The paper by Heule et al. about hyper-binary resolution using intree-based lookahead is pretty funky. The idea is actually quite simple (and as usual, not exactly trivial to come up with): we re-use past propagations by reversing the order in which literals are normally enqueued.

A simple example

First, a queue is built that starts with a leaf literal and then follows it up through binary clauses until it can. Then it backtracks (adds to the queue a special, * element) and continues. The point of the queue is to have an example order that we can use to dequeue literals from in reverse propagation order. Obviously, there are many different orders in which we can build this queue and I wouldn’t be surprised if there are some nice heuristics one can use. Let’s just assume we have such a queue.

For example if y leads to x, then an example queue will have first element x and then y. So we first enqueue x, propagate, and then we enqueue y. If x already fails, there is no point in enqueuing y (and y is failed along with x). If both y and z lead to x but only z fails, then we don’t have to perform the propagations done by x twice: We enqueue x, propagate, create new decision level, enqueue y, propagate (nothing fails), backtrack 1 decision level, enqueue z, and now we fail. Notice that we didn’t have to propagate x twice even though we probed two literals (y and z) that both entailed x.

Failing mid-way

The paper mentions failed literals that fail mid-way while dequeueing elements. We obviously cannot simply enqueue these literals, as they would be unset next time we backtrack. So these have to be kept in an array and set later, when we are at decision level 0. Further, once we are in a failed state, anything dequeued that is at the same or lower level also fails, so we need to keep an indicator of failure for these literals.

Keeping reasons updated

Let’s suppose we enqueued x and propagated it. Next is y. We enqueue y… but we need to know what is the reason why x got set. The reason is of course the binary clause that we examined when we built the queue: (x, ~y). The reason is needed to be set because we will be jumping backwards through the implication graph to the deepest common ancestor to attach the new hyper-binary clause there. When jumping back, we might need to go back all the way to y, through x. In order to perform transitive reduction (as explained later), we need to know if the binary clause (x, ~y) was redundant or irredundant. This information needs to be stored in the queue and every time we dequeue a new literal y the reason of the previously enqueued literal needs to be set to the inverse of the currently enqueued literal i.e. ~y.

Transitive reduction

Updating reasons becomes a real problem in case we wish to perform transitive reduction. Transitive reduction removes binary clauses that are useless from a binary implication graph reachability perspective. However, if it removes a binary clause that is later used by the queue to update a reason, we encounter a problem. We may update a literal with a reason that is no longer valid as the corresponding binary clause has been replaced by a chain of binary clauses. Later transitive reductions will take into account that this binary clause exists (it doesn’t) and will make further reductions that may be incorrect. In particular, further transitive reductions might remove an element of the chain itself — kind of like biting our own tail.

There seems to be a couple of options to fix the problem:

1. Not to perform transitive reduction at all. This may have been the intention of the designers, as the BCP_NHBR function does not perform transitive reduction.
2. Update the queue to reflect the changed set of binary clauses. Unfortunately this would be very expensive and thus basically not doable in reasonable about of time as far as I can tell.
3. Never remove binary clauses that are used for the queue. This means we need to mark such clauses and then check for markings when removing binary clauses. This is the implementation that I chose. We can immediately unmark a clause once the corresponding element has been dequeued, making it possible to remove it later. In CryptoMiniSat I simply unmark all binary clauses at the end — it’s faster.

Conclusions

I remember some people always asking me why I haven’t yet implemented intree-based probing. It is much faster than normal probing. However, it’s not perfect. For example, it cannot be used to perform a fast depth-first walk of the tree and as such stamping is not really possible while doing it — always updating closing times for already dequeued elements seems to defeat the purpose of the whole idea (i.e. reversing the propagation order). Secondly, I haven’t yet found a way to efficiently perform Stalmarck while doing intree probing. Thirdly, it’s not exactly trivial to implement — as explained above.

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