Resolution Based Search John Hooker GSIA 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: x j – atomic proposition x j – negated proposition  – (inclusive) “or” x j, x j – literals x 1  x 2 – clause

Resolution x 1  x 2  x 4 x 1  x 2  x x 2  x 3  x 4 Parents of resolvent Resolvent Absorption x1  x2x1  x2 absorbs (is contained in) x1  x2  x3x1  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.

Sample Problem

Replace negative variables with complements = 1 – x 3

Resolution-Based Search Consider the partial assignment (x 1, x 2, x 3, x 4, x 5 ) = (0,0,0,0,?) It is redundant; i.e., already violates a constraint x 3 + x 4  1 Add a nogood to avoid repeating this partial assignment: Minimal nogood: x 1  x 2  x 3  x 4 Dependent nogood: x 3  x 4

Partial assignment Nogoods generated Nogoods after resolution 1234 x3  x4x3  x4 x3  x4x3  x4 x3  x4x3  x4 x3x3 x3  x4x3  x4 x3x4x3x x1  x5x1  x5 x3x4x1  x5x3x4x1  x5 x1  x3x1  x3 x3x4x1x3x4x1 x1  x3x1  x3  Resolution-Based Search (x 1, x 2, x 3, x 4, x 5 ) = (0,0,0,0,?) 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  x4x3  x4 x3  x4x3  x4 x3  x4x3  x4 x3x3 x3  x4x3  x4 x3x4x3x x1  x5x1  x5 x3x4x1  x5x3x4x1  x5 x1  x3x1  x3 x3x4x1x3x4x1 x1  x3x1  x3  Resolution-Based Search Feasible solutions in green. Dependent nogoods based on objective function.

Depth-first branching = Resolution-Based Search with Parallel Resolution and Minimal Nogoods x 1  x 2  x 4 x 2  x 3  x x 1  x 2  x 3 penultimate literals in blue ultimate literals in red resolve on ultimate literals only parallel resolvent

Parallel Absorption parallel absorbs x1  x2x1  x2 x2  x3  x4x2  x3  x4 Ultimate literal in first clause is penultimate in second clause

Depth-first branching tree

Leaf node Partial assignment Nogoods generated Nogoods after parallel resolution x1x2x3x4x5x1x2x3x4x5 x1x2x3x4x5x1x2x3x4x5 7 x1x2x3x4x5x1x2x3x4x5 x1x2x3x4x1x2x3x x1x2x3x4x1x2x3x4 x1x2x3x1x2x x1x2x3x4x1x2x3x4 x1x2x3x1x2x3x4x1x2x3x1x2x3x x1x2x3x4x1x2x3x4 x1x2x1x x1x2x5x1x2x5 x1x2x1x2x5x1x2x1x2x x1x2x5x3x4x1x2x5x3x4 x1x2x1x2x5x1x2x5x3x4x1x2x1x2x5x1x2x5x3x x1x2x5x3x4x1x2x5x3x4 x1x2x1x2x5x1x2x5x3x1x2x1x2x5x1x2x5x3 Partial assignments must conform to nogoods. Must begin with values opposite to signs of penultimate literals.

Leaf node Partial assignment Nogoods generated Nogoods after parallel resolution x1x2x3x4x5x1x2x3x4x5 x1x2x3x4x5x1x2x3x4x5 7 x1x2x3x4x5x1x2x3x4x5 x1x2x3x4x1x2x3x x1x2x3x4x1x2x3x4 x1x2x3x1x2x x1x2x3x4x1x2x3x4 x1x2x3x1x2x3x4x1x2x3x1x2x3x x1x2x3x4x1x2x3x4 x1x2x1x x1x2x5x1x2x5 x1x2x1x2x5x1x2x1x2x x1x2x5x3x4x1x2x5x3x4 x1x2x1x2x5x1x2x5x3x4x1x2x1x2x5x1x2x5x3x x1x2x5x3x4x1x2x5x3x4 x1x2x1x2x5x1x2x5x3x1x2x1x2x5x1x2x5x

Leaf node Partial assignment Nogoods generated Nogoods after parallel resolution x1x2x5x3x4x1x2x5x3x4 x1x2x1x2x5x1x2x5x3x1x2x5x3x4x1x2x1x2x5x1x2x5x3x1x2x5x3x x1x2x5x3x4x1x2x5x3x4 x1x x1x5x3x4x2x1x5x3x4x2 x1x1x5x3x4x2x1x1x5x3x4x x1x5x3x4x2x1x5x3x4x2 x1x1x5x3x4x1x1x5x3x x1x5x3x4x1x5x3x4 x1x1x5x3x1x1x5x x1x5x3x4x1x5x3x4 x1x1x5x3x1x5x3x4x1x1x5x3x1x5x3x x1x5x3x4x1x5x3x4 x1x1x5x1x1x x1x5x1x5 

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.

Replace minimal nogoods x 1  x 2  x 3  x 4  x 5 with dependent nogoods x 1  x 5 x 1  x 2  x 5 “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 x3x4x3x4 x3x4x3x x3x4x3x4 x3x x3x4x3x4 x3x3x4x3x3x x1x5x1x5 x3x3x4x1x5x3x3x4x1x optimal x1 x3x1 x3 x3x1x3x x3x4x3x4 x3x1x3x4x3x1x3x x3x1x3x1  Must choose x 4 as ultimate literal Either literal can be ultimate Only x 3 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 x1x2x5x1x2x5 x1x2x5x1x2x5 x 1 x 2 x 3 x 4 x x1x5x1x5 x1x5x2x1x1x5x2x1 x 1 x 5 x 2 x 3 x x3x4x3x4 x1x5x2x1x3x4x1x5x2x1x3x4 x 1 x 5 x 2 x 3 x x3x4x3x4 x1x5x2x1x3x1x5x2x1x3 215 x 1  x 5 x 2  x 1 x 3 x 1  x 5 x 2  x 5 x 1 x 5 x 2 x 3 x 4 Generalized partial order dynamic backtracking x 1 and x 2 fixed x 1 precedes x 5, but both can become ultimate. This merges two groups.

Partial assignment Nogoods generated Nogoods after parallel resolution Ultimate variable groups 125 x1x2x5x1x2x5 x1x2x5x1x2x5 x 1 x 2 x 3 x 4 x x1x5x1x5 x1x5x2x1x1x5x2x1 x 1 x 5 x 2 x 3 x x3x4x3x4 x1x5x2x1x3x4x1x5x2x1x3x4 x 1 x 5 x 2 x 3 x x3x4x3x4 x1x5x2x1x3x1x5x2x1x3 215 x 1  x 5 x 2  x 1 x 3 x 1  x 5 x 2  x 5 x 1 x 5 x 2 x 3 x 4 x 1  x 5 resolves with x 1  x 2  x 5 x 2 fixed ultimate variables must be x 4 only, or x 3 and x 4.. The latter merges 2 groups

Partial assignment Nogoods generated Nogoods after parallel resolution Ultimate variable groups x3x4x3x4 x 1  x 5 x 2  x 1 x 3 x 1  x 5 x 2  x 5 x 4 x 1 x 5 x 2 x 3 x optimal x3x1x3x1 x 1  x 5 x 3 x 1  x 5 x 4 x 2 x 1 x 5 x 2 x 3 x optimal x3x1x3x1 x 1  x 5 x 3 x 1  x 5 x 4 x 2 x 3  x 1 x 3  x 5 x 1 x 5 x 2 x 3 x x3x1x3x1  x 1 x 5 x 2 x 3 x 4 Completion of the algorithm