# Davis Putnam Procedure (DP)
```
while True:
	if [] in formula:
		return UNSAT
	//remove all variables that occur only in one phase
	vars = one_phased(formula)
	for var in vars:
		for clause in formula:
			if var in formula:
				remove(clause)
	
	var = decide(formula)
	
	// Add non trivial resolvants
	formula += resolution(var, formula)
	
	for clause in formula:
			if var in formula:
				remove(clause)

	if formula is []:
		return SAT
```
- Notice that 
	- Adding a resolvant always preservers SAT (resolution rule)
	- Adding any clause always preserves UNSAT
	
	- Removing any clause always preserves SAT
	- Removing a clause containing a 'one phase variable' preserves UNSAT (reverse of resolution rule)
# Davis Putnam Logemann Loveland DPLL
## Boolean Constraint Propagation (BCP)
- Given a unit clause $l$
1. Remove all subsumed clauses, i.e. $C$ where $l \in C$
2. Remove $\lnot l$ from all $C$ (which contain it) [Unit Prop](SAT_in_general#Special Case Unit Resolution)
```
def dpll(formula):
	if [] in formula:
		return UNSAT
		vars = one_phased(formula)
	for var in vars:
		for clause in formula:
			if var in formula:
				remove(clause)
	if formula is []:
		return SAT
		
	var = decide()
	if dpll(formula + [var]) == SAT:
		return SAT
	return dpll(formula + [-var])
```
- Correct because of theorem of Shanon
- $f(x) = (x \land f(1)) \lor (\overline x \land f(0))$ 
- 
# Backjumping
- If some conflict has been derived without using specific assignments/ decision 
- We can skip over that decision when flipping: ![[backjump.png]]
- If conflict stems from for example from (1 2) (1 -2) 
- Removes 'bad decisions'/ 'useless' from the trail
- Can reduce search space from exponential to quadratic in number of bad decisions
	- Whatever that means

# CDCL
- DPLL with learning 
- On conflict, learn a conflict clause which will help in further exploration
	- Usually the first UIP of the [implication graph](SAT_in_general#Implication Graph) is used as conflict clause 
- After learning a conflict:
	- Back-jump to the lowest level in the implication graph supporting the UIP 
	- Assign negated UIP literal
	- this is why first UIP is good (“pulls in” as few decision levels as possible)
## Reduction
- Not all clauses are useful forever
	- Useless via Subsumption, or trivially satisfied by units
-  Try to reduce the amount of clauses you need to consider in BCP
- Schedule throwing away clauses 
	- geometric, Luby, arithmetic scheme
- Consider various heuristics 
	 - Size
	 - Last Recently Used (LRU), 
	 - Clause Move To Front (CMTF)
	 - Clause Scores similar to VSIDS
	 - Glucose/ Literal Block Distance (LBD)
		 - LBD = number of distinct decision levels among a (learned) clause's literals
		 - Low LBD ("glue clause"): literals tightly related across few levels -> likely useful, keep
		 - High LBD: literals span many levels, weakly related -> good deletion candidate
	 - Used flag
## Restarts
- Schedule restarts every so often can improve performance
- Throw away the trail
- Keep (most) learned clauses, phase saving, other state
- Idea:
	- The solver might have zoomed in on an efficient part of the search space
		- For SAT instances might stuck in a UNSAT part
		- For UNSAT instances might miss a shorter proof
	- Avoid going down the same path by keeping what was learned
		- Clauses, phase saving, variable scoring

## Phase saving
- Whenever assigning a variable, save the phase it is assigned
- When deciding on a variable assign the saved phase
- Idea: Go down the right branch of the search tree