Resolution Based Search

John HookerGSIA

Feb 2003

Previous Work

This talk generalizes & unifies contributions of:

• Gaschnig 1977 (backjumping, backmarking, backchecking)

• Ginsberg 1993 (dynamic backtracking)

• McAllester 1993 (partial order dynamic backtracking)

• Ginsberg & McAllester 1994 (generalized dynamic backtracking)

Some Definitions

In propositional logic:

xj – atomic proposition

xj – negated proposition

– (inclusive) “or”

xj, xj – literals

x1 x2 – clause

Resolution

x1 x2 x4

x1 x2 x3

-------------------------------

x2 x3 x4

Parents of resolvent

Resolvent

Absorption

x1 x2 absorbs (is contained in) x1 x2 x3

For clauses, absorption implication

Resolution Algorithm

To apply the resolution algorithm to a clause set S:

While there are 2 clauses in S with a resolvent R that no clause in S absorbs:

Remove from S all clauses absorbed by R.

Add R to S.

Prime Implicates

Clause C is a prime implicate of clause set S if C is a strongest clause implied by S.

That is, if S implies a clause C that absorbs C then C = C.

Theorem (Quine): The resolution algorithm converts S to the set of prime implicates of S.

}1,0{1234

032

1

3324

332

3223subject tominimize

543

4321

43

521

541

5321

31

jxxxx

xxxx

xx

xxx

xxx

xxxxxx

Sample Problem

3234

332

1

3324

232

2223subject tominimize

543

4321

43

521

541

5321

31

xxx

xxxx

xx

xxx

xxx

xxxxxx

Replace negative variables with complements

= 1 – x3

Resolution-Based Search

Consider the partial assignment(x1, x2, x3, x4, x5) = (0,0,0,0,?)

It is redundant; i.e., already violates a constraint

x3 + x4 1

Add a nogood to avoid repeating this partial assignment:

Minimal nogood: x1 x2 x3 x4

Dependent nogood: x3 x4

Partial assignment

Nogoods generated

Nogoods after

resolution

1234 x3 x4 x3 x4

1234 x3 x4 x3

1234 x3 x4

x3

x4

12345 x1 x5

x3

x4

x1 x5

12345 x1 x3

x3

x4

x1

12345 x1 x3

Resolution-Based Search

(x1, x2, x3, x4, x5) = (0,0,0,0,?)

3234

332

1

3324

232

2223subject tominimize

543

4321

43

521

541

5321

31

xxx

xxxx

xx

xxx

xxx

xxxxxx

Since nogoods after resolution are prime implicates, we can find a partial assignment consistent with them without backtracking

Partial assignment

Nogoods generated

Nogoods after

resolution

1234 x3 x4 x3 x4

1234 x3 x4 x3

1234 x3 x4

x3

x4

12345 x1 x5

x3

x4

x1 x5

12345 x1 x3

x3

x4

x1

12345 x1 x3

Resolution-Based Search

3234

332

1

3324

232

2223subject tominimize

543

4321

43

521

541

5321

31

xxx

xxxx

xx

xxx

xxx

xxxxxx

Feasible solutions in green.

Dependent nogoods based on objective function.

Depth-first branching = Resolution-Based Search with Parallel Resolution and Minimal Nogoods

x1 x2 x4

x2 x3 x4

-------------------------------

x1 x2 x3

penultimate literals in blue

ultimate literals in red

resolve on ultimate literals only

parallel resolvent

Parallel Absorption

parallel absorbsx1 x2 x2 x3 x4

Ultimate literal in first clause is penultimate in second clause

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31

1

2 2

3 3

4 4 4 4

5 5

5 5

3 3

4 4 4 4

5 5

3 3

4 4 4 4

2 2

1

Depth-first branching tree

Leaf node

Partial assignment

Nogoods generated

Nogoods after parallel resolution

6 12345 x1x2x3x4x5 x1x2x3x4x5

7 12345 x1x2x3x4x5 x1x2x3x4

8 1234 x1x2x3x4 x1x2x3

10 1234 x1x2x3x4

x1x2x3

x1x2x3x4

11 1234 x1x2x3x4 x1x2

13 125 x1x2x5

x1x2

x1x2x5

16 12534 x1x2x5x3x4

x1x2

x1x2x5

x1x2x5x3x4

17 12534 x1x2x5x3x4

x1x2

x1x2x5

x1x2x5x3

Partial assignments must conform to nogoods.

Must begin with values opposite to signs of penultimate literals.

Leaf node

Partial assignment

Nogoods generatedNogoods after

parallel resolution

6 12345 x1x2x3x4x5 x1x2x3x4x5

7 12345 x1x2x3x4x5 x1x2x3x4

8 1234 x1x2x3x4 x1x2x3

10 1234 x1x2x3x4

x1x2x3

x1x2x3x4

11 1234 x1x2x3x4 x1x2

13 125 x1x2x5

x1x2

x1x2x5

16 12534 x1x2x5x3x4

x1x2

x1x2x5

x1x2x5x3x4

17 12534 x1x2x5x3x4

x1x2

x1x2x5

x1x2x5x3

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2

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5

6 7

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9

10 11

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13 14

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19 20

1

2 2

3 3

4 4 4 4

5 5

5 5

3 3

4 4 4 4

21

22

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24

25 26

27

28

29 30

31

5 5

3 3

4 4 4 4

2 2

Leaf nodePartial

assignmentNogoods generated

Nogoods after parallel resolution

19 12534 x1x2x5x3x4

x1x2

x1x2x5

x1x2x5x3

x1x2x5x3x4

20 12534 x1x2x5x3x4 x1

25 15342 x1x5x3x4x2

x1

x1x5x3x4x2

26 15342 x1x5x3x4x2

x1

x1x5x3x4

27 1534 x1x5x3x4

x1

x1x5x3

29 1534 x1x5x3x4

x1

x1x5x3

x1x5x3x4

30 1534 x1x5x3x4

x1

x1x5

31 15 x1x5

1

2

3

4

5

6 7

8

9

10 11

12

13 14

15

16 17

18

19 20

21

22

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24

25 26

27

28

29 30

31

1

2 2

3 3

4 4 4 4

5 5

5 5

3 3

4 4 4 4

5 5

3 3

4 4 4 4

2 2

1

violates

2223 5321 xxxx

232 541 xxx

violates

Backtrack to here

Since infeasibility established by this point.

Backjumping

Backjumping = Parallel resolution search that uses minimal nogoods, except that dependent nogoods are used whenever two consecutive nogoods have a parallel resolvent.

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1

2 2

3 3

4 4 4 4

5 5

5 5

3 3

4 4 4 4

5 5

3 3

4 4 4 4

2 2

1Replace minimal nogoods

x1x2x3x4x5

x1x2x3x4x5

with dependent nogoods

x1x5

x1x2x5

“missing” literals must be added prior to last penultimate literal

Backchecking and Backmarking

These also have interpretations as forms of parallel resolution search.

Partial Order Dynamic Backtracking

Partial order = dynamic backtracking

Parallel resolution search that uses minimal nogoods.

The ultimate literals in the nogoods imply a partial order on the variables in which penultimate variables must precede ultimate ones.

The choice of ultimate literal in a new nogood must be consistent with this partial order.

Partial assignment

Nogoods generated

Nogoods after parallel

resolution

1234 x3x4 x3x4

3124 x3x4 x3

1234 x3x4

x3

x3x4

31245 x1x5

x3

x3x4

x1x5

13245optimal

x1 x3

x3

x1

12345 x3x4

x3

x1

x3x4

31245 x3x1

Must choose x4 as ultimate literal

Either literal can be ultimate

Only x3 is fixed; others chosen as desired

Partial Order Dynamic Backtracking

Note that a nogood can occur twice

Theorem. Any partial assignment that conforms to the current nogoods in parallel resolution search is nonredundant.

Theorem. Parallel resolution search terminates after an exhaustive implicit enumeration.

Theorem. The complexity of parallel resolution in the context of parallel resolution search is O(N), where N is the number of literals in the current nogoods.

Generalized Partial Order Dynamic Backtracking

Partition variables into groups of ultimate variables.

Associate a group with each new nogood to define its ultimate variables (possibly > 1).

Ultimate variables define partial order: penultimate variables precede them.

Groups merge when their representatives are chosen to be ultimate in the same clause.

Parallel resolve on any ultimate variable. Parallel absorption also generalized.

Fewer penultimate variables more freedom to search.

All variables ultimate full resolution search.

Partial assignment

Nogoods generated

Nogoods after parallel

resolution

Ultimate variable groups

125 x1x2x5 x1x2x5 x1 x2 x3 x4 x5

125 x1x5

x1x5

x2x1

x1x5 x2 x3 x4

234 x3x4

x1x5

x2x1

x3x4

x1x5 x2 x3 x4

234 x3x4

x1x5

x2x1

x3

x1x5 x2 x3x4

215 x1 x5

x1x5

x2x1

x3

x1 x5

x2x5

x1x5 x2 x3x4

Generalized partial order dynamic backtracking

x1 and x2 fixed

x1 precedes x5, but both can become ultimate. This merges two groups.

Partial assignment

Nogoods generated

Nogoods after parallel

resolution

Ultimate variable groups

125 x1x2x5 x1x2x5 x1 x2 x3 x4 x5

125 x1x5

x1x5

x2x1

x1x5 x2 x3 x4

234 x3x4

x1x5

x2x1

x3x4

x1x5 x2 x3 x4

234 x3x4

x1x5

x2x1

x3

x1x5 x2 x3x4

215 x1 x5

x1x5

x2x1

x3

x1 x5

x2x5

x1x5 x2 x3x4

x1x5 resolves with x1x2x5

x2 fixed

ultimate variables must be x4 only, or x3 and x4..

The latter merges 2 groups

Partial assignment

Nogoods generated

Nogoods after parallel

resolution

Ultimate variable groups

2134 x3x4

x1x5

x2x1

x3

x1 x5

x2x5

x4

x1x5 x2 x3x4

21345optimal

x3x1

x1x5

x3

x1 x5

x4

x2

x1x5 x2 x3x4

12345optimal

x3x1

x1x5

x3

x1 x5

x4

x2

x3x1

x3x5

x1x5 x2 x3x4

31245 x3x1 x1x5 x2 x3x4

Completion of the algorithm