1. Stronger learning and higher backjumping
Clause minimization
Stronger learning and higher backjumping
Recursive conflict clause reduction (deep /shallow)
Glucose-style
Volunteers and inverse-arcs
ContraMinisat

2. Learnt Conflict Clauses, and minimization
𝑟𝑒𝑎𝑠𝑜𝑛 𝑥 3 = 𝜔 4 , 𝑟𝑒𝑎𝑠𝑜𝑛 𝑥 6 = 𝜔 5 , …
Local / General implication graph 4 1
Learnt Conflict clause: (x10 Ç :x4 Ç x11) 6 
conflict
Decision 3 3 2 5
Redundant – does not change the satisfiability 2 5
𝑥 10 → 𝑥 11
or, 𝑥 11 → 𝑥 10
Clause minimization: Drop 𝑥 11

3. Learnt Conflict Clauses, and minimization
More generally: if we have a c.c. 𝑐=( 𝑥 0 ∨⋯∨ 𝑥 𝑛 ), and
... all reverse paths on the implication graph hit 𝑐 literals... 4 6 
conflict 5
Redundant – does not change the satisfiability 5
𝑥 𝑖 ∈𝑐
𝑥 𝑖 ∧ 𝑥 𝑗 ∧ 𝑥 𝑘 → 𝑥 𝑙
or, 𝑥 𝑙 → 𝑥 𝑖 ∨ 𝑥 𝑗 ∨ 𝑥 𝑘
Clause minimization: Drop 𝑥 𝑙
𝑥 𝑙
𝑥 𝑗 ∈𝑐
𝑥 𝑘 ∈𝑐

4. More generally... Learnt clause: 𝑐= 𝑥 0 ∨⋯∨ 𝑥 𝑛
For 0≤𝑗≤𝑛, we can remove 𝑥 𝑗 if 𝜑∧ 𝑥 𝑗 → 𝑐∖𝑥 𝑗
Or, more generally, for any 𝑐 ′ ⊆ 𝑐∖ 𝑥 𝑗 , 𝜑∧ 𝑥 𝑗 →𝑐′
equivalently, if 𝜑∧ 𝑐 ′ → 𝑥 𝑗
E.g., in the previous slide: 𝑥 𝑖 ∧ 𝑥 𝑗 ∧ 𝑥 𝑘 → 𝑥 𝑙
Order of checking matters. Can we optimize ?

5. Clause minimization: minisat
Deep mode (recursive conflict-clause reduction):
Traverse recursively 𝑟𝑒𝑎𝑠𝑜𝑛( 𝑥 𝑗 )
Stop if a literal is not in the decision levels of 𝑐 .
Stop if hitting a decision variable.
If all paths hit a variable from 𝑐, remove 𝑥 𝑗 .
e.g., we hit 𝑥 2 , 𝑥 5 , 𝑥 7 , hence 𝑥 2 ∧ 𝑥 5 ∧ 𝑥 7 → 𝑥 𝑗 , or equivalently 𝑥 𝑗 → 𝑥 2 ∨ 𝑥 5 ∨ 𝑥 7
Instead of maintaining the list of decision levels of c’s literals and checking it each time, they maintain an abstraction of that list; hence they might end up visiting such a level, which will anyway result in hitting a decision variable and aborting.
* Minimizing Learned clauses/Sörensson, Biere, SAT’09

6. Clause minimization: minisat
Deep mode (recursive conflict-clause reduction), optimization:
Suppose 𝐿= 𝑥∈𝑐 𝑑.𝑙.(𝑥)
Don’t: maintain 𝐿 as a list and search it.
Do: maintain an abstraction of 𝐿 in a 32- bit int;
Suppose we hit 𝑙∉𝐿 but 𝑙∈𝑎𝑏𝑠(𝐿) => hit a decision variable and abort.
Hence, cannot hinder correctness
* Minimizing Learned clauses/Sörensson, Biere, SAT’09

7. Clause minimization: minisat
Shallow mode: same as deep mode, but only search in current d.l.
Literals seen in the current analysis
* Minimizing Learned clauses/Sörensson, Biere, SAT’09

8. Clause minimization: Glucose
Use binary clauses:
If (𝑐 0 ∨ 𝑥 𝑖 ) is a binary clause, remove 𝑥 𝑖 from 𝑐.
asserting literal
Instead of maintaining the list of decision levels of c’s literals and checking it each time, they maintain an abstraction of that list; hence they might end up visiting such a level, which will anyway result in hitting a decision variable and aborting.
* Chanseok Ho’s Glucose patch to minisat.

9. Volunteers and inverse arcs
Let c = ( 𝑥 0 ∨…∨ 𝑥 𝑛 ) = 𝑟𝑒𝑎𝑠𝑜𝑛(𝑥 0 )
i.e., 𝑥 1 ∧⋯∧ 𝑥 𝑛 → 𝑥 0
𝑐 has the following two properties:
𝑥 1 … 𝑥 𝑛 are currently false
𝑥 0 is the latest in the trail
In a volunteer of 𝑥 0 , item#2 no longer holds:
For some 1≤𝑖≤𝑛, 𝑙𝑒𝑣𝑒𝑙 𝑥 𝑖 >𝑙𝑒𝑣𝑒𝑙( 𝑥 0 )
i.e., some false literal discovered after true literal.

10. Volunteers and inverse arcs
Let c ′ = 𝑥 0 ∨ 𝑦 1 …∨ 𝑦 𝑛
i.e., 𝑦 1 ∧⋯∧ 𝑦 𝑛 → 𝑥 0
𝑐′ has the following two properties:
𝑦 1 … 𝑦 𝑛 are currently false
𝑥 0 is not the latest in the trail // (possibly earlier d.l., possibly was a decision)
e.g., current level: 𝑥 1 ∧ 𝑥 2 ∧⋯∧ 𝑥 𝑛 → 𝑥 0
𝑐 ′ is called a volunteer* for 𝑥 0 : gives us an alternative reason clause.
Volunteers are generalization of inverse-arcs** 6 5 3
level:
* Generalized Conflict-Clause Strengthening for Satisfiability Solvers/Van-Gelder, SAT’11
** generalized framework for conflict analysis / Audemard et al. SAT’08.

11. Inverse arcs Conflict clause A=( α∨ 𝑥 𝑚 ∨ 𝑦 𝑖 )
Conflict clause A=( α∨ 𝑥 𝑚 ∨ 𝑦 𝑖 )
𝛼 – literals with level ≤𝑖
𝑥 – asserting literal
𝑖<𝑚 – back-jumping level
4. Find a volunteer for 𝑦 𝑖 𝑐′=( 𝛽 𝑗 ∨ 𝛾 𝑘 ∨ 𝑦 𝑖 )
𝐴 ′ = 𝑟𝑒𝑠 𝐴,𝑐′,𝑦 = 𝛼∨ 𝛽 𝑗 ∨ 𝛾 𝑘 ∨ 𝑥 𝑚
Ideally: 𝛾 𝑘 =0
Then 𝐴′ is an asserting clause with back-jumping level 𝑗<𝑖.
Otherwise… (next slide)
𝑗<𝑖
𝑘≥𝑖 1
Note that we do not start the process if in A there is more than one literal at level i.
** generalized framework for conflict analysis / Audemard et al. SAT’08.

12. Inverse arcs
Otherwise 𝛾 𝑘 >0 : repeat with 𝐴′ being the conflicting clause.
Not necessarily will be better. But we can apply recursively.
Heuristics for choosing the inverse arc 𝑐′:
Do not start the process if in 𝐴 there is more than one literal at level 𝑖.
Otherwise, allow only if ∀𝑙∈ 𝛾 𝑘 . 𝑙𝑒𝑣𝑒𝑙 𝑙 =𝑚 and 𝑙 seen on the implication graph so far
This guarantees that it will be eliminated during conflict analysis,
… and that we jump to 𝑗 < 𝑖
** generalized framework for conflict analysis / Audemard et al. SAT’08.

13. Inverse arcs can create cycles
Generally adding inverse arcs can lead to cycles in the implication graph, e.g.:
𝑐 0 = 𝑥 0 𝑥 1 𝑥 𝑐 1 =( 𝑥 𝑥 𝑥 3 ) 𝑐 2 = 𝑥 𝑥 2
This, in turn, may lead to cycles in the resolution process.
Such cycles are expensive to detect.
𝑥 0
𝑥 1
𝑥 3
𝑥 2
𝑥 4
** generalized framework for conflict analysis / Audemard et al. SAT’08.

14. Precosat’s solution to volunteers without cycles
Precosat considers volunteers of a specific type:
When learning 𝑐 with asserting literal 𝑥 0 (𝑟𝑒𝑎𝑠𝑜𝑛 𝑥 0 =𝑐)
Consider a clause c′=( 𝑥 0 𝑦 1 ⋯ 𝑦 𝑛 ) a volunteer if 𝑥 0 was last to be assigned in 𝑐’.
⇒ 𝑥 0 did not lead to propagating any 𝑦 𝑖 ,
⇒ no cycles
𝑐′≠𝑟𝑒𝑎𝑠𝑜𝑛( 𝑥 0 ) because of the literal order in the trail.
Precosat apparently has the same goal but looks only at a specific type of volunteers.
* Precosat (unpublished; open source)

15. Example Suppose 𝑐= 𝑥 0 𝑧 , 𝑐′={ 𝑥 0 𝑦 } The trail:
Had 𝑦 been processed first… 𝑟𝑒𝑎𝑠𝑜𝑛 𝑥 0 =𝑐′ 𝑥 … 𝑦
𝑥 0
𝑟𝑒𝑎𝑠𝑜𝑛 𝑥 0 =𝐶
qhead
* Precosat (unpublished; open source)

16. ContraMiniSat BCP is not done directly on the trail as in Minisat
Instead, maintain a priority-queue of implications
Implication = clause + priority
𝑙𝑒𝑣𝑒𝑙, 𝑡𝑟𝑎𝑖𝑙_𝐼𝑛𝑑𝑒𝑥 // lower is better
Rationale: prefer antecedents with low decision levels; among those, prefer stronger literals.
Max. dec. level of the non-watched literals
Index of the implying literal in the local trail
* Contrasat – A contrarian SAT solver (system description) /Van-Gelder