog Neng-Fa Zhou CUNY Brooklyn College and Graduate Center Neng-Fa Zhou at CSPAST'10 in Fukuoka

  2. Outline • Global constraints • element, all_distinct, circuit, cumulative • Implementation language • Action rules • Implementation of propagators with AR • element, all_distinct, circuit, cumulative • Parallelization of propagators • all_distinct Neng-Fa Zhou at CSPAST'10 in Fukuoka

  3. The element Constraint • element(I,L,X) • L=[X1,X2,…,Xn]. The Ith element of L is X, counting from 1. L[I]=X. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  4. Example An Assignment Problem A factory has four workers w1,w2,w3,w4 and four products p1,p2,p3,p4. The problem is to assign workers to products so that each worker is assigned to one product, each product is assigned to one worker, and the profit maximized. The profit made by each worker working on each product is given in the matrix. Profit matrix is: Neng-Fa Zhou at CSPAST'10 in Fukuoka

  5. A Solution That Uses element go:- Vars=[W1,W2,W3,W4], Vars :: 1..4, all_distinct(Vars), element(W1,[7,1,3,4],Val1), element(W2,[8,2,5,1],Val2), element(W3,[4,3,7,2],Val3), element(W4,[3,1,6,3],Val4), Profit #= Val1+Val2+Val3+Val4, maxof(labeling(Vars),Profit), writeln(Profit). Neng-Fa Zhou at CSPAST'10 in Fukuoka

  6. The all_distinct(L) Constraint • all_distinct([V1,V2,…,Vn]) • The elements V1, V2, …, Vn are pairwise different. For all i,j1..n (i<j) Vi Vj. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  7. Example-1Graph Coloring • Model-1 (neq) • For each two neighbors i and j, CiCj • Model-2 (all_distinct) • For each complete subgraph {i1,i2,…,ik}, all_distinct([Ci1, Ci2,…, Cik]) • post_neqs(Neqs) in B-Prolog Neng-Fa Zhou at CSPAST'10 in Fukuoka

  8. Example-1 (cont.)Benchmarking Results Source of benchmarks: Agostino Dovier, Andrea Formisano, and Enrico Pontelli, A comparison of CLP(FD) and ASP solutions to NP-complete problems, ICLP’05. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  9. Example-2The N-Queens Problem • Model-1 (neq) • For i,j1..n (i < j) Qi Qj Qi-Qj (j-i) Qj-Qi (j-i) • Model-2 (all_distinct) • all_distinct([Q1,…,Qn]),all_distinct([Q1-1,Q2-2,…,Qn-n]),all_distinct([Q1+1,Q2+2,…,Qn+n]) Neng-Fa Zhou at CSPAST'10 in Fukuoka

  10. Example-2 (cont.)Encodings in B-Prolog • Model-1 • Model-2 queens(N,Qs):- length(Qs,N), Qs in 1..N, foreach(I in 1..N-1, J in I+1..N, (Qs[I] #\= Qs[J], abs(Qs[I]-Qs[J]) #\= J-I)). queens(N,Qs):- length(Qs,N), Qs in 1..N, Qs1 @= [Qs[I]-I : I in 1..N], Qs2 @= [Qs[I]+I : I in 1..N], all_distinct(Qs),all_distinct(Qs1), all_distinct(Qs2). Neng-Fa Zhou at CSPAST'10 in Fukuoka

  11. The circuit(L) Constraint • Let L=[X1,…,Xn], where Xi1..n. An assignment (X1/a1,…,Xn/an) satisfies this constraint if {1 a1 ,…,n an} forms a Hamiltonian cycle. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  12. Example-1The TSP Problem tsp(Vars):- Vars=[V1,V2,…,V8], V1 :: [2,3,4], V2 :: [1,3,6,7], … V8 :: [5,6,7], circuit(Vars). Neng-Fa Zhou at CSPAST'10 in Fukuoka

  13. Example-2The Knight Tour Problem ktour(Vars):- Vars=[V1,V2,…,V64], V1 :: [11,18], … V64 :: [47,54], circuit(Vars). Neng-Fa Zhou at CSPAST'10 in Fukuoka

  14. The cumulative Constraint • cumulative(Starts,Durations,Resources,Limit) Starts = [S1,…,Sn], Durations = [D1,…,Dn], Resources = [R1,…,Rn], The resource limit cannot be exceeded at any time • When Resources=[1,…,1] and Limit=1serialized(Starts,Durations) Neng-Fa Zhou at CSPAST'10 in Fukuoka

  15. Example Disjunctive Scheduling • Model-1: use disjunctive constraints • Model-2: use global constraints • post_disjunctive_tasks(Disjs) where Disjs=[disj_tasks(S1,D1,S2,D2),…] • B-Prolog converts disjunctive constraints into serialized VV5+97#=<VV17#\/VV17+52#=<VV5, VV5+97#=<VV26#\/VV26+59#=<VV5, VV5+97#=<VV32#\/VV32+41#=<VV5, VV5+97#=<VV49#\/VV49+63#=<VV5, ….. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  16. Disjunctive Scheduling • Benchmarking results Source of benchmarks:www.cril.univ-artois.fr/~lecoutre/research/benchmarks/benchmarks.html Neng-Fa Zhou at CSPAST'10 in Fukuoka

  17. Outline • Global constraints • element, all_distinct, circuit, cumulative • Implementation language • Action rules • Implementation of propagators with AR • element, all_distinct, circuit, cumulative • Parallelization of propagators • all_distinct Neng-Fa Zhou at CSPAST'10 in Fukuoka

  18. Action Rules • Events • Instantiation: ins(X) • Domain • bound(X), dom(X), dom(X,E), dom_any(X), and dom_any(X,E) • Time : time(X) • GUI • actionPerformed(X), mouseClicked(X,E)… • General • event(X,O) Agent, Condition, {EventSet} => Action Neng-Fa Zhou at CSPAST'10 in Fukuoka

  19. Applications of Action Rules • Lazy evaluation • Constraint propagators freeze(X,G), var(X), {ins(X)} => true. freeze(X,G) => call(G). 'X in C-Y_ac'(X,Y,C),var(X),var(Y), {dom(Y,Ey)} => Ex is C-Ey, domain_set_false(X,Ex). 'X in C-Y_ac'(X,Y,C) => true. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  20. Outline • Global constraints • element, all_distinct, circuit, cumulative • Implementation language • Action rules • Implementation of propagators with AR • element, all_distinct, circuit, cumulative • Parallelization of propagators • all_distinct Neng-Fa Zhou at CSPAST'10 in Fukuoka

  21. Implementation of element • element(I,L,X) • L=[X1,X2,…,Xn]. The Ith element of L is X, counting from 1. • Maintaining consistency I in 1..n,X in min(L)..max(L), X ≠ X1 → I ≠ 1, … X ≠ Xn → I ≠ n Neng-Fa Zhou at CSPAST'10 in Fukuoka

  22. element(I,L,X) : L is ground • from X to I • From I to X element_X_to_I(X,I,Map),var(X), {dom_any(X,E)} => map_get_indexes(Map,E,Indexes), I notin Indexes. element_X_to_I(X,I,Map) => true. element_I_to_X(I,X,Vect,Map),var(I), {dom_any(I,E)} => arg(E,Vect,Le), decrement_counter(Map,X,Le). element_I_to_X(I,X,Vect,Counters) => true. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  23. Implementation of all_distinct • Hall-set finding For the constraint all_distinct([X1,X2,…,Xn] where Xi’s domain is Di. A set H is a Hall set if the number of subsets of H among D1, D2, …, and Dn is greater than or equal to |H|. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  24. Implementation of all_distinct in B-Prolog • Check if an updated domain is a Hall set • Weakness • X1 in [1,2], X2 in [1,3], X3 in [2,3], X4 in [1,2,3,4] • Fail to find the Hall set [1,2,3] outof(X,Left,Right), var(X), {generated,ins(X),bound(X),dom(X)} => check_hall_set(X,Left,Right). outof(X,Left,Right) => exclude_list(X,Left),exclude_list(X,Right). Neng-Fa Zhou at CSPAST'10 in Fukuoka

  25. Channeling Constraints • Primal and dual constraints • More pruning power • X1 in [1,2], X2 in [1,3], X3 in [2,3, X4 in [1,2,3,4] • Y1 in [1,2,4], Y2 in [1,3,4], Y3 in [2,3,4], Y4 in [4] ↓ • X1 in [1,2], X2 in [1,3], X3 in [2,3, X4=4 • Y1 in [1,2], Y2 in [1,3], Y3 in [2,3], Y4=4 • all_distinct([X1,X2,…,Xn]) where Xi in 1..n • all_distinct([Y1,Y2,…,Yn]) where Yi in 1..n Neng-Fa Zhou at CSPAST'10 in Fukuoka

  26. Encoding Channeling Constraints in B-Prolog • The primal-dual relationship is maintained using 2n propagators primal_dual(Xi,I,DualVarVector),var(Xi), {dom_any(Xi,J)} => DualVarVector[J]#\= I. primal_dual(Xi,I,DualVarVector) => true. Neng-Fa Zhou at CSPAST'10 in Fukuoka

  27. Effectiveness of Channeling Constraints N-Queens problem Neng-Fa Zhou at CSPAST'10 in Fukuoka

  28. Implementation of circuit • circuit([X1,X2,…,Xn]) • all_distinct([X1,X2,…,Xn]) • Test reachability whenever the domain of any Xi is updated • Ensure non-existence of sub-cycles whenever any Xi is instantiated • How to test reachability? Neng-Fa Zhou at CSPAST'10 in Fukuoka

  29. Implementation of serialized(Starts,Durations) • Edge finding and not-first/not-last rules • d(Ω ∪ {T}) > lct(Ω ∪ {T})-est(Ω)task T must be started before all tasks in Ω • d(Ω ∪ {T}) > lct(Ω)-est(Ω ∪ {T}) task T must be started after all tasks in Ω • d(Ω ∪ {T}) > lct(Ω)-est(T) task T cannot start first • d(Ω ∪ {T}) > lct(T)-est(Ω) task T cannot start last Neng-Fa Zhou at CSPAST'10 in Fukuoka

  30. Outline • Global constraints • element, all_distinct, circuit, cumulative • Implementation language • Action rules • Implementation of propagators with AR • element, all_distinct, circuit, cumulative • Parallelization of propagators • all_distinct Neng-Fa Zhou at CSPAST'10 in Fukuoka

  31. Parallelizing Propagators for all_distinct • Parallelism • Hall-set finding can be applied to different sets in parallel • For one set, counting of subsets can be done in parallel • Synchronization models • Add a mutex variable into each domain variable • Postpone updates until the next choice point and only let the main thread perform updates • Implementation • Task queues and update queues • Use pthreads Neng-Fa Zhou at CSPAST'10 in Fukuoka

  32. Future Work • More experiments and benchmarking • Integrating with SAT technology • Complete the project on parallelizing all_distinct • Parallelization of other global constraints • Incorporating and/or parallelism Neng-Fa Zhou at CSPAST'10 in Fukuoka

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