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CLP Principles

CLP Principles. H. Simonis COSYTEC SA 4, rue Jean Rostand F-91893 Orsay Cedex helmut@cosytec.fr. Outline. Background on CLP Simple Examples Global Constraints Search. What is common among. the production of Mirage 2000 fighter aircraft

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CLP Principles

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  1. CLP Principles H. Simonis COSYTEC SA 4, rue Jean Rostand F-91893 Orsay Cedex helmut@cosytec.fr CHIP Overview

  2. Outline • Background on CLP • Simple Examples • Global Constraints • Search CHIP Overview

  3. What is common among • the production of Mirage 2000 fighter aircraft • the personnel planning for the guards in all French jails • the production of Belgian chocolates • the selection of the music programme of a Pop music radio station • the design of advanced signal processing chips • the print engine controller in Xerox copiers They all use constraint programming to solve their problem CHIP Overview

  4. Constraint Programming - in a nutshell • Declarative description of problems with • Variables which range over (finite) sets of values • Constraints over subsets of variables which restrict possible value combinations • A solution is a value assignment which satisfies all constraints • Constraint propagation/reasoning • Removing inconsistent values for variables • Detect failure if constraint can not be satisfied • Interaction of constraints via shared variables • Incomplete • Search • User controlled assignment of values to variables • Each step triggers constraint propagation • Different domains require/allow different methods CHIP Overview

  5. Techniques behind CLP Predicate logic Unification Non-determinism Logic Programming Constraint propagation Consistency checking Demons CLP tools Artificial Intelligence Operations Research Simplex & Linear programming Integer programming Implicit enumeration Branch & bound Flow algorithms Scheduling methods Graph theory Combinatorics Spatial data structures Equation solving methods Mathematics CHIP Overview

  6. Example: Problem • Solve the cryptarithmetic puzzle • Each character represents a digit • Different characters have different values • Numbers do not start with 0 SEND +MORE ------ MONEY CHIP Overview

  7. Example: Model top:- [S, E, N, D, M, O, R, Y] :: 0..9, alldifferent([S, E, N, D, M, O, R, Y]), S \= 0, M \= 0, 1000 * S + 100*E + 10*N + D + 1000*M + 100*O + 10*R + E = 10000 *M + 1000*O + 100*N + 10*E + Y, labeling([S, E, N, D, M, O, R, Y]). Variable definition Constraints between variables Search routine CHIP Overview

  8. Constraint reasoning • Simplification (each variable occurs once) • 1000*S in {1..9}+ 91*E in {0..9} + D in {0..9} + 10*R in {0..9} = 9000*M in {1..9} +900*O in {0..9} + 90*N in {0..9} + Y in {0..9} • Evaluation lhs/rhs • lhs in 1000..9918 • rhs in 9000..89919 • Merging of sides • constraint in 9000..9918 CHIP Overview

  9. Reasoning • Consequence • M = 1 • S = 9 • O in {0..1} • Propagation of alldifferent • O = 0 CHIP Overview

  10. Reasoning • Re-evaluation of equality • 1000*9+ 91*E in {2..8} + D in {2..8} + 10*R in {2..8} = 9000*1 +900*0 + 90*N in {2..8} + Y in {2..8} • lhs in 9204..9816, rhs in 9182..9728, eq in 9204..9728 • N \= 2, E \= 8 • Re-evaluation, ... • Continuing the process gives • M = 1, S = 9, O = 0, E in 4..7, N in 5..8, D in 2..8, R in 2..8, Y in 2..8 CHIP Overview

  11. Starting labeling • First variable is E, first value is 4 • Propagation on equality gives • 1000*9+ 91*4 + D in {2..8} + 10*R in {2..8} = 9000*1 +900*0 + 90*N in {5..8} + Y in {2..8} • results in N = 5, D = 8, R = 8, Y = 2 • Propagation on alldifferent fails • Backtracking to last choice CHIP Overview

  12. First alternative • Next value for E is 5 • Propagation on equality gives • 1000*9+ 91*5 + D in {2..8} + 10*R in {2..8} = 9000*1 +900*0 + 90*N in {5..8} + Y in {2..8} • results in N = 6, R = 8 • Constraint propagation (alldifferent + equality) gives • D = 7, Y = 2 CHIP Overview

  13. Points to remember • Even small problems create complex propagation chains • The same constraint can be woken several times in the same propagation loop • The order in which constraints are woken has an influence on the speed • Propagation continues until no further information is obtained • Search (under user control) required to find ground solution • Basic structure of finite domain program always the same • define variables • define constraints • user defined search • typical: pick variable and find value CHIP Overview

  14. Example 2: The N-queens problem • Place queens on a NxN chessboard so that they do not attack each other • The classical constraints example • Not a hard problem: possible to construct generic solutions • Used here show the impact of the search routine CHIP Overview

  15. Model run(N):- length(L, N), L :: 1..N, create_dif(L, 1, N, L1, L2), alldifferent(L), alldifferent(L1), alldifferent(L2), labeling(L). create_dif([], N, M, [], []). create_dif([H|T], N, M, [H+N|R], [H+M|S]):- N1 is N+1, M1 is M-1, create_dif(T, N1, M1, R, S). CHIP Overview

  16. Naïve search labeling([]). labeling([X|Y]) :- indomain(X), labeling(Y). CHIP Overview

  17. Results first hard instance 22 CHIP Overview

  18. Searchtree (N=22) CHIP Overview

  19. First fail label([]). label([X|Y]) :- delete(Var,[X|Y],Rest,0,first_fail), indomain(Var), label(Rest). CHIP Overview

  20. Results First hard instance 80 CHIP Overview

  21. Searchtree (N=80) CHIP Overview

  22. run(N):- length(L, N), L :: 1..N, create_dif(L, 1, N, L1, L2), alldifferent(L), alldifferent(L1), alldifferent(L2), reorder(L, LL), label(LL). reorder(L, L1):- front_rear(L, L, [], F, R), merge_it(F, R, L1). front_rear(R, [], F, F, R). front_rear(R, [_], F, F, R). front_rear([H|T], [_, _|Q], F, Fend, Rear):- front_rear(T, Q, [H|F], Fend, Rear). merge_it([], [], []). merge_it([], [A], [A]). merge_it([A|A1], [B|B1], [A, B|C1]):- merge_it(A1, B1, C1). label([]). label([X|Y]) :- delete(Var, [X|Y], Rest, 0, first_fail), indomain(Var, middle), label(Rest). Heuristic reordering CHIP Overview

  23. Results Exceptional hard instances 108, 168 CHIP Overview

  24. Searchtree (N=108) CHIP Overview

  25. Credit based partial search label(L):- length(L,K), Credit is K*K, credit(L, Credit, K, choose, choice, 5, part(1,2)). choose(Term, LTerm, RTerm):- delete(Term, LTerm, RTerm, 0,first_fail). choice(X):- indomain(X,middle). CHIP Overview

  26. Results runs up to several thousand queens with less than 10 backtracking steps CHIP Overview

  27. Searchtree (N=108) CHIP Overview

  28. Global Constraints CHIP Overview

  29. Need for global constraints 1 X X in {2,3} Y 2 Y in {2,3}  U in {1,2,3,4} Z 3 Z in {1,3} U 4 local reasoning, no action global reasoning, detect implications by bi-partite matching CHIP Overview

  30. Global constraints • Work on sets of variables • Global conditions, not local constraints • Semantic methods • Operations Research • Spatial algorithms • Graph theory • Network flows • Building blocks (high-level constraint primitives) • Multi-purpose • As general as possible • Usable with other constraints • Very strong propagation • Acceptable algorithmic complexity CHIP Overview

  31. Constraint morphology precedence diffn cumulative sequence cycle case among alldifferent setup disjunctive permutation prod/cons \= >=, distance atmost, atleast circuit element Different Order Resource Tour Dependency CHIP Overview

  32. cumulative The Cumulative global constraint • Cumulative constraint • Resource limits over periods of time • Upper/lower limits • Soft/hard limits • Gradual constraint relaxation • Application • Resource restrictive scheduling, producer consumer constraints, disjunctive schedule, manpower constraints, overtime CHIP Overview

  33. Cumulative • Methods • obligatory parts • task intervals • available space • many more (25000 lines of C code) • Concepts • one constraint may be used for different purposes CHIP Overview

  34. diffn The Diffn global constraint • Diffn constraint • non overlapping areas on n-dimensional rectangles • distances between rectangles • limit use of areas • relaxation • Application • layout, packing, resource assignment, setup, distribution planning, time-tabling CHIP Overview

  35. Diffn • Methods • obligatory parts • region intervals • max flow on assignment • available space • many others (32000 lines of C code) • Concepts CHIP Overview

  36. cycle The Cycle global constraint • Cycle constraint • Finds cycles in directed graphs with minimal cost • Assign resources, find compatible start dates • Applications • Tour planning, personnel rotation, distribution problems, production sequencing CHIP Overview

  37. Cycle • Methods • connected components • bi-partite matching • non-oriented graph concepts • shortest/longest paths • micro-rules • many others (38000 lines of C code) • Concepts CHIP Overview

  38. among 1 of 3 2 of 5 1 of 6 The Among global constraint • Among constraint • How often do values occur in (sub)sequences • based on counting arguments • interaction between sequences • Applications • production sequencing, time tabling, coloring problems, set covering CHIP Overview

  39. precedence The Precedence global constraint • Precedence constraint • Combine resource constraints and precedence networks • Reasoning on latency (position in network) • Co-operation between multiple resources • Applications • resource restricted scheduling, channel routing, frequency allocation CHIP Overview

  40. The Sequence global constraint sequence <= 40 hours 2 days off after day with more than 10 hours, next day must have less than 8 hours • Sequence constraint • constraints on pattern inside sequences • combinatorial pattern matching • counting arguments • Applications • Time tabling, personnel assignment, • work rules, scheduling with daily working time limits CHIP Overview

  41. cycle cumulative diffn The power of global constraints plan schedule • Multi-functional tools • Building blocks assign CHIP Overview

  42. Controlling Search CHIP Overview

  43. Search tree visualization • Generation of search tree representation at run-time • Shows parent child relation, failed sub-trees, success nodes • In examples here • leaf failure nodes suppressed • failure trees not collapsed • Interface a set of simple meta-call predicates • Supported by work in DISCIPL Esprit project • Declarative debugging • Visualization of constraint programming results • Understand behaviour of different strategies CHIP Overview

  44. Search tree tool Menubar Panel Tree view Other Views, here domain state Info Area CHIP Overview

  45. Search strategies • How to find values for variables • Central to application of strategies/heuristics • Chronological backtracking • explores full search tree • complete • often stuck in one part of tree • Partial search meta-heuristics • different search methods • not complete • polynomial complexity • used to explore different parts of search tree in systematic fashion • uses normal variable and value selection criteria CHIP Overview

  46. Credit based search • Systematic search at top of tree • Limited amount of backtracking = credit • typical N, N2, N3 • Distribute credit to children in different ways • preference on first child • equal credit to all children • If credit runs out, perform deterministic search • Allow small amount of local search to overcome problems CHIP Overview

  47. Example of tree search credit([X1..X10],8, 10, my_delete, my_indomain,4,part(1,2)), CHIP Overview

  48. Credit Example CHIP Overview

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