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BPSolver’s Solutions to the ASP Competition Problems

BPSolver’s Solutions to the ASP Competition Problems. Neng-Fa Zhou 周　能法 The City University of New York zhou@sci.brooklyn.cuny.edu. Outline. Overview of the ASP competition B-Prolog’s features BPSolver’s winning solutions BPSolver’s hopeful solutions BPSolver’s losing solutions

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BPSolver’s Solutions to the ASP Competition Problems

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  1. BPSolver’s Solutions to the ASP Competition Problems Neng-Fa Zhou 周　能法 The City University of New York zhou@sci.brooklyn.cuny.edu N.F. Zhou, KR'12

  2. Outline • Overview of the ASP competition • B-Prolog’s features • BPSolver’s winning solutions • BPSolver’s hopeful solutions • BPSolver’s losing solutions • Observations N.F. Zhou, KR'12

  3. Overview of ASP Competition (Model & Solve) • Principles • To foster open comparison of ASP with any other declarative paradigm • To foster development of new language constructs • To foster development of new heuristics and/or algorithms N.F. Zhou, KR'12

  4. Overview of ASP Competition (Model & Solve) • Participants • Aclasp (Gringo + Clasp + Gecode) • BPSolver (B-Prolog) • EZCSP (Gringo + Clasp + B-Prolog) • Fastdownward (PDDL) • IDP (grounder Gidl + MinisatID) • Potassco (Gringo + Clasp + Gecode) N.F. Zhou, KR'12

  5. Overview of ASP Competition (Model & Solve) • Benchmarks (34) • P-problems (7) • NP-problems (19) • Beyond NP problems (2) • Optimization problems (6) N.F. Zhou, KR'12

  6. Another View of the Results(Clasp Vs. B-Prolog) N.F. Zhou, KR'12

  7. Outline • Overview of the ASP competition • B-Prolog’s features • BPSolver’s winning solutions • BPSolver’s hopeful solutions • BPSolver’s losing solutions • Observations N.F. Zhou, KR'12

  8. B-Prolog =Prolog + Tabling + CLP(FD) • Prolog enhanced with • Array subscripts • Loop constructs • Tabling • Memorize and reuse intermediate results • Suitable for dynamic programming problems • CLP(FD) • Constraint Logic Programming over Finite Domains • Suitable for constraint satisfaction problems (NP-complete) N.F. Zhou, KR'12

  9. Array Subscripts in B-Prolog • In arithmetic expressions • In arithmetic constraints • In calls to @= and @:= • In any other context, X[I1,…,In] is the same as X^[I1,…,In] S is X[1]+X[2]+X[3] X[1]+X[2] #= X[3] X[1,2] @= 100X[1,2] @:= 100 N.F. Zhou, KR'12

  10. Loop Constructs in B-Prolog • foreach • foreach(E1inD1, . . ., EninDn, LVars, Goal) • Example: • foreach(A in [a,b], I in 1..2, writeln((A,I)) • List comprehension • [T : E1inD1, . . ., EninDn, LVars, Goal] • Examples: • L @= [(A,I): A in [a,b], I in 1..2]. • sum([A[I,J] : I in 1..N, J in 1..N]) #= N*N N.F. Zhou, KR'12

  11. More Examples • Create a list of 10 random integers • Create a matrix of random integers L @= [X : I in 1..10, [X], X is random] matrix(NRow,NCols,M):- M @= [R : I in 1..NRows, [R], (R @= [X : J in 1..NCols, [X],X is random])]. N.F. Zhou, KR'12

  12. Tabling in B-Prolog • Eliminate infinite loops • Reduce redundant computations • :-table path/2. • path(X,Y):-edge(X,Y).path(X,Y):-edge(X,Z),path(Z,Y). • :-table fib/2.fib(0,1).fib(1,1).fib(N,F):- N>1, N1 is N-1,fib(N1,F1), N2 is N-2,fib(N2,F2), F is F1+F2. N.F. Zhou, KR'12

  13. The Table-All Approach • Characteristics • All the arguments of a tabled subgoal are used in variant checking • All answers are tabled • Problems • The number of answers may be too large or even infinite for DP and ML problems • When computing aggregates, tabling noncontributing answers is a waste N.F. Zhou, KR'12

  14. Mode-Directed Tabling in B-Prolog • Table mode declaration • C: Cardinality limit • Modes • + : input • - : output • min: minimized • max: maximized :-table p(M1,...,Mn):C. N.F. Zhou, KR'12

  15. Example: Shortest Path Problem • sp(X,Y,P,W) • P is a path between X and Y with minimal weight W. :-table sp(+,+,-,min). sp(X,Y,[(X,Y)],W) :- edge(X,Y,W). sp(X,Y,[(X,Z)|Path],W) :- edge(X,Z,W1), sp(Z,Y,Path,W2), W is W1+W2. N.F. Zhou, KR'12

  16. CLP(FD) in B-Prolog • A rich set of built-in constraints • Unification and arithmetic constraints (#=, #\=, #>, #>=, #<, #=<) • Global constraints • A glass-box approach to the implementation • All propagators are described in Action Rules (TPLP’06) • Action Rules is open to the user for describing problem-specific propagators • A rich set of labeling options N.F. Zhou, KR'12

  17. Global Constraints in B-Prolog • all_different(L) & all_distinct(L) • post_neqs(L) • circuit(L) • count(V,L,RelOp,N) • exactly(N,L,V) • atleast(N,L,V) • atmost(N,L,V) • cumulative(Starts,Durations,Resources,Limit) • serialized(Starts,Durations) • diffn(L) • element(I,L,V) • path_from_to(From,To,L)& path_from_to(From,To,L,Lab) N.F. Zhou, KR'12

  18. Outline • Overview of the ASP competition • B-Prolog’s features • BPSolver’s winning solutions • BPSolver’s hopeful solutions • BPSolver’s losing solutions • Observations N.F. Zhou, KR'12

  19. BPSolver’s Winning Solutions N.F. Zhou, KR'12

  20. Winning Solutions in Prolog • Grammar-Based Information Extraction • Parsing • Labyrinth • State space search • Tomography • Network covering N.F. Zhou, KR'12

  21. Winning Solutions With Tabling • Reachability • Hydraulic Planning • Hydraulic Leaking • Airport Pickup • Hanoi Tower N.F. Zhou, KR'12

  22. Reachability :-table reach/1. reach(X):- start(X). reach(Y):- reach(X), edge(X,Y). N.F. Zhou, KR'12

  23. Hydraulic Planning :-table pressurize(+,-,min). pressurize(Node,Plan,Len):- full(Node),!, Plan=[],Len=0. pressurize(Node,[Valve|Plan],Len):- link(AnotherNode,Node,Valve), \+ stuck(Valve), pressurize(AnotherNode,Plan,Len1), Len is Len1+1. N.F. Zhou, KR'12

  24. Hydraulic Leaking :-table pressurize(+,-,min). pressurize(Node,Plan,(Leaks,Len)):- full(Node),!, Plan=[],Leaks=0,Len=0. pressurize(Node,[Valve|Plan],(Leaks,Len)):- link(AnotherNode,Node,Valve), \+ stuck(Valve), pressurize(AnotherNode,Plan,(Leaks1,Len1)), Len is Len1+1, (leaking(Valve)-> Leaks is Leaks1+1 ; Leaks is Leaks1 ). N.F. Zhou, KR'12

  25. Airport Pickup • Passengers at Airport #1 want to move to Airport #2 and vice versa • A certain number of vehicles are available • Some of the locations are gas stations CITY MAP N.F. Zhou, KR'12

  26. Solution to Airport Pickup :-table move_vehicle(+,+,+,+,max,-). move_vehicle(X,X,_Cap,GasLevel,Objective,Steps):- Objective=(GasLevel,0),Steps=[]. move_vehicle(X,Y,Cap,GasLevel,Objective,[drive(Z)|Steps]):- (driveway(X,Z,GasNeeded);driveway(Z,X,GasNeeded)), GasLevel>=GasNeeded, GasLevel1 is GasLevel-GasNeeded, move_vehicle(Z,Y,Cap,GasLevel1,Objective1,Steps), Objective1=(AfterGasLevel,MLen), MLen1 is MLen-1, Objective=(AfterGasLevel,MLen1). move_vehicle(X,Y,Cap,GasLevel,Objective,[refuel|Steps]):- gasstation(X), GasLevel<Cap, move_vehicle(X,Y,Cap,Cap,Objective1,Steps), Objective1=(AfterGasLevel,MLen), MLen1 is MLen-1, Objective=(AfterGasLevel,MLen1). N.F. Zhou, KR'12

  27. Hanoi Tower (4-Pegs) A B C D A B C D Two snapshots from the sequence by the Frame-Stewart algorithm N.F. Zhou, KR'12

  28. Problem Reduction • If the largest disk is in its final position, remove it. A B C D A B C D N.F. Zhou, KR'12

  29. Create an Intermediate State • Subproblems Sub-prob-1 A B C D A B C D Sub-prob-2 A B C D A B C D N.F. Zhou, KR'12

  30. The Solution(4-Peg Hanoi Tower) :-table plan4(+,+,+,-,min). plan4(N,_CState,_GState,Plan,Len):-N=:=0,!,Plan=[],Len=0. plan4(N,CState,GState,Plan,Len):- reduce_prob(N,CState,GState,CState1,GState1),!, N1 is N-1, plan4(N1,CState1,GState1,Plan,Len). plan4(N,CState,GState,Plan,Len):- partition_disks(N,CState,GState,ItState,Mid,Peg), remove_larger_disks(CState,Mid,CState1), plan4(Mid,CState1,ItState,Plan1,Len1), % sub-prob1 remove_smaller_or_equal_disks(CState,Mid,CState2), remove_smaller_or_equal_disks(GState,Mid,GState2), N1 is N-Mid, plan3(N1,CState2,GState2,Peg,Plan2,Len2), % sub-prob2 remove_larger_disks(GState,Mid,GState1), plan4(Mid,ItState,GState1,Plan3,Len3), % sub-prob3 append(Plan1,Plan2,Plan3,Plan), Len is Len1+Len2+Len3. N.F. Zhou, KR'12

  31. Winning Solutions in CLP(FD) • Tangram • Magic Square Sets (all_different) • Weight-Assignment Tree (element) • Knight Tour (circuit) • Disjunctive Scheduling (post_disjunctive_tasks) • Maximal Clique N.F. Zhou, KR'12

  32. Magic Square Sets semi(Board,N,Magic):- foreach(I in 1..N, sum([Board[I,J] : J in 1..N])#=Magic), foreach(J in 1..N, sum([Board[I,J] : I in 1..N])#=Magic). normal(Board,N,Magic):- sum([Board[I,I] : I in 1..N]) #= Magic, sum([Board[I,N-I+1] : I in 1..N]) #= Magic. N.F. Zhou, KR'12

  33. Outline • Overview of the ASP competition • B-Prolog’s features • BPSolver’s winning solutions • BPSolver’s hopeful solutions • BPSolver’s losing solutions • Observations N.F. Zhou, KR'12

  34. BPSolver’s Hopeful Solutions • Graph Coloring (Ranking = 5) • Sokoban Optimization (Ranking = 4) N.F. Zhou, KR'12

  35. Graph Coloring • Model-1 (neq) • For each two neighbors i and j, CiCj • Model-2 (all_distinct) • For each complete subgraph (clique) {i1,i2,…,ik}, all_distinct([Ci1, Ci2,…, Cik]) • post_neqs(Neqs) in B-Prolog • Model-3 (Model-2 + symmetry breaking) • Allen Van Gelder: Another look at graph coloring via propositional satisfiability N.F. Zhou, KR'12

  36. Graph Coloring in B-Prolog Go:- create_vars(Vars), generate_neqs(Vars,Neqs), post_neqs(Neqs,Cliques), largest_clique(Cliques,LClique), (labeling(LClique)-> labeling_ffc(Vars),!;fail), output(Vars). N.F. Zhou, KR'12

  37. Performance Comparison(Model-2 Vs. Model-3) N.F. Zhou, KR'12

  38. The Sokoban Problem [push(c6r3,down,c6r5), push(c5r4,left,c3r4), push(c3r4,down,c3r5), push(c5r5,up,c5r4), push(c6r5,left,c5r5), push(c5r4,right,c6r4), push(c5r5,up,c5r4), push(c6r4,up,c6r3), push(c5r4,left,c4r4), push(c6r3,down,c6r4), push(c3r5,up,c3r3), push(c4r4,left,c3r4), push(c6r4,left,c4r4)] N.F. Zhou, KR'12

  39. BPSolver’s Solution to Sokoban N.F. Zhou, KR'12

  40. The Competition Results N.F. Zhou, KR'12

  41. BPSolver’s Losing Solutions(Score) • Partner Units (0) • Reverse Folding (0) • Strategic Companies (0) • Company Controls Optimize (0) • Stable Marriage (5) • Solitaire (28) • Maze Generation (49) N.F. Zhou, KR'12

  42. A Common Interface to SAT and LP/MIP queens(N):- length(Qs,N), Qs :: 1..N, foreach(I in 1..N-1, J in I+1..N, (Qs[I] $\= Qs[J], Qs[I]-Qs[J] $\= J-I, Qs[J]-Qs[I] $\= J-I)), sat_solve(Qs), writeln(Qs). N.F. Zhou, KR'12

  43. Numberlink N.F. Zhou, KR'12

  44. B-Prolog =Prolog + Tabling + CLP(FD)+SAT • Prolog enhanced with • Array subscripts • Loop constructs • Tabling • Memorize and reuse intermediate results • Suitable for dynamic programming problems • CLP(FD) & SAT • Constraint Logic Programming over Finite Domains • Suitable for constraint satisfaction problems (NP-complete) N.F. Zhou, KR'12

  45. Observations • ASP vs B-Prolog • ASP encodings are generally shorter than BP encodings • Grounding vs. top-down tabled evaluation • SAT vs. CP N.F. Zhou, KR'12

  46. Resources • BPSolver’s solutions • www.sci.brooklyn.cuny.edu/~zhou/asp11/ • www.probp.com/asp11/ • B-Prolog • www.probp.com • Papers • www.sci.brooklyn.cuny.edu/~zhou/ N.F. Zhou, KR'12

  47. Thanks • ASP competition organization committee • BPSolver team members • Agostino Dovier • Yuanlin Zhang • NSF and PSC-CUNY N.F. Zhou, KR'12

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