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+# 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
+