Tag Archives: model counting

How Approximate Model Counting Works

Approximate model counting allows to count the number of solutions (or “models”) to propositional satisfiability problems. This problem seems trivial at first given a propositional solver that can find a single solution: find one solution, ban it, ask for another one, ban it, etc. until all solutions are counted. The issue is that sometimes, the number of solutions is 2^50 and so counting this way is too slow. There are about 2^266 atoms in the universe, so counting anywhere near that is impossible using this method.

Exact Counting

Since we cannot count 1-by-1, we are then left with trying to count in some smarter way. There are a bunch of methods to count exactly, the simplest is to cut the problem on a variable, count when the variable is True, count when the variable is False, recursively, and add them all up. Given caching of components that recur while “cutting” away, this can be quite successful, as implemented by sharpSAT (see Marc Thurley’s paper).

These counters can scale quite well, but have some downsides. In particular, when the memory runs out, the cache needs to be groomed, sometimes resorting back to 1-by-1 counting, which we know will fail as there is no way 2^200 can be counted 1-by-1 in any reasonable amount of time. The caching systems used are smart, though, retaining last-used entries when the cache needs to be groomed. However, sometimes this grooming algorithm can lead to cyclic behaviour that effectively simulates 1-by-1 counting.

Approximate Counting

What Chakraborty, Meel and Vardi did in their paper, was to create a counter that counts not exactly, but “probably approximately correctly”. This jumble of terms basically means: there is a certain probability that the counting is correct, within a given threshold. We can both improve the probability and the threshold given more CPU time spent. In practical terms, the probability can be set to be over 99.99% and the threshold can be set to be under 20%, still beating exact counters. Note that 20% is not so much, as e.g. 2^60*1.2 = 2^(60.3).

A Galton Box

So what’s the trick? How can we approximately count and give guarantees about the count? The trick is in fact quite simple. Let’s say you have to count balls, and there are thousands of them. One way to do it is to count 1-by-1. But, if you have a machine that can approximately half the number of balls you have, it can be done a lot faster: you half the balls, then check if you have at least 5 doing 1-by-1 counting. If you do, you half them again, and check if you have at least 5, etc. Eventually, let’s say you halved it 11 times and now you are left with 3 balls. So approximately, you must have had 3*2^11 = 6144 balls to begin with. In the end you had to execute the 1-by-1 count only 11*5+3+1 =59 times — a lot less than 6144! This is the idea used by ApproxMC.

Approximately Halving Using XORs

The “approximate halving” function used by ApproxMC is the plain XOR function, populated with variables picked with 50% probability. So for example if we have variables v1…v10, we pick each variable with 50% probability and add them into the same XOR. Let’s say we picked v1,v2,v5 and v8. The XOR would then be: v1⊕v2⊕v5⊕v8=1. This XOR is satisfied if an odd number of variables from the set {v1,v2,v5,v8} are 1. This intuitively forbids about half the solutions. The “intuitively” part of course is not enough, and if you read the original paper you will find the rigorous mathematical proof for the approximate halving of solutions this XOR function provides.

OK, so all we need to do is add these XORs to our original problem and we are done! This sounds easy, but there is a small hurdle and there is a big hurdle associated with this.

The small hurdle is that the original problem is a CNF, i.e. a conjunction of disjunctions, looking like “(v1 OR v2) AND (v2 OR not v3 OR not v4) AND…”. The XOR obviously does not look like this. The straightforward translation of XOR into CNF is exponential, so we need to add some variables to cut them smaller. It’s not that hard to figure this out and eventually add all the XORs into the CNF.

XORs, CDCL, and Gauss-Jordan Elimination

The larger hurdle is that once the XORs are in the CNF using the translation, the CNF becomes exponentially hard to solve using standard CDCL as used in most SAT solvers. This is because Gauss-Jordan elimination is exponentially hard for the standard CDCL to perform — but we need Gauss-Jordan elimination because the XORs will interact with each other, as we expect there to be many of them. Without being able to resolve the XORs with each other efficiently and derive information from them, it will be practically impossible to solve the CNF.

The solution is to tightly integrate Gauss-Jordan elimination into the solving process. CryptoMiniSat was the first solver to do this tight integration (albeit only for Gaussian elimination, which is sufficient). Other solvers have followed, in particular the work by Tero Laitinen and work by Cheng-Shen Han and Jie-Hong Roland Jiang. CryptoMiniSat currently uses the code by Han and Jiang after some cleanup and updates.

The latest work on ApproxMC and CryptoMiniSat has added one more thing besides the tight integration of the CDCL cycle: it now allows in- and pre-processing to occur while the XORs are inside the system. This brought some serious speedups as pre- and inprocessing are important factors in SAT solving. In fact, all modern SAT solvers strongly depend on them being active and working.

Concluding Remarks

We have gone from what model counting is, through how approximate counting works from a high-level perspective, all the way to the detail of running such a system inside a modern SAT solver. In case you want to try it out, you can do it by downloading the pre-built binaries or building it from source.

ApproxMCv3, a modern approximate model counter

This blogpost and its underlying work has been brewing for many years, and I’m extremely happy to be able to share it with you now. Kuldeep Meel and myself have been working very hard on speeding up approximate model counting for SAT and I think we have made real progress. The research paper, accepted at AAAI-19 is available here. The code is available here (release with static binary here). The main result is that we can solve a lot more problems than before. The speed of solving is orders(!) of magnitude faster than the previous best system:


The idea of approximate model counting, originally by Chakraborty, Meel and Vardi was a huge hit back in 2013, and many papers have followed it, trying to improve its results. All of them were basically tied to CryptoMiniSat, the SAT solver that I maintain, as all of them relied on XOR constraints being added to the regular CNF of a typical SAT problem.

So it made sense to examine what CryptoMiniSat could do to improve the speed of approximate counting. This time interestingly coincided with me giving up on XORs in CryptoMiniSat. The problem was the following. A lot of new in- and preprocessing systems were being invented, mostly by Armin Biere et al, and I quickly realised that I simply couldn’t keep adding them, because they didn’t take into account XOR constraints. They handled CNF just fine, but not XORs. So XORs became a burden, and I removed them in versions 3 and 4 of CryptoMiniSat. But there was need, and Kuldeep made it very clear to me that this is an exciting area. So, they had to come back.


But how to both have and not have XOR constraints? Re-inventing all the algorithms for XORs was not a viable option. The solution I came up with was a rather trivial one: forget the XORs during inprocessing and recover them after. The CNF would always remain the source of truth. Extracting all the XORs after in- and preprocessing would allow me to run the Gauss-Jordan elimination on the XORs post-recovery. So I can have the cake and eat it too.

The process is conceptually quite easy:

  1. Blast all XORs into clauses that are in the input using intermediate variables. I had all the setup for this, as I was doing Bounded Variable Addition  (also by Biere et al.) so I didn’t have to write code to “hide” these additional variables.
  2. Perform pre- or inprocessing. I actually only do inprocessing nowadays (as it has faster startup time). But preprocessing is just inprocessing at the start ;)
  3. Recover the XORs from the CNF. There were some trivial methods around. They didn’t work as well as one would have hoped, but more on that later
  4. Run the CDCL and Gauss-Jordan code at the same time.
  5. Destroy the XORs and goto 2.

This system allows for everything to be in CNF form, lifting the XORs out when necessary and then forgetting them when it’s convenient. All of these steps are rather trivial, except, as I later found out, recovery.

XOR recovery

Recovering XORs sounds like a trivial task. Let’s say we have the following clauses

This is conceptually equivalent to the XOR v1+v2+v3=1. So recovering this is trivial, and has been done before, by Heule in particular, in his PhD thesis. The issue with the above is the following: a stronger system than the above still implies the XOR, but doesn’t look the same. Let me give an example:

This is almost equivalent to the previous set of clauses, but misses a literal from one of the clauses. It still implies the XOR of course. Now what? And what to do when missing literals mean that an entire clause can be missing? The algorithm to recover XORs in such cases is non-trivial. It’s non-trivial not only because of the complexity of how many combinations of missing literals and clauses there can be (it’s exponential) but because one must do this work extremely fast because SAT solvers are sensitive to time.

The algorithm that is in the paper explains all the bit-fiddling and cache-friendly data layout used along with some fun algorithms that I’m sure some people will like. We even managed to use compiler intrinsics to use target-specific assembly instructions for hamming weight calculation. It’s a blast. Take a look.

The results

The results, as shown above, speak for themselves. Problems that took thousands of seconds to solve can now be solved under 20. The reason for such incredible speedup is basically the following. CryptoMiniSatv2 was way too clunky and didn’t have all the fun stuff that CryptoMiniSatv5 has, plus the XOR handling was incorrect, loosing XORs and the like. The published algorithm solves the underlying issue and allows CNF pre- and inprocessing to happen independent of XORs, thus enabling CryptoMiniSatv5 to be used in all its glory. And CryptoMiniSatv5 is fast, as per the this year’s SAT Competition results.

Some closing words

Finally, I want to say thank you to Kuldeep Meel who got me into the National University of Singapore to do the work above and lots of other cool work, that we will hopefully publish soon. I would also like to thank the National Supercomputing Center Singapore  that allowed us to run a ton of benchmarks on their machines, using at least 200 thousand CPU hours to make this paper. This gave us the chance to debug all the weird edge-cases and get this system up to speed where it beats the best exact counters by a wide margin. Finally, thanks to all the great people I had the chance to meet and sometimes work with at NUS, it was a really nice time.