DLL DLL AlgorithmAlgorithm

How to realize search in hardware1. We need to realize hardware stack2. Tree search is organized with hardware stack.3. The node of the tree is variable.4. The item on the stack is a sum of three literals (in 3SAT,

can be more in general POS SAT).5. Encoding: 10 = negated variable, 01 = positive variable, 11

= no variable, 00 = mark or not used.6. Example: 1010011101 with order of variables abcd from

left means a’ + b’ + c + e (d variable missing)7. This encoding is standard to logic, is used in Cube Calculus.

DLL Algorithm

• Davis, Logemann and Loveland– M. Davis, G. Logemann and D. Loveland, “A Machine

Program for Theorem-Proving”, Communications of ACM, Vol. 5, No. 7, pp. 394-397, 1962

• Basic framework for many modern SAT solvers• Also known as DPLL for historical reasons

• DFS – depth first search

Possible Possible representationrepresentation

a b c d Marker, etc

10 01 01 11 1=used

01 11 01 01 0=unused

01 11 01 10

01 11 10 01

etc

• Used = converted to 1, satisfiedUsed = converted to 1, satisfied• Non-used = not satisfied yetNon-used = not satisfied yet

Think how to check conflict in hardware Think how to check conflict in hardware using our implementation on stack??!using our implementation on stack??!

Problems to think about

1.1. How can I improve my SAT oracle to allow for How can I improve my SAT oracle to allow for implementing DLL in hardware?implementing DLL in hardware?

2.2. How can I represent this problem with How can I represent this problem with Pipelining?Pipelining?

3.3. Can I invent other ideas for backtracking?Can I invent other ideas for backtracking?

4.4. How to use stack for backtracking?How to use stack for backtracking?

5.5. How to represent set of clauses on the stack?How to represent set of clauses on the stack?

You can assume that you solve 3SAT. You can assume that you solve 3SAT. This will simplify the problem.This will simplify the problem.