New Encodings of Pseudo-Boolean Constraints into CNF

Transcription

1 New Encodings of Pseudo-Boolean Constraints into CNF Olivier Bailleux, Yacine Boufkhad, Olivier Roussel New Encodings ofpseudo-boolean Constraintsinto CNF 1

2 Outline Pseudo-Boolean constraints Encodings into CNF Unit Propagation and Generalized Arc Consistency The problem with Unit Propagation The new encodings Some preliminary experiments New Encodings ofpseudo-boolean Constraintsinto CNF 2

3 Pseudo-Boolean (PB) Constraints A (linear) pseudo-boolean constraint may be defined over boolean variables by a i.l i d with a i, d Z l i {x i, x i }, x i B {<, >,,, =} Example: 3x 1 3x x 3 + x 4 + x 5 5 Extends both clauses and cardinality constraints cardinalities:, all a i = 1 and d > 1 clauses:, all a i = 1 and d = 1 i New Encodings ofpseudo-boolean Constraintsinto CNF 3

4 PB Constraints stronger than clauses? Some facts suggest that PB constraints are more expressive than clauses PB constraints use basic arithmetic many problems are more easily encoded as PB (e.g. adder) NP-complete problems can be encoded as one single PB constraint (e.g. variants of Knapsack) any CNF can be encoded as one single non-linear PB constraint without additional variables, encoding a PB constraint into CNF is exponential the Pigeon-Hole problem can be solved polynomially when encoded as PB constraints (with learning). but is it so sure? New Encodings ofpseudo-boolean Constraintsinto CNF 4

5 Encoding PB constraints into CNF 1 more or less specialized direct encoding doesn t introduce additional variables exponential 2 BDD (Binary Decision Diagrams) requires additional variables exponential 3 Adder+Comparator (Warner s encoding) requires additional variables polynomial 4 and a few other encodings New Encodings ofpseudo-boolean Constraintsinto CNF 5

6 Generalized Arc Consistency (GAC) Let C be a PB constraint, l and l i be literals Whenever C {l 1, l 2,... l n } = l, we expect that l will be generated by the inference process When this is the case for any set of literals, the inference process is said to maintain Generalized Arc Consistency (GAC). Basic PB inference rules maintain GAC. Depending on the encoding into CNF, Unit Propagation (UP) may or may not maintain GAC New Encodings ofpseudo-boolean Constraintsinto CNF 6

7 Example 4x 1 + 3x 2 + 3x 3 + x 4 + x 5 < 7 As soon as x 1 becomes true, x 2 and x 3 must be set to false (otherwise the constraint will be falsified) New Encodings ofpseudo-boolean Constraintsinto CNF 7

8 Encoding PB constraints into CNF 1 more or less specialized direct encoding doesn t introduce additional variables exponential UP generally maintains GAC 2 BDD (Binary Decision Diagrams) requires additional variables exponential UP maintains GAC 3 Adder+Comparator (Warner s encoding) requires additional variables polynomial UP does not maintain GAC New Encodings ofpseudo-boolean Constraintsinto CNF 8

9 The trouble with UP on the adder encoding 4x 1 + 3x 2 + 3x 3 + x 4 + x 5 < 7 x 1 = T, all other variables are unknown (U), UP doesn t infer anything x 1 = x 2 = x 3 = T, all other variables are unknown (U), UP doesn t even detect inconsistency T T T 4x 1 3x 2 3x U + U < 7 U U x 4 U x 5 New Encodings ofpseudo-boolean Constraintsinto CNF 9

10 The big question: Does there exist an encoding which is both polynomial and such that UP maintains GAC? Mainly a theoretical question (such an encoding may not be efficient in practice). But the existence of this encoding would narrow the gap between PB constraints and clauses. And the answer is... New Encodings ofpseudo-boolean Constraintsinto CNF 10

11 YES! It does exist, and it is rather easy. Sketch: 1 normalize constraints to use only < 2 for each literal l j, transform the constraint i a i.l i < d into watchdogs i j a i.l i d a j = l j 4 decompose each coefficient a i into binary 5 for each power of two occurring in the binary decomposition, use a unary encoding to sum the variables having a coefficient with this bit set to 1 6 compute the half of each sum and use it as a carry for the next stage 7 compare the result of the final stage with d a j (UP doesn t work well on this) New Encodings ofpseudo-boolean Constraintsinto CNF 11

12 YES! It does exist, and it is rather easy. Sketch: 1 normalize constraints to use only < 2 for each literal l j, transform the constraint i a i.l i < d into watchdogs i j a i.l i d a j = l j 3 add an offset so that the new right term d is a multiple of the power of 2 corresponding to the last stage 4 decompose each coefficient a i into binary 5 for each power of two occurring in the binary decomposition, use a unary encoding to sum the variables having a coefficient with this bit set to 1 6 compute the half of each sum and use it as a carry for the next stage 7 compare the result of the final stage with d (one single bit to compare, this is the trick!) New Encodings ofpseudo-boolean Constraintsinto CNF 11

13 The big picture input variables input variables in the bucket in the bucket input variables in the bucket input variables in the bucket result of the comparison half half half bucket for 2 0 bucket for 2 1 bucket for 2 2 bucket for 2 3 New Encodings ofpseudo-boolean Constraintsinto CNF 12

14 Rewrite coefficient in binary 10x 1 + 7x 2 + 3x 3 is rewritten as (8 + 2)x 1 + ( )x 2 + (2 + 1)x 3 New Encodings ofpseudo-boolean Constraintsinto CNF 13

15 Rewrite coefficient in binary 10x 1 + 7x 2 + 3x 3 is rewritten as (8 + 2)x 1 + ( )x 2 + (2 + 1)x 3 and then as (x 1 ) (x 2 ) (x 1 + x 2 + x 3 ) (x 2 + x 3 ).2 0 New Encodings ofpseudo-boolean Constraintsinto CNF 13

16 Rewrite coefficient in binary 10x 1 + 7x 2 + 3x 3 is rewritten as (8 + 2)x 1 + ( )x 2 + (2 + 1)x 3 and then as (x 1 ) (x 2 ) (x 1 + x 2 + x 3 ) (x 2 + x 3 ).2 0 input variables multiplied by a given power of 2 form a bucket variables in a bucket are added and represented in unary the half operator reports a carry from one bucket (for 2 i ) to the next one (for 2 i+1 ) the last bucket represents the sum (almost) New Encodings ofpseudo-boolean Constraintsinto CNF 13

17 Unary representation n bits encode an integer x between 0 and n X = 0 X = 1 X = 2 X = x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 4 x 5 New Encodings ofpseudo-boolean Constraintsinto CNF 14

18 Convention 1s must be at the beginning, 0s must be at the end so, the unary representation 1 1 U U 0 x 1 x 2 x 3 x 4 x 5 encodes a number which is 2 and 4 any input vector of Booleans can be converted to this representation by a cascade of unary adders/sorters. UP actually does the conversion (sorts the bits). New Encodings ofpseudo-boolean Constraintsinto CNF 15

19 Unary adder (Totalizer) sum of two numbers in unary notation X + Y = Z 1 1 U U 0 x 1 x 2 x 3 x 4 x U U 0 0 y 1 y 2 y 3 y 4 y 5 = U U U U z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 z 10 can be encoded with simple clauses x i y j = z i+j (meaning if X i and Y j then Z i + j) to deal with 1s. Same principle with 0s but we don t need them in our context. or by using sorting networks (asymptotically more efficient) New Encodings ofpseudo-boolean Constraintsinto CNF 16

20 Half operator Retaining bits of even indices in the unary representation of X gives the unary representation of X/2 Example x 1 x 2 x 3 x 4 x 5 x x 2 x 4 x 6 New Encodings ofpseudo-boolean Constraintsinto CNF 17

21 Comparator and offset Unit Propagation doesn t work well on a usual comparator Solution: add the same constant to both side of the constraint so that the right term becomes a multiple of 2 max (the weight of the last bucket) this only adds a constant term to the buckets (easy) but most importantly it makes the comparator trivial (the result of the comparator is just one output bit of the last adder) New Encodings ofpseudo-boolean Constraintsinto CNF 18

22 Properties of this encoding Unit Propagation maintains GAC O(n 2 log(n) log(a max )) variables O(n 3 log(n) log(a max )) clauses New Encodings ofpseudo-boolean Constraintsinto CNF 19

23 Variant instead of generating each literal implied by a constraint, we may just want to identify inconsistencies as soon as they appear it simplifies the encoding (becauses unit refutation is more powerful than unit propagation) can make sense with a CDCL solver: the solver will learn the relevant conflicts and will use them later for unit propagation New Encodings ofpseudo-boolean Constraintsinto CNF 20

24 Experiments some encouraging preliminary experiments on Bin-packing instances (see paper) Olivier Bailleux submitted his solver BoolVar to the PB09 competition (only deals with decision problems). This solver translates PB constraints into CNF (using different encodings) and then calls minisat. New Encodings ofpseudo-boolean Constraintsinto CNF 21

25 Preview of results in category DEC-SMALLINT-LIN approximately 200 instances solved Rank Solver #instances solved (relative) 1 solver A solver B solver C solver D solver E solver F +2 7 BoolVar 0 8 solver G -4 9 solver H solver I -124 New Encodings ofpseudo-boolean Constraintsinto CNF 22

26 Conclusion Contributions of this paper a new encoding of PB constraints into CNF which is both polynomial and which allows UP to maintain GAC a variant of this encoding which is shorter and which allows UP to detect inconsistencies narrowing the (theoretical) gap between PB constraints and clauses Perspective optimize the encoding by identifying and factoring frequent sums Open question could it be possible that such a SAT encoding be more efficient in practice than a dedicated PB solver? New Encodings ofpseudo-boolean Constraintsinto CNF 23