BPSolver’s Winning Solutions to ASP Competition Problems

Neng-Fa Zhou at TUWIEN 1

BPSolver’s Winning Solutions to ASP Competition Problems

Neng-Fa Zhou

The City University of New Yorkzhou@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 Observations

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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

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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)

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Overview of ASP Competition (Model & Solve) Benchmarks (34)

– P-problems (7)– NP-problems (19)– Beyond NP problems (2)– Optimization problems (6)

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Another View of the Results(Clasp Vs. B-Prolog)

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Overview of the ASP competition B-Prolog’s features BPSolver’s winning solutions BPSolver’s hopeful solutions BPSolver’s losing solutions Observations

Outline

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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)

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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

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Loop Constructs in B-Prolog

foreach– foreach(E1 in D1, . . ., En in Dn, LVars, Goal)

– Example:• foreach(A in [a,b], I in 1..2, writeln((A,I))

List comprehension– [T : E1 in D1, . . ., En in Dn, 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

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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.

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

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Mode-Directed Tabling in B-Prolog Table mode declaration

– C: Cardinality limit

– Modes• + : input• - : output• min: minimized• max: maximized

:-table p(M1,...,Mn):C.

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Example: Shortest Path Problem

sp(X,Y,P,W)– P is a shortest 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.

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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

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)

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Overview of the ASP competition B-Prolog’s features BPSolver’s winning solutions BPSolver’s hopeful solutions BPSolver’s losing solutions Observations

Outline

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BPSolver’s Winning Solutions

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Winning Solutions in Prolog

Grammar-Based Information Extraction– Parsing

Labyrinth– State space search

Tomography– Set covering

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Winning Solutions With Tabling

Reachability Hydraulic Planning Hydraulic Leaking Airport Pickup Hanoi Tower

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Reachability

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:-table reach/1.reach(X):- start(X).reach(Y):- reach(X), edge(X,Y).

Hydraulic Planning

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:-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.

Hydraulic Leaking

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:-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 ).

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

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CITY MAP

Solution to Airport Pickup

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:-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).

Hanoi Tower (4-Pegs)

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A B C D A B C D

Two snapshots from the sequence by the Frame-Stewart algorithm

Problem Reduction

If the largest disk is in its final position, remove it.

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A B C D A B C D

Create an Intermediate State

Subproblems

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A B C D A B C D

A B C D A B C D

Sub-prob-1

Sub-prob-2

The Solution(4-Peg Hanoi Tower)

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:-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.

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

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Magic Square Sets

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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.

Overview of the ASP competition B-Prolog’s features BPSolver’s winning solutions BPSolver’s hopeful solutions BPSolver’s losing solutions Observations

Outline

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BPSolver’s Hopeful Solutions

Graph Coloring (Ranking = 5) Sokoban Optimization (Ranking = 4)

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Graph Coloring

Model-1 (neq)• For each two neighbors i and j, CiCj

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 satisfiabilityNeng-Fa Zhou at TUWIEN 34

Graph Coloring in B-Prolog

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Go:- create_vars(Vars), generate_neqs(Vars,Neqs), post_neqs(Neqs,Cliques), % new built-in largest_clique(Cliques,LClique), (labeling(LClique)-> labeling_ffc(Vars),!;fail), output(Vars).

Performance Comparison(Model-2 Vs. Model-3)

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BPSolver’s Solution to Sokoban

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The Competition Results

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BPSolver’s Losing Solutions(Score) Partner Units (0) Reverse Folding (0) Solitaire (0) Strategic Companies (0) Company Controls Optimize (0) Stable Marriage (5) Maze Generation (49)

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Observations

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