z3/PARALLEL_PROJECT_NOTES.md

Parallel project notes

We track notes for updates to smt_parallel.cpp and possibly solver/parallel_tactic.cpp

Variable selection heuristics

• Lookahead solvers:
  • lookahead in the smt directory performs a simplistic lookahead search using unit propagation.
  • lookahead in the sat directory uses custom lookahead solver. They both proxy on a cost model where the most useful variable to branch on is the one that _minimizes_ the set of new clauses maximally through unit propagation. In other words, if a literal _p_ is set to true, and _p_ occurs in clause \\neg p \\vee q \\vee r, then it results in reducing the clause from size 3 to 2 (because \\neg p will be false after propagating _p_).
• VSIDS:
  • As referenced in Matteo and Antti's solvers.
  • Variable activity is a proxy for how useful it is to case split on a variable during search. Variables with a higher VSIDS are split first.
  • VSIDS is updated dynamically during search. It was introduced in the paper with Moscovitz, Malik, et al in early 2000s. A good overview is in Armin's tutorial slides (also in my overview of SMT).
  • VSIDS does not keep track of variable phases (if the variable was set to true or false).
• Proof prefix:
  • Collect the literals that occur in learned clauses. Count their occurrences based on polarity. This gets tracked in a weighted score.
  • The weight function can be formulated to take into account clause sizes.
  • The score assignment may also decay similar to VSIDS.
  • We could also use a doubly linked list for literals used in conflicts and keep reinsert literals into the list when they are used. This would be a "Variable move to front" (VMTF) variant.
• From local search:
  • Note also that local search solvers can be used to assign variable branch priorities.
  • We are not going to directly run a local search solver in the mix up front, but let us consider this heuristic for completeness.
  • The heuristic is documented in Biere and Cai's journal paper on integrating local search for CDCL.
  • Roughly, it considers clauses that move from the UNSAT set to the SAT set of clauses. It then keeps track of the literals involved.
• Assignment trails:
  • We could also consider the assignments to variables during search.
  • Variables that are always assigned to the same truth value could be considered to be safe to assign that truth value.
  • The cubes resulting from such variables might be a direction towards finding satisfying solutions.

Algorithms

This section considers various possible algorithms. In the following, F refers to the original goal, T is the number of CPU cores or CPU threads.

Base algorithm

The existing algorithm in smt_parallel is as follows:

1. Run a solver on F with a bounded number of conflicts.
2. If the result is SAT/UNSAT, or UNKNOWN with an interrupt or timeout, return. If the maximal number of conflicts were reached continue.
3. Spawn T solvers on F with a bounded number of conflicts, wait until a thread returns UNSAT/SAT or all threads have reached a maximal number of conflicts.
4. Perform a similar check as in 2.
5. Share unit literals learned by each thread.
6. Compute unit cubes for each thread T.
7. Spawn T solvers with F \wedge \ell, where \ell is a unit literal determined by lookahead function in each thread.
8. Perform a similar check as in 2.
9. Shared unit literals learned by each thread, increase the maximal number of conflicts, go to 3.

Algorithm Variants

• Instead of using lookahead solving to find unit cubes use the proof-prefix based scoring function.
• Instead of using independent unit cubes, perform a systematic (where systematic can mean many things) cube and conquer strategy.
• Change the synchronization barrier discipline.
• [Future] Include in-processing