Projection, Inference, and Consistency
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1 Projection, Inference, and Consistency John Hooker Carnegie Mellon University IJCAI 2016, New York City
2 A high-level presentation. Don t worry about the details. 2
3 Projection as a Unifying Concept Projection underlies inference, optimization, and consistency maintenance 3
4 Projection as a Unifying Concept Projection underlies inference, optimization, and consistency maintenance Logical inference is projection onto a subset of variables. 4
5 Projection as a Unifying Concept Projection underlies inference, optimization, and consistency maintenance Logical inference is projection onto a subset of variables. Optimization is projection onto one or more variables. 5
6 Projection as a Unifying Concept Projection underlies inference, optimization, and consistency maintenance Logical inference is projection onto a subset of variables. Optimization is projection onto one or more variables. Consistency maintenance lies at the heart of constraint programming and is a form of projection. 6
7 Projection as a Unifying Concept Projection underlies inference, optimization, and consistency maintenance Logical inference is projection onto a subset of variables. Optimization is projection onto one or more variables. Consistency maintenance lies at the heart of constraint programming and is a form of projection. Let s start with George Boole whose probability logic brings together projection, inference, and optimization. 7
8 Probability Logic George Boole is best known for propositional logic and Boolean algebra. But he proposed a highly original approach to probability logic. 8
9 Logical Inference The problem: Given a set S of propositions Each with a given probability. And a proposition P that can be deduced from S What probability can be assigned to P? 9
10 Boole s Probability Logic In 1970s, Theodore Hailperin offered an interpretation of Boole s probability logic based on modern concept of linear programming. LP first clearly formulated in 1930s by Kantorovich. Hailperin (1976) 10
11 Boole s Probability Logic if if Clause 1 Eample then 1 2 then 2 3 Probability Deduce probability range for 3
12 Boole s Probability Logic if if Clause 1 Eample then 1 2 then 2 3 Probability Deduce probability range for 3 Interpret if-then statements as material conditionals
13 Boole s Probability Logic Clause 1 Eample Probability Deduce probability range for 3 Interpret if-then statements as material conditionals
14 Boole s Probability Logic Clause 1 Eample Probability Deduce probability range for 3 Linear programming model min/ ma p p p p p 000 = probability that ( 1, 2, 3 ) = (0,0,0) 14
15 Boole s Probability Logic Clause 1 Eample Probability Deduce probability range for 3 Linear programming model min/ ma p p p p p 000 = probability that ( 1, 2, 3 ) = (0,0,0) Solution: 0 [0.1, 0.4] 15
16 Boole s Probability Logic This LP model was re-invented in AI community. Nilsson (1986) Column generation methods are now available. To deal with eponential number of variables. 16
17 Boole s Probability Logic Inference: infer probability of a proposition. Projection: project feasible set onto a variable To derive range of probabilities Optimization: formulate as a linear programming problem. 17
18 Optimization as Projection An LP can be solved by Fourier elimination The only known method in Boole s day This is a projection method. Fourier (1827) 18
19 Optimization as Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b
20 Optimization as Projection Eliminate variables we want to project out. To solve min/ ma y y ay, Ay b 0 0 project out all variables y j ecept y 0. To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j where c, d
21 Optimization as Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d
22 Optimization as Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d
23 Optimization as Projection Eliminate variables we want to project out. To solve project out all variables y j ecept y 0. min/ ma y y ay, Ay b 0 0 To project out y j, eliminate it from pairs of inequalities: cy c0y j dy d y 0 j c d y y j y c c d d c d y c0 d0 c0 d0 23
24 Inference as Projection Project onto propositional variables of interest Suppose we wish to infer from these clauses everything we can about propositions 1, 2, 3 24
25 Inference as Projection Project onto propositional variables of interest Suppose we wish to infer from these clauses everything we can about propositions 1, 2, 3 We can deduce This is a projection onto 1, 2, 3 25
26 Inference as Projection Resolution is a projection method! Similar to Fourier elimination Actually, identical to Fourier elimination + rounding! To project out j, eliminate it from pairs of clauses: C j D j C D Quine (1952,1955) JH (1992,2012) 26
27 Inference as Projection Projection methods similar to Fourier elimination Resolution for logical clauses Quine (1952,1955) for cardinality clauses JH (1988) for 0-1 linear inequalities JH (1992) for general integer linear inequalities Williams & JH (2015) 27
28 Inference as Projection Elimination-based methods tend to be slow. Another approach is an optimization method: logic-based Benders decomposition 28
29 Inference as Projection Elimination-based methods tend to be slow. Another approach is an optimization method: logic-based Benders decomposition For problems that radically simplify when certain variables are fied. Resulting subproblem may be easy to solve. Solve master problem to search over ways to fi variables. Solution of subproblem yields a Benders cut that is added to master problem and helps direct search. Repeat until problem is solved. 29
30 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem 1 2 Benders cut from previous iteration 30
31 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem 1 2 solution of master ( 1, 2, 3 ) = (0,1,0) Resulting subproblem 31
32 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem 1 2 solution of master ( 1, 2, 3 ) = (0,1,0) Resulting subproblem Subproblem is infeasible. ( 1, 3 )=(0,0) creates infeasibility 32
33 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,0) Benders cut (nogood) Resulting subproblem Subproblem is infeasible. ( 1, 3 )=(0,0) creates infeasibility 33
34 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Resulting subproblem 34
35 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Resulting subproblem Subproblem is feasible 35
36 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Enumerative Benders cut Subproblem is feasible Resulting subproblem 36
37 Benders Decomposition as Projection Benders decomposition computes a projection! Benders cuts describe projection onto master problem variables. Current Master problem solution of master ( 1, 2, 3 ) = (0,1,1) Enumerative Benders cut Resulting subproblem JH (2000) JH and Ottosson (2003) Continue until master is infeasible. Black Benders cuts describe projection. 37
38 Projection in Satisfiability Solvers Satisfiability solvers solve inference problems by a projection method. Using conflict clauses.
39 Projection in Satisfiability Solvers Satisfiability solvers solve inference problems by a projection method. Using conflict clauses. They do so by generating Benders cuts! Conflict clauses = Benders cuts
40 Projection in Satisfiability Solvers Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2, 3 first. 40
41 Projection in Satisfiability Solvers Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2, 3 first. Conflict clauses 41
42 Projection in Satisfiability Solvers Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2, 3 first. Conflict clauses Backtrack to 3 at feasible leaf nodes 42
43 Projection in Satisfiability Solvers Benders cuts = conflict clauses in a SAT algorithm Branch on 1, 2, 3 first. Conflict clauses containing 1, 2, 3 describe projection 43
44 Consistency as Projection Consistency is a fundamental concept in constraint programming (CP). 44
45 Consistency as Projection Consistency is a fundamental concept in constraint programming (CP). CP models use high-level global constraints Such as all-different( 1,, n ) to require that 1,, n take different values. rather than writing i j for all i, j 45
46 Consistency as Projection Filtering algorithms remove values from variable domains that are inconsistent with individual global constraints. This ideally achieves domain consistency. 46
47 Consistency as Projection Filtering algorithms remove values from variable domains that are inconsistent with individual global constraints. This ideally achieves domain consistency. Reduced domains are propagated to the net constraint for further filtering. Smaller domains simplify search. 47
48 Consistency as Projection Achieving domain consistency is a form of projection. Project onto each individual variable j. Eample: Constraint set alldiff,, a, b a, b b, c 48
49 Consistency as Projection Achieving domain consistency is a form of projection. Project onto each individual variable j. Eample: Constraint set alldiff,, a, b a, b b, c Solutions,, ( a, b, c) ( bac,, ) 49
50 Consistency as Projection Achieving domain consistency is a form of projection. Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, a, b a, b b, c,, ( a, b, c) ( bac,, ) a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 50
51 Consistency as Projection Achieving domain consistency is a form of projection. Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, a, b a, b b, c,, ( a, b, c) ( bac,, ) This achieves domain consistency. a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 51
52 Consistency as Projection Achieving domain consistency is a form of projection. Project onto each individual variable j. Eample: Constraint set Solutions Projection onto alldiff,, a, b a, b b, c,, ( a, b, c) ( bac,, ) This achieves domain consistency. We will regard a projection as a constraint set. a b 1, Projection onto 2 a b 2, Projection onto 3 3 c 52
53 Consistency as Projection Once we view domain consistency as projection, we see how to etend it to a stronger property. Project onto sets of variables rather than single variables. Constraint set is J-consistent if it contains its projection onto a set J of variables. Domain consistent = { j }-consistent for each j. Resolution and logic-based Benders achieve J-consistency for SAT. 53
54 Consistency as Projection J-consistency and propagation. J-consistency can be useful if we propagate through a richer structure than domains. such as decision diagrams Andersen, Hadžić JH, Tiedemann (2007) Bergman, Ciré, van Hoeve, JH (2014) 54
55 Consistency as Projection Problem: projection generally results in greater compleity. Such as projecting a polyhedron. Lifting into a higher space tends to reduce compleity. For eample, resolution vs. etended resolution. Disaggregation of variables. Let s see what happens for some common global constraints. 55
56 Consistency as Projection Constraint How hard to project? among sequence regular alldiff Easy and fast. More complicated but fast. Easy and basically same labor as domain consistency. Surprisingly complicated but practical for small domains. 56
57 Projecting Among Constraint Requires that a certain number of variables in a set take specified values. Many applications. among ( 1,, n ), V, t, u requires that at least t and at most u variables among 1,, n take values in V. 57
58 Projection of Projecting Among Constraint among (,, ), V, t, u where 1 among ( 1,, n 1), V, t, u onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise n 58
59 Eample Projection of among (,, ), V, t, u where D D D D D Projecting Among Constraint 1 onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u
60 Eample Projection of among (,, ), V, t, u where D D D D D Projecting Among Constraint 1 onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 60
61 Eample Projection of among (,, ), V, t, u where D D D D D Projecting Among Constraint 1 onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 61
62 Eample Projection of among (,, ), V, t, u where D D D D D Projecting Among Constraint 1 onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} 62
63 Eample Projection of among (,, ), V, t, u where D D D D D Projecting Among Constraint 1 onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} 63
64 Projecting Among Constraint Eample Projection of among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} among (),{ c, d},( t 4),min{ u 2,0} 64
65 Projecting Among Constraint Eample Projection of among (,, ), V, t, u where D D D D D onto 1,, n1 is ( t 1), u 1 if D n V ( t, u) t,min{ u, n 1} if Dn V ( t 1),min{ u, n 1} otherwise a, b a, b, c a,d c, d d among ( 1,, n 1), V, t, u n among (,,,, ),{ c, d}, t, u among ( 1, 2, 3, 4 ),{ c, d},( t 1), u 1 among ( 1, 2, 3 ),{ c, d},( t 2), u 2 among ( 1, 2),{ c, d},( t 3),min{ u 2,2} among ( 1),{ c, d},( t 4),min{ u 2,1} among (),{ c, d},( t 4),min{ u 2,0} Feasible if and only if ( t 4) minu 2,0 65
66 Projecting Sequence Constraint Sequence constraint used for assembly line load balancing and the like. For eample, at most 3 of every 10 cars on the line require an air conditioner. Equivalent to overlapping among constraints. 66
67 Projecting Sequence Constraint Projection is based on an integrality property. The coefficient matri of the inequality formulation has consecutive ones property. So projection of the conve hull of the feasible set is an integral polyhedron. Polyhedral projection therefore suffices. Straightforward (but tedious) application of Fourier elimination yields the projection. 67
68 Projecting Sequence Constraint Projection is based on an integrality property. The coefficient matri of the inequality formulation has consecutive ones property. So projection of the conve hull of the feasible set is an integral polyhedron. Polyhedral projection therefore suffices. Straightforward (but tedious) application of Fourier elimination yields the projection. Projection onto any subset of variables is a generalized sequence constraint. Compleity of projecting out k is O(kq), where q = length of the overlapping sequences. 68
69 69
70 Projecting Sequence Constraint Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1,
71 Projecting Sequence Constraint Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0,
72 Projecting Sequence Constraint Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0, To project out 4, add constraints among ( ),{1},1,1 among (,, ),{1},1,2 among (, ),{1},0,1 among ( ),{1},0,
73 Projecting Sequence Constraint Eample t3 t t,,, 0,1,, 1 among (,, ),{1},2,2, 4,5, To project out 6, add constraint among (,, ),{1},1, To project out 5, add constraints among (,, ),{1},1,1 among (, ),{1},0, To project out 4, add constraints among ( ),{1},1,1 among (,, ),{1},1,2 among (, ),{1},0,1 among ( ),{1},0, To project out 3, fi ( 1, 2 ) = (1,1) 73
74 Projecting Regular Constraint Versatile constraint used for employee shift scheduling, etc. For eample, a worker can change shifts only after a day off. Formulates constraint as deterministic finite automaton. Or as regular language epression. 74
75 Projecting Regular Constraint Projection can be read from state transition graph. Compleity of projecting onto 1,, k for all k is O(nm 2 ), where n = number of variables, m = ma number of states per stage. 75
76 Projecting Regular Constraint Projection can be read from state transition graph. Compleity of projecting onto 1,, k for all k is O(nm 2 ), where n = number of variables, m = ma number of states per stage. Shift scheduling eample Assign each worker to shift i {a,b,c} on each day i = 1,,7. Must work any given shift 2 or 3 days in a row. No direct transition between shifts a and c. Variable domains: D 1 = D 5 = {a,c}, D 2 = {a,b,c}, D 3 = D 6 = D 7 = {a,b}, D 4 = {b,c} 76
77 Projecting Regular Constraint Deterministic finite automaton for this problem instance: = absorbing state Regular language epression: (((aa aaa)(bb bbb))* ((cc ccc)(bb bbb))*)*(c (aa aaa) (cc ccc)) 77
78 Projecting Regular Constraint State transition graph for 7 stages Dashed lines lead to unreachable states. 78
79 Projecting Regular Constraint State transition graph for 7 stages Dashed lines lead to unreachable states. Filtered domains 79
80 Projecting Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. 80
81 Projecting Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. 81
82 Projecting Regular Constraint To project onto 1, 2, 3, truncate the graph at stage 4. Resulting graph can be viewed as a constraint that describes the projection. Constraint is easily propagated through a relaed decision diagram. 82
83 Projecting Alldiff Constraint Used for sequencing and much else. Domain consistency easily achieved by matching algorithm and duality theory. 83
84 Projecting Alldiff Constraint Projection is inherently complicated. But it can simplify for small domains. The result is a disjunction of constraint sets, each of which contains an alldiff constraint and some atmost constraints. 84
85 Projecting Alldiff Constraint Eample alldiff,,,, ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g
86 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, 86
87 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. 87
88 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. 88
89 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get alldiff,,, atmost (,, ),{ a, f, g},
90 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get alldiff,,, atmost (,, ),{ a, f, g}, If 4 {a,f,g}, we get 4 = e, and we remove e from other domains. 90
91 Eample alldiff,,,, Projecting out 5, we get Projecting Alldiff Constraint ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g a f g alldiff,,,, atmost (,,, ),{,, }, because 5 must take one of the values in {a,f,g}, leaving 2 for other i s. Projecting out 4, we note that 4 {a,f,g} or 4 {a,f,g}. If 4 {a,f,g}, we get alldiff,,, atmost (,, ),{ a, f, g}, If 4 {a,f,g}, we get 4 = e, and we remove e from other domains. So the projection is D1 a, b, c alldiff 1, 2, 3 D2 c, d atmost ( 1, 2, 3 ),{ a, f, g},1 D 3 d, f 91
92 Projecting Alldiff Constraint Eample alldiff,,,, ,,,,,,,,,,,,,, D a b c D c d e D d e f D e f g D a f g Projecting out 3, we get simply alldiff, 1 2 Projecting out 2, we get the original domain for 1 D1 a, b, c 92
93 93
94 That s it. 94
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