Searching for RH Counterexamples — Exploring Data

We’re ironically searching for counterexamples to the Riemann Hypothesis.

  1. Setting up Pytest
  2. Adding a Database
  3. Search Strategies
  4. Unbounded integers
  5. Deploying with Docker
  6. Performance Profiling
  7. Scaling up
  8. Productionizing

In the last article we added a menagerie of “production readiness” features like continuous integration tooling (automating test running and static analysis), alerting, and a simple deployment automation. Then I let it loose on AWS, got extremely busy with buying a house, forgot about this program for a few weeks (no alerts means it worked flawlessly!), and then saw my AWS bill.

So I copied the database off AWS using pg_dump (piped to gzip), terminated the instances, and inspected the results. A copy of the database is here. You may need git-lfs to clone it. If I wanted to start it back up again, I could spin them back up, and use gunzip | psql to restore the database, and it would start back up from where it left off. A nice benefit of all the software engineering work done thus far.

This article will summarize some of the data, show plots, and try out some exploratory data analysis techniques.

Summary

We stopped the search mid-way through the set of numbers with 136 prime divisors.

The largest number processed was

1255923956750926940807079376257388805204
00410625719434151527143279285143764977392
49474111379646103102793414829651500824447
17178682617437033476033026987651806835743
3694669721205424205654368862231754214894
07691711699791787732382878164959602478352
11435434547040000

Which in factored form is the product of these terms

  2^8   3^7   5^4   7^4  11^3  13^3  17^2  19^2  23^2  29^2
 31^2  37^2  41^2  43^1  47^1  53^1  59^1  61^1  67^1  71^1
 73^1  79^1  83^1  89^1  97^1 101^1 103^1 107^1 109^1 113^1
127^1 131^1 137^1 139^1 149^1 151^1 157^1 163^1 167^1 173^1
179^1 181^1 191^1 193^1 197^1 199^1 211^1 223^1 227^1 229^1
233^1 239^1 241^1 251^1 257^1 263^1 269^1 271^1 277^1 281^1
283^1 293^1 307^1 311^1 313^1 317^1 331^1 337^1 347^1 349^1
353^1 359^1 367^1 373^1 379^1 383^1 389^1 397^1 401^1 409^1
419^1 421^1 431^1 433^1 439^1 443^1 449^1 457^1 461^1 463^1
467^1 479^1 487^1 491^1 499^1 503^1 509^1 521^1 523^1 541^1
547^1 557^1 563^1 569^1 571^1 577^1

The best witness—the number with the largest witness value—was

38824169178385494306912668979787078930475
9208283469279319659854547822438432284497
11994812030251439907246255647505123032869
03750131483244222351596015366602420554736
87070007801035106854341150889235475446938
52188272225341139870856016797627204990720000

which has witness value 1.7707954880001586, which is still significantly smaller than the needed 1.782 to disprove RH.

The factored form of the best witness is

 2^11   3^7   5^4   7^3  11^3  13^2  17^2  19^2  23^2  29^2 
 31^2  37^1  41^1  43^1  47^1  53^1  59^1  61^1  67^1  71^1 
 73^1  79^1  83^1  89^1  97^1 101^1 103^1 107^1 109^1 113^1 
127^1 131^1 137^1 139^1 149^1 151^1 157^1 163^1 167^1 173^1 
179^1 181^1 191^1 193^1 197^1 199^1 211^1 223^1 227^1 229^1 
233^1 239^1 241^1 251^1 257^1 263^1 269^1 271^1 277^1 281^1 
283^1 293^1 307^1 311^1 313^1 317^1 331^1 337^1 347^1 349^1 
353^1 359^1 367^1 373^1 379^1 383^1 389^1 397^1 401^1 409^1 
419^1 421^1 431^1 433^1 439^1 443^1 449^1 457^1 461^1 463^1 
467^1 479^1 487^1 491^1 499^1 503^1 509^1 521^1 523^1 541^1 
547^1 557^1 563^1 

The average search block took 4m15s to compute, while the max took 7m7s and the min took 36s.

The search ran for about 55 days (hiccups included), starting at 2021-03-05 05:47:53 and stopping at 2021-04-28 15:06:25. The total AWS bill—including development, and periods where the application was broken but the instances still running, and including instances I wasn’t using but forgot to turn off—was $380.25. When the application was running at its peak, the bill worked out to about $100/month, though I think I could get it much lower by deploying fewer instances, after we made the performance optimizations that reduced the need for resource-heavy instances. There is also the possibility of using something that integrates more tightly with AWS, such as serverless jobs for the cleanup, generate, and process worker jobs.

Plots

When in doubt, plot it out. I started by writing an export function to get the data into a simpler CSV, which for each n only stored \log(n) and the witness value.

I did this once for the final computation results. I’ll call this the “small” database because it only contains the largest witness value in each block. I did it again for an earlier version of the database before we introduced optimizations (I’ll call this the “large” database), which had all witness values for all superabundant numbers processed up to 80 prime factors.. The small database was only a few dozen megabytes in size, and the large database was ~40 GiB, so I had to use postgres cursors to avoid loading the large database into memory. Moreover, then generated CSV was about 8 GiB in size, and so it required a few extra steps to sort it, and get it into a format that could be plotted in a reasonable amount of time.

First, using GNU sort to sort the file by the first column, \log(n)

sort -t , -n -k 1 divisor_sums.csv -o divisor_sums_sorted.csv

Then, I needed to do some simple operations on massive CSV files, including computing a cumulative max, and filtering down to a subset of rows that are sufficient for plotting. After trying to use pandas and vaex, I realized that the old awk command line tool would be great at this job. So I wrote a simple awk script to process the data, and compute data used for the cumulative max witness value plots below.

Then finally we can use vaex to create two plots. The first is a heatmap of witness value counts. The second is a plot of the cumulative max witness value. For the large database:

Witness value heatmap for the large database
The cumulative maximum witness value for the large database.

And for the small database

A heatmap for the witness values for the small database
The cumulative maximum witness value for the small database.

Note, the two ridges disagree slightly (the large database shows a longer flat line than the small database for the same range), because of the way that the superabundant enumeration doesn’t go in increasing order of n. So larger witness values in the range 400-500 are found later.

Estimating the max witness value growth rate

The next obvious question is whether we can fit the curves above to provide an estimate of how far we might have to look to find the first witness value that exceeds the desired 1.782 threshold. Of course, this will obviously depend on the appropriateness of the underlying model.

A simple first guess would be split between two options: the real data is asymptotic like a + b / x approaching some number less than 1.782 (and hence this approach cannot disprove RH), or the real data grows slowly (perhaps doubly-logarithmic) like a + b \log \log x, but eventually surpasses 1.782 (and RH is false). For both cases, we can use scipy’s curve fitting routine as in this pull request.

For the large database (roughly using log n < 400 since that’s when the curve flatlines due to the enumeration order), we get a reciprocal fit of

\displaystyle f(x) \approx 1.77579122 - 2.72527824 / x

and a logarithmic fit of

\displaystyle f(x) \approx 1.65074314 + 0.06642373 \log(\log(x))

The fit of the large database to a + b/x. Note the asymptote of 1.7757 suggests this will not disprove RH.
The fit of the large database to a + b log log x. If this is accurate, we would find the counterexample around log(n) = 1359.

The estimated asymptote is around 1.7757 in the first case, and the second case estimates we’d find an RH counterexample at around log(n) = 1359.

For the small database of only sufficiently large witness values, this time going up to about log(n) \approx 575, the asymptotic approximation is

\displaystyle 1.77481154 -2.31226382 / x

And the logarithmic approximation is

\displaystyle 1.70825262 + 0.03390312 \log(\log(x))

The reciprocal approximation of the small database with asymptote 1.77481154
The logarithmic approximation of the small database with RH counterexample estimate at log(n) = 6663

Now the asymptote is slightly lower, at 1.7748, and the logarithmic model approximates the counterexample can be found at approximately \log(n) = 6663.

Both of the logarithmic approximations suggest that if we want to find an RH counterexample, we would need to look at numbers with thousands of prime factors. The first estimate puts a counterexample at about 2^{1960}, the second at 2^{9612}, so let’s say between 1k and 10k prime factors.

Luckily, we can actually jump forward in the superabundant enumeration to exactly the set of candidates with m prime factors. So it might make sense to jump ahead to, say, 5k prime factors and search in that region. However, the size of a level set of the superabundant enumeration still grows exponentially in m. Perhaps we should (heuristically) narrow down the search space by looking for patterns in the distribution of prime factors for the best witness values we’ve found thus far. Perhaps the values of n with the best witness values tend to have a certain concentration of prime factors.

Exploring prime factorizations

At first, my thought was to take the largest witness values, look at their prime factorizations, and try to see a pattern when compared to smaller witness values. However, other than the obvious fact that the larger witness values correspond to larger numbers (more and larger prime factors), I didn’t see an obvious pattern from squinting at plots.

To go in a completely different direction, I wanted to try out the UMAP software package, a very nice and mathematically sophisticated for high dimensional data visualization. It’s properly termed a dimensionality reduction technique, meaning it takes as input a high-dimensional set of data, and produces as output a low-dimensional embedding of that data that tries to maintain the same shape as the input, where “shape” is in the sense of a certain Riemannian metric inferred from the high dimensional data. If there is structure among the prime factorizations, then UMAP should plot a pretty picture, and perhaps that will suggest some clearer approach.

To apply this to the RH witness value dataset, we can take each pair (n, \sigma(n)/(n \log \log n)), and associate that with a new (high dimensional) data point corresponding to the witness value paired with the number’s prime factorization

\displaystyle (\sigma(n)/(n \log \log n), k_1, k_2, \dots, k_d),

where n = 2^{k_1} 3^{k_2} 5^{k_3} \dots p_d^{k_d}, with zero-exponents included so that all points have the same dimension. This pull request adds the ability to factorize and export the witness values to a CSV file as specified, and this pull request adds the CSV data (using git-lfs), along with the script to run UMAP, the resulting plots shown below, and the saved embeddings as .npy files (numpy arrays).

When we do nothing special to the data and run it through UMAP we see this plot.

UMAP plotted on the raw prime factorization and witness value dataset.

It looks cool, but if you stare at it for long enough (and if you zoom in when you generate the plot yourself in matplotlib) you can convince yourself that it’s not finding much useful structure. The red dots dominate (lower witness values) and the blue dots are kind of spread haphazardly throughout the red regions. The “ridges” along the chart are probably due to how the superabundant enumeration skips lots of numbers, and that’s why it thins out on one end: the thinning out corresponds to fewer numbers processed that are that large since the enumeration is not uniform.

It also seemed like there is too much data. The plot above has some 80k points on it. After filtering down to just those points whose witness values are bigger than 1.769, we get a more manageable plot.

Witness values and prime factors processed with UMAP, where the witness value is at least 1.769.

This is a bit more reasonable. You can see a stripe of blue dots going through the middle of the plot.

Before figuring out how that blue ridge might relate to the prime factor patterns, let’s take this a few steps further. Typically in machine learning contexts, it helps to normalize your data, i.e., to transform each input dimension into a standard Z-score with respect to the set of values seen in that dimension, subtracting the mean and dividing by the standard deviation. Since the witness values are so close to each other, they’re a good candidate for such normalization. Here’s what UMAP plots when we normalize the witness value column only.

UMAP applied to the (normalized) witness values and prime factorizations. Applied to all witness values.

Now this is a bit more interesting! Here the colormap on the right is in units of standard deviation of witness values. You can see a definite bluest region, and it appears that the data is organized into long brushstrokes, where the witness values increase as you move from one end of the stroke to the other. At worst, this suggests that the dataset has structure that a learning algorithm could discover.

Going even one step further, what if we normalize all the columns? Well, it’s not as interesting.

UMAP when normalizing all columns, not just the witness value.

If you zoom in, you can see that the same sort of “brushstroke” idea is occurring here too, with blue on one end and red on the other. It’s just harder to see.

The previous image, zoomed in around a cluster of data

We would like to study the prettiest picture and see if we can determine what pattern of prime numbers the blue region has, if any. The embedding files are stored on github, and I put up (one version of) the UMAP visualization as an interactive plot via this pull request.

I’ve been sitting on this draft for a while, and while this article didn’t make a ton of headway, the pictures will have to do while I’m still dealing with my new home purchase.

Some ideas for next steps:

  • Squint harder at the distributions of primes for the largest witness values in comparison to the rest.
  • See if a machine learning algorithm can regress witness values based on their prime factorizations (and any other useful features I can derive). Study the resulting hypothesis to determine which features are the most important. Use that to refine the search strategy.
  • Try searching randomly in the superabundant enumeration around 1k and 10k prime factors, and see if the best witness values found there match the log-log regression.
  • Since witness values above a given threshold seem to be quite common, and because the UMAP visualization shows some possible “locality” structure for larger witness values, it suggests if there is a counterexample to RH then there are probably many. So a local search method (e.g., local neighborhood search/discrete gradient ascent with random restarts) might allow us to get a better sense for whether we are on the right track.

Until next time!

Optimization Models for Subset Cover

In a recent newsletter article I complained about how researchers mislead about the applicability of their work. I gave SAT solvers as an example. People provided interesting examples in response, but what was new to me was the concept of SMT (Satisfiability Modulo Theories), an extension to SAT. SMT seems to have more practical uses than vanilla SAT (see the newsletter for details).

I wanted to take some time to explore SMT solvers, and I landed on Z3, an open-source SMT solver from Microsoft. In particular, I wanted to compare it to ILP (Integer Linear Programing) solvers, which I know relatively well. I picked a problem that I thought would work better for SAT-ish solvers than for ILPs: subset covering (explained in the next section). If ILP still wins against Z3, then that would be not so great for the claim that SMT is a production strength solver.

All the code used for this post is on Github.

Subset covering

A subset covering is a kind of combinatorial design, which can be explained in terms of magic rings.

An adventurer stumbles upon a chest full of magic rings. Each ring has a magical property, but some pairs of rings, when worn together on the same hand, produce a combined special magical effect distinct to that pair.

The adventurer would like to try all pairs of rings to catalogue the magical interactions. With only five fingers, how can we minimize the time spent trying on rings?

Mathematically, the rings can be described as a set X of size n. We want to choose a family F of subsets of X, with each subset having size 5 (five fingers), such that each subset of X of size 2 (pairs of rings) is contained in some subset of F. And we want F to be as small as possible.

Subset covering is not a “production worthy” problem. Rather, I could imagine it’s useful in some production settings, but I haven’t heard of one where it is actually used. I can imagine, for instance, that a cluster of machines has some bug occurring seemingly at random for some point-to-point RPCs, and in tracking down the problem, you want to deploy a test change to subsets of servers to observe the bug occurring. Something like an experiment design problem.

If you generalize the “5” in “5 fingers” to an arbitrary positive integer k, and the “2” in “2 rings” to l < k, then we have the general subset covering problem. Define M(n, k, l) to be the minimal number of subsets of size k needed to cover all subsets of size l. This problem was studied by Erdős, with a conjecture subsequently proved by Vojtěch Rödl, that asymptotically M(n,k,l) grows like \binom{n}{l} / \binom{k}{l}. Additional work by Joel Spencer showed that a greedy algorithm is essentially optimal.

However, all of the constructive algorithms in these proofs involve enumerating all \binom{n}{k} subsets of X. This wouldn’t scale very well. You can alternatively try a “random” method, incurring a typically \log(r) factor of additional sets required to cover a 1 - 1/r fraction of the needed subsets. This is practical, but imperfect.

To the best of my knowledge, there is no exact algorithm, that both achieves the minimum and is efficient in avoiding constructing all \binom{n}{k} subsets. So let’s try using an SMT solver. I’ll be using the Python library for Z3.

Baseline: brute force Z3

For a baseline, let’s start with a simple Z3 model that enumerates all the possible subsets that could be chosen. This leads to an exceedingly simple model to compare the complex models against.

Define boolean variables \textup{Choice}_S which is true if and only if the subset S is chosen (I call this a “choice set”). Define boolean variables \textup{Hit}_T which is true if the subset T (I call this a “hit set”) is contained in a chosen choice set. Then the subset cover problem can be defined by two sets of implications.

First, if \textup{Choice}_S is true, then so must all \textup{Hit}_T for T \subset S. E.g., for S = \{ 1, 2, 3 \} and l=2, we get

\displaystyle \begin{aligned} \textup{Choice}_{(1,2,3)} &\implies \textup{Hit}_{(1,2)} \\ \textup{Choice}_{(1,2,3)} &\implies \textup{Hit}_{(1,3)} \\ \textup{Choice}_{(1,2,3)} &\implies \textup{Hit}_{(2,3)} \end{aligned}

In Python this looks like the following (note this program has some previously created lookups and data structures containing the variables)

for choice_set in choice_sets:
  for hit_set_key in combinations(choice_set.elements, hit_set_size):
    hit_set = hit_set_lookup[hit_set_key]
    implications.append(
      z3.Implies(choice_set.variable, hit_set.variable))

Second, if \textup{Hit}_T is true, it must be that some \textup{Choice}_S is true for some S containing T as a subset. For example,

\displaystyle \begin{aligned} \textup{Hit}_{(1,2)} &\implies \\ & \textup{Choice}_{(1,2,3,4)} \textup{ OR} \\ & \textup{Choice}_{(1,2,3,5)} \textup{ OR} \\ & \textup{Choice}_{(1,2,4,5)} \textup{ OR } \cdots \\ \end{aligned}

In code,

for hit_set in hit_sets.values():
  relevant_choice_set_vars = [
    choice_set.variable
    for choice_set in hit_set_to_choice_set_lookup[hit_set]
  ]
  implications.append(
    z3.Implies(
      hit_set.variable, 
      z3.Or(*relevant_choice_set_vars)))

Next, in this experiment we’re allowing the caller to specify the number of choice sets to try, and the solver should either return SAT or UNSAT. From that, we can use a binary search to find the optimal number of sets to pick. Thus, we have to limit the number of \textup{Choice}_S that are allowed to be true and false. Z3 supports boolean cardinality constraints, apparently with a special solver to handle problems that have them. Otherwise, the process of encoding cardinality constraints as SAT formulas is not trivial (and the subject of active research). But the code is simple enough:

args = [cs.variable for cs in choice_sets] + [parameters.num_choice_sets]
choice_sets_at_most = z3.AtMost(*args)
choice_sets_at_least = z3.AtLeast(*args)

Finally, we must assert that every \textup{Hit}_T is true.

solver = z3.Solver()
for hit_set in hit_sets.values():
  solver.add(hit_set.variable)

for impl in implications:
  solver.add(impl)

solver.add(choice_sets_at_most)
solver.add(choice_sets_at_least)

Running it for n=7, k=3, l=2, and seven choice sets (which is optimal), we get

>>> SubsetCoverZ3BruteForce().solve(
  SubsetCoverParameters(
    num_elements=7,
    choice_set_size=3,
    hit_set_size=2,
    num_choice_sets=7))
[(0, 1, 3), (0, 2, 4), (0, 5, 6), (1, 2, 6), (1, 4, 5), (2, 3, 5), (3, 4, 6)]
SubsetCoverSolution(status=<SolveStatus.SOLVED: 1>, solve_time_seconds=0.018305063247680664)

Interestingly, Z3 refuses to solve marginally larger instances. For instance, I tried the following and Z3 times out around n=12, k=6 (about 8k choice sets):

from math import comb

for n in range(8, 16):
  k = int(n / 2)
  l = 3
  max_num_sets = int(2 * comb(n, l) / comb(k, l))
  params = SubsetCoverParameters(
    num_elements=n,
    choice_set_size=k,
    hit_set_size=l,                                   
    num_choice_sets=max_num_sets)

    print_table(
      params, 
      SubsetCoverZ3BruteForce().solve(params), 
      header=(n==8))

After taking a long time to generate the larger models, Z3 exceeds my 15 minute time limit, suggesting exponential growth:

status               solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SolveStatus.SOLVED   0.0271              8             4                3             28
SolveStatus.SOLVED   0.0346              9             4                3             42
SolveStatus.SOLVED   0.0735              10            5                3             24
SolveStatus.SOLVED   0.1725              11            5                3             33
SolveStatus.SOLVED   386.7376            12            6                3             22
SolveStatus.UNKNOWN  900.1419            13            6                3             28
SolveStatus.UNKNOWN  900.0160            14            7                3             20
SolveStatus.UNKNOWN  900.0794            15            7                3             26

An ILP model

Next we’ll see an ILP model for the sample problem. Note there are two reasons I expect the ILP model to fall short. First, the best solver I have access to is SCIP, which, despite being quite good is, in my experience, about an order of magnitude slower than commercial alternatives like Gurobi. Second, I think this sort of problem seems to not be very well suited to ILPs. It would take quite a bit longer to explain why (maybe another post, if you’re interested), but in short well-formed ILPs have easily found feasible solutions (this one does not), and the LP-relaxation of the problem should be as tight as possible. I don’t think my formulation is very tight, but it’s possible there is a better formulation.

Anyway, the primary difference in my ILP model from brute force is that the number of choice sets is fixed in advance, and the members of the choice sets are model variables. This allows us to avoid enumerating all choice sets in the model.

In particular, \textup{Member}_{S,i} \in \{ 0, 1 \} is a binary variable that is 1 if and only if element i is assigned to be in set S. And \textup{IsHit}_{T, S} \in \{0, 1\} is 1 if and only if the hit set T is a subset of S. Here “S” is an index over the subsets, rather than the set itself, because we don’t know what elements are in S while building the model.

For the constraints, each choice set S must have size k:

\displaystyle \sum_{i \in X} \textup{Member}_{S, i} = k

Each hit set T must be hit by at least one choice set:

\displaystyle \sum_{S} \textup{IsHit}_{T, S} \geq 1

Now the tricky constraint. If a hit set T is hit by a specific choice set S (i.e., \textup{IsHit}_{T, S} = 1) then all the elements in T must also be members of S.

\displaystyle \sum_{i \in T} \textup{Member}_{S, i} \geq l \cdot \textup{IsHit}_{T, S}

This one works by the fact that the left-hand side (LHS) is bounded from below by 0 and bounded from above by l = |T|. Then \textup{IsHit}_{T, S} acts as a switch. If it is 0, then the constraint is vacuous since the LHS is always non-negative. If \textup{IsHit}_{T, S} = 1, then the right-hand side (RHS) is l = |T| and this forces all variables on the LHS to be 1 to achieve it.

Because we fixed the number of choice sets as a parameter, the objective is 1, and all we’re doing is looking for a feasible solution. The full code is here.

On the same simple example as the brute force

>>> SubsetCoverILP().solve(
  SubsetCoverParameters(
    num_elements=7,
    choice_set_size=3,
    hit_set_size=2,
    num_choice_sets=7))
[(0, 1, 3), (0, 2, 6), (0, 4, 5), (1, 2, 4), (1, 5, 6), (2, 3, 5), (3, 4, 6)]
SubsetCoverSolution(status=<SolveStatus.SOLVED: 1>, solve_time_seconds=0.1065816879272461)

It finds the same solution in about 10x the runtime as the brute force Z3 model, though still well under one second.

On the “scaling” example, it fares much worse. With a timeout of 15 minutes, it solves n=8, decently fast, n=9,12 slowly, and times out on the rest.

status               solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SolveStatus.SOLVED   1.9969              8             4                3             28
SolveStatus.SOLVED   306.4089            9             4                3             42
SolveStatus.UNKNOWN  899.8842            10            5                3             24
SolveStatus.UNKNOWN  899.4849            11            5                3             33
SolveStatus.SOLVED   406.9502            12            6                3             22
SolveStatus.UNKNOWN  902.7807            13            6                3             28
SolveStatus.UNKNOWN  900.0826            14            7                3             20
SolveStatus.UNKNOWN  900.0731            15            7                3             26

A Z3 Boolean Cardinality Model

The next model uses Z3. It keeps the concept of Member and Hit variables, but they are boolean instead of integer. It also replaces the linear constraints with implications. The constraint that forces a Hit set’s variable to be true when some Choice set contains all its elements is (for each S, T)

\displaystyle \left ( \bigwedge_{i \in T} \textup{Member}_{S, i} \right ) \implies \textup{IsHit}_T

Conversely, A Hit set’s variable being true implies its members are in some choice set.

\displaystyle \textup{IsHit}_T \implies \bigvee_{S} \bigwedge_{i \in T} \textup{Member}_{S, i}

Finally, we again use boolean cardinality constraints AtMost and AtLeast so that each choice set has the right size.

The results are much better than the ILP: it solves all of the instances in under 3 seconds

status              solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SolveStatus.SOLVED  0.0874              8             4                3             28
SolveStatus.SOLVED  0.1861              9             4                3             42
SolveStatus.SOLVED  0.1393              10            5                3             24
SolveStatus.SOLVED  0.2845              11            5                3             33
SolveStatus.SOLVED  0.2032              12            6                3             22
SolveStatus.SOLVED  1.3661              13            6                3             28
SolveStatus.SOLVED  0.8639              14            7                3             20
SolveStatus.SOLVED  2.4877              15            7                3             26

A Z3 integer model

Z3 supports implications on integer equation equalities, so we can try a model that leverages this by essentially converting the boolean model to one where the variables are 0-1 integers, and the constraints are implications on equality of integer formulas (all of the form “variable = 1”).

I expect this to perform worse than the boolean model, even though the formulation is almost identical. The details of the model are here, and it’s so similar to the boolean model above that it needs no extra explanation.

The runtime is much worse, but surprisingly it still does better than the ILP model.

status              solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SolveStatus.SOLVED  2.1129              8             4                3             28
SolveStatus.SOLVED  14.8728             9             4                3             42
SolveStatus.SOLVED  7.6247              10            5                3             24
SolveStatus.SOLVED  25.0607             11            5                3             33
SolveStatus.SOLVED  30.5626             12            6                3             22
SolveStatus.SOLVED  63.2780             13            6                3             28
SolveStatus.SOLVED  57.0777             14            7                3             20
SolveStatus.SOLVED  394.5060            15            7                3             26

Harder instances

So far all the instances we’ve been giving the solvers are “easy” in a sense. In particular, we’ve guaranteed there’s a feasible solution, and it’s easy to find. We’re giving roughly twice as many sets as are needed. There are two ways to make this problem harder. One is to test on unsatisfiable instances, which can be harder because the solver has to prove it can’t work. Another is to test on satisfiable instances that are hard to find, such as those satisfiable instances where the true optimal number of choice sets is given as the input parameter. The hardest unsatisfiable instances are also the ones where the number of choice sets allowed is one less than optimal.

Let’s test those situations. Since M(7, 3, 2) = 7, we can try with 7 choice sets and 6 choice sets.

For 7 choice sets (the optimal value), all the solvers do relatively well

method                    status  solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SubsetCoverILP            SOLVED  0.0843              7             3                2             7
SubsetCoverZ3Integer      SOLVED  0.0938              7             3                2             7
SubsetCoverZ3BruteForce   SOLVED  0.0197              7             3                2             7
SubsetCoverZ3Cardinality  SOLVED  0.0208              7             3                2             7

For 6, the ILP struggles to prove it’s infeasible, and the others do comparatively much better (at least 17x better).

method                    status      solve_time_seconds  num_elements  choice_set_size  hit_set_size  num_choice_sets
SubsetCoverILP            INFEASIBLE  120.8593            7             3                2             6
SubsetCoverZ3Integer      INFEASIBLE  3.0792              7             3                2             6
SubsetCoverZ3BruteForce   INFEASIBLE  0.3384              7             3                2             6
SubsetCoverZ3Cardinality  INFEASIBLE  7.5781              7             3                2             6

This seems like hard evidence that Z3 is better than ILPs for this problem (and it is), but note that the same test on n=8 fails for all models. They can all quickly prove 8 < M(8, 3, 2) \leq 11, but time out after twenty minutes when trying to determine if M(8, 3, 2) \in \{ 9, 10 \}. Note that k=3, l=2 is the least complex choice for the other parameters, so it seems like there’s not much hope to find M(n, k, l) for any seriously large parameters, like, say, k=6.

Thoughts

These experiments suggest what SMT solvers can offer above and beyond ILP solvers. Disjunctions and implications are notoriously hard to model in an ILP. You often need to add additional special variables, or do tricks like the one I did that only work in some situations and which can mess with the efficiency of the solver. With SMT, implications are trivial to model, and natively supported by the solver.

Aside from reading everything I could find on Z3, there seems to be little advice on modeling to help the solver run faster. There is a ton of literature for this in ILP solvers, but if any readers see obvious problems with my SMT models, please chime in! I’d love to hear from you. Even without that, I am pretty impressed by how fast the solves finish for this subset cover problem (which this experiment has shown me is apparently a very hard problem).

However, there’s an elephant in the room. These models are all satisfiability/feasibility checks on a given solution. What is not tested here is optimization, in the sense of having the number of choice sets used be minimized directly by the solver. In a few experiments on even simpler models, z3 optimization is quite slow. And while I know how I’d model the ILP version of the optimization problem, given that it’s quite slow to find a feasible instance when the optimal number of sets is given as a parameter, it seems unlikely that it will be fast when asked to optimize. I will have to try that another time to be sure.

Also, I’d like to test the ILP models on Gurobi, but I don’t have a personal license. There’s also the possibility that I can come up with a much better ILP formulation, say, with a tighter LP relaxation. But these will have to wait for another time.

In the end, this experiment has given me some more food for thought, and concrete first-hand experience, on the use of SMT solvers.

Earthmover Distance

Problem: Compute distance between points with uncertain locations (given by samples, or differing observations, or clusters).

For example, if I have the following three “points” in the plane, as indicated by their colors, which is closer, blue to green, or blue to red?

example-points.png

It’s not obvious, and there are multiple factors at work: the red points have fewer samples, but we can be more certain about the position; the blue points are less certain, but the closest non-blue point to a blue point is green; and the green points are equally plausibly “close to red” and “close to blue.” The centers of masses of the three sample sets are close to an equilateral triangle. In our example the “points” don’t overlap, but of course they could. And in particular, there should probably be a nonzero distance between two points whose sample sets have the same center of mass, as below. The distance quantifies the uncertainty.

same-centers.png

All this is to say that it’s not obvious how to define a distance measure that is consistent with perceptual ideas of what geometry and distance should be.

Solution (Earthmover distance): Treat each sample set A corresponding to a “point” as a discrete probability distribution, so that each sample x \in A has probability mass p_x = 1 / |A|. The distance between A and B is the optional solution to the following linear program.

Each x \in A corresponds to a pile of dirt of height p_x, and each y \in B corresponds to a hole of depth p_y. The cost of moving a unit of dirt from x to y is the Euclidean distance d(x, y) between the points (or whatever hipster metric you want to use).

Let z_{x, y} be a real variable corresponding to an amount of dirt to move from x \in A to y \in B, with cost d(x, y). Then the constraints are:

  • Each z_{x, y} \geq 0, so dirt only moves from x to y.
  • Every pile x \in A must vanish, i.e. for each fixed x \in A, \sum_{y \in B} z_{x,y} = p_x.
  • Likewise, every hole y \in B must be completely filled, i.e. \sum_{y \in B} z_{x,y} = p_y.

The objective is to minimize the cost of doing this: \sum_{x, y \in A \times B} d(x, y) z_{x, y}.

In python, using the ortools library (and leaving out a few docstrings and standard import statements, full code on Github):

from ortools.linear_solver import pywraplp

def earthmover_distance(p1, p2):
    dist1 = {x: count / len(p1) for (x, count) in Counter(p1).items()}
    dist2 = {x: count / len(p2) for (x, count) in Counter(p2).items()}
    solver = pywraplp.Solver('earthmover_distance', pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)

    variables = dict()

    # for each pile in dist1, the constraint that says all the dirt must leave this pile
    dirt_leaving_constraints = defaultdict(lambda: 0)

    # for each hole in dist2, the constraint that says this hole must be filled
    dirt_filling_constraints = defaultdict(lambda: 0)

    # the objective
    objective = solver.Objective()
    objective.SetMinimization()

    for (x, dirt_at_x) in dist1.items():
        for (y, capacity_of_y) in dist2.items():
            amount_to_move_x_y = solver.NumVar(0, solver.infinity(), 'z_{%s, %s}' % (x, y))
            variables[(x, y)] = amount_to_move_x_y
            dirt_leaving_constraints[x] += amount_to_move_x_y
            dirt_filling_constraints[y] += amount_to_move_x_y
            objective.SetCoefficient(amount_to_move_x_y, euclidean_distance(x, y))

    for x, linear_combination in dirt_leaving_constraints.items():
        solver.Add(linear_combination == dist1[x])

    for y, linear_combination in dirt_filling_constraints.items():
        solver.Add(linear_combination == dist2[y])

    status = solver.Solve()
    if status not in [solver.OPTIMAL, solver.FEASIBLE]:
        raise Exception('Unable to find feasible solution')

    return objective.Value()

Discussion: I’ve heard about this metric many times as a way to compare probability distributions. For example, it shows up in an influential paper about fairness in machine learning, and a few other CS theory papers related to distribution testing.

One might ask: why not use other measures of dissimilarity for probability distributions (Chi-squared statistic, Kullback-Leibler divergence, etc.)? One answer is that these other measures only give useful information for pairs of distributions with the same support. An example from a talk of Justin Solomon succinctly clarifies what Earthmover distance achieves

Screen Shot 2018-03-03 at 6.11.00 PM.png

Also, why not just model the samples using, say, a normal distribution, and then compute the distance based on the parameters of the distributions? That is possible, and in fact makes for a potentially more efficient technique, but you lose some information by doing this. Ignoring that your data might not be approximately normal (it might have some curvature), with Earthmover distance, you get point-by-point details about how each data point affects the outcome.

This kind of attention to detail can be very important in certain situations. One that I’ve been paying close attention to recently is the problem of studying gerrymandering from a mathematical perspective. Justin Solomon of MIT is a champion of the Earthmover distance (see his fascinating talk here for more, with slides) which is just one topic in a field called “optimal transport.”

This has the potential to be useful in redistricting because of the nature of the redistricting problem. As I wrote previously, discussions of redistricting are chock-full of geometry—or at least geometric-sounding language—and people are very concerned with the apparent “compactness” of a districting plan. But the underlying data used to perform redistricting isn’t very accurate. The people who build the maps don’t have precise data on voting habits, or even locations where people live. Census tracts might not be perfectly aligned, and data can just plain have errors and uncertainty in other respects. So the data that district-map-drawers care about is uncertain much like our point clouds. With a theory of geometry that accounts for uncertainty (and the Earthmover distance is the “distance” part of that), one can come up with more robust, better tools for redistricting.

Solomon’s website has a ton of resources about this, under the names of “optimal transport” and “Wasserstein metric,” and his work extends from computing distances to computing important geometric values like the barycenter, computational advantages like parallelism.

Others in the field have come up with transparency techniques to make it clearer how the Earthmover distance relates to the geometry of the underlying space. This one is particularly fun because the explanations result in a path traveled from the start to the finish, and by setting up the underlying metric in just such a way, you can watch the distribution navigate a maze to get to its target. I like to imagine tiny ants carrying all that dirt.

Screen Shot 2018-03-03 at 6.15.50 PM.png

Finally, work of Shirdhonkar and Jacobs provide approximation algorithms that allow linear-time computation, instead of the worst-case-cubic runtime of a linear solver.

Binary Search on Graphs

Binary search is one of the most basic algorithms I know. Given a sorted list of comparable items and a target item being sought, binary search looks at the middle of the list, and compares it to the target. If the target is larger, we repeat on the smaller half of the list, and vice versa.

With each comparison the binary search algorithm cuts the search space in half. The result is a guarantee of no more than \log(n) comparisons, for a total runtime of O(\log n). Neat, efficient, useful.

There’s always another angle.

What if we tried to do binary search on a graph? Most graph search algorithms, like breadth- or depth-first search, take linear time, and they were invented by some pretty smart cookies. So if binary search on a graph is going to make any sense, it’ll have to use more information beyond what a normal search algorithm has access to.

For binary search on a list, it’s the fact that the list is sorted, and we can compare against the sought item to guide our search. But really, the key piece of information isn’t related to the comparability of the items. It’s that we can eliminate half of the search space at every step. The “compare against the target” step can be thought of a black box that replies to queries of the form, “Is this the thing I’m looking for?” with responses of the form, “Yes,” or, “No, but look over here instead.”

binarysearch1

As long as the answers to your queries are sufficiently helpful, meaning they allow you to cut out large portions of your search space at each step, then you probably have a good algorithm on your hands. Indeed, there’s a natural model for graphs, defined in a 2015 paper of Emamjomeh-Zadeh, Kempe, and Singhal that goes as follows.

You’re given as input an undirected, weighted graph G = (V,E), with weights w_e for e \in E. You can see the entire graph, and you may ask questions of the form, “Is vertex v the target?” Responses will be one of two things:

  • Yes (you win!)
  • No, but e = (v, w) is an edge out of v on a shortest path from v to the true target.

Your goal is to find the target vertex with the minimum number of queries.

Obviously this only works if G is connected, but slight variations of everything in this post work for disconnected graphs. (The same is not true in general for directed graphs)

When the graph is a line, this “reduces” to binary search in the sense that the same basic idea of binary search works: start in the middle of the graph, and the edge you get in response to a query will tell you in which half of the graph to continue.

binarysearch2.png

And if we make this example only slightly more complicated, the generalization should become obvious:

binarysearch3

Here, we again start at the “center vertex,” and the response to our query will eliminate one of the two halves. But then how should we pick the next vertex, now that we no longer have a linear order to rely on? It should be clear, choose the “center vertex” of whichever half we end up in. This choice can be formalized into a rule that works even when there’s not such obvious symmetry, and it turns out to always be the right choice.

Definition: median of a weighted graph G with respect to a subset of vertices S \subset V is a vertex v \in V (not necessarily in S) which minimizes the sum of distances to vertices in S. More formally, it minimizes

\Phi_S(v) = \sum_{u \in S} d(v, u),

where d(u,v) is the sum of the edge weights along a shortest path from v to u.

And so generalizing binary search to this query-model on a graph results in the following algorithm, which whittles down the search space by querying the median at every step.

Algorithm: Binary search on graphs. Input is a graph G = (V,E).

  • Start with a set of candidates S = V.
  • While we haven’t found the target and |S| > 1:
    • Query the median v of S, and stop if you’ve found the target.
    • Otherwise, let e = (v, w) be the response edge, and compute the set of all vertices x \in V for which e is on a shortest path from v to x. Call this set T.
    • Replace S with S \cap T.
  • Output the only remaining vertex in S

Indeed, as we’ll see momentarily, a python implementation is about as simple. The meat of the work is in computing the median and the set T, both of which are slight variants of Dijkstra’s algorithm for computing shortest paths.

The theorem, which is straightforward and well written by Emamjomeh-Zadeh et al. (only about a half page on page 5), is that this algorithm requires only O(\log(n)) queries, just like binary search.

Before we dive into an implementation, there’s a catch. Even though we are guaranteed only \log(n) many queries, because of our Dijkstra’s algorithm implementation, we’re definitely not going to get a logarithmic time algorithm. So in what situation would this be useful?

Here’s where we use the “theory” trick of making up a fanciful problem and only later finding applications for it (which, honestly, has been quite successful in computer science). In this scenario we’re treating the query mechanism as a black box. It’s natural to imagine that the queries are expensive, and a resource we want to optimize for. As an example the authors bring up in a followup paper, the graph might be the set of clusterings of a dataset, and the query involves a human looking at the data and responding that a cluster should be split, or that two clusters should be joined. Of course, for clustering the underlying graph is too large to process, so the median-finding algorithm needs to be implicit. But the essential point is clear: sometimes the query is the most expensive part of the algorithm.

Alright, now let’s implement it! The complete code is on Github as always.

Always be implementing

We start with a slight variation of Dijkstra’s algorithm. Here we’re given as input a single “starting” vertex, and we produce as output a list of all shortest paths from the start to all possible destination vertices.

We start with a bare-bones graph data structure.

from collections import defaultdict
from collections import namedtuple

Edge = namedtuple('Edge', ('source', 'target', 'weight'))

class Graph:
    # A bare-bones implementation of a weighted, undirected graph
    def __init__(self, vertices, edges=tuple()):
        self.vertices = vertices
        self.incident_edges = defaultdict(list)

        for edge in edges:
            self.add_edge(
                edge[0],
                edge[1],
                1 if len(edge) == 2 else edge[2]  # optional weight
            )

    def add_edge(self, u, v, weight=1):
        self.incident_edges[u].append(Edge(u, v, weight))
        self.incident_edges[v].append(Edge(v, u, weight))

    def edge(self, u, v):
        return [e for e in self.incident_edges[u] if e.target == v][0]

And then, since most of the work in Dijkstra’s algorithm is tracking information that you build up as you search the graph, we define the “output” data structure, a dictionary of edge weights paired with back-pointers for the discovered shortest paths.

class DijkstraOutput:
    def __init__(self, graph, start):
        self.start = start
        self.graph = graph

        # the smallest distance from the start to the destination v
        self.distance_from_start = {v: math.inf for v in graph.vertices}
        self.distance_from_start[start] = 0

        # a list of predecessor edges for each destination
        # to track a list of possibly many shortest paths
        self.predecessor_edges = {v: [] for v in graph.vertices}

    def found_shorter_path(self, vertex, edge, new_distance):
        # update the solution with a newly found shorter path
        self.distance_from_start[vertex] = new_distance

        if new_distance < self.distance_from_start[vertex]:
            self.predecessor_edges[vertex] = [edge]
        else:  # tie for multiple shortest paths
            self.predecessor_edges[vertex].append(edge)

    def path_to_destination_contains_edge(self, destination, edge):
        predecessors = self.predecessor_edges[destination]
        if edge in predecessors:
            return True
        return any(self.path_to_destination_contains_edge(e.source, edge)
                   for e in predecessors)

    def sum_of_distances(self, subset=None):
        subset = subset or self.graph.vertices
        return sum(self.distance_from_start[x] for x in subset)

The actual Dijkstra algorithm then just does a “breadth-first” (priority-queue-guided) search through G, updating the metadata as it finds shorter paths.

def single_source_shortest_paths(graph, start):
    '''
    Compute the shortest paths and distances from the start vertex to all
    possible destination vertices. Return an instance of DijkstraOutput.
    '''
    output = DijkstraOutput(graph, start)
    visit_queue = [(0, start)]

    while len(visit_queue) > 0:
        priority, current = heapq.heappop(visit_queue)

        for incident_edge in graph.incident_edges[current]:
            v = incident_edge.target
            weight = incident_edge.weight
            distance_from_current = output.distance_from_start[current] + weight

            if distance_from_current <= output.distance_from_start[v]:
                output.found_shorter_path(v, incident_edge, distance_from_current)
                heapq.heappush(visit_queue, (distance_from_current, v))

    return output

Finally, we implement the median-finding and T-computing subroutines:

def possible_targets(graph, start, edge):
    '''
    Given an undirected graph G = (V,E), an input vertex v in V, and an edge e
    incident to v, compute the set of vertices w such that e is on a shortest path from
    v to w.
    '''
    dijkstra_output = dijkstra.single_source_shortest_paths(graph, start)
    return set(v for v in graph.vertices
               if dijkstra_output.path_to_destination_contains_edge(v, edge))

def find_median(graph, vertices):
    '''
    Compute as output a vertex in the input graph which minimizes the sum of distances
    to the input set of vertices
    '''
    best_dijkstra_run = min(
         (single_source_shortest_paths(graph, v) for v in graph.vertices),
         key=lambda run: run.sum_of_distances(vertices)
    )
    return best_dijkstra_run.start

And then the core algorithm

QueryResult = namedtuple('QueryResult', ('found_target', 'feedback_edge'))

def binary_search(graph, query):
    '''
    Find a target node in a graph, with queries of the form "Is x the target?"
    and responses either "You found the target!" or "Here is an edge on a shortest
    path to the target."
    '''
    candidate_nodes = set(x for x in graph.vertices)  # copy

    while len(candidate_nodes) > 1:
        median = find_median(graph, candidate_nodes)
        query_result = query(median)

        if query_result.found_target:
            return median
        else:
            edge = query_result.feedback_edge
            legal_targets = possible_targets(graph, median, edge)
            candidate_nodes = candidate_nodes.intersection(legal_targets)

    return candidate_nodes.pop()

Here’s an example of running it on the example graph we used earlier in the post:

'''
Graph looks like this tree, with uniform weights

    a       k
     b     j
      cfghi
     d     l
    e       m
'''
G = Graph(['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i',
           'j', 'k', 'l', 'm'],
          [
               ('a', 'b'),
               ('b', 'c'),
               ('c', 'd'),
               ('d', 'e'),
               ('c', 'f'),
               ('f', 'g'),
               ('g', 'h'),
               ('h', 'i'),
               ('i', 'j'),
               ('j', 'k'),
               ('i', 'l'),
               ('l', 'm'),
          ])

def simple_query(v):
    ans = input("is '%s' the target? [y/N] " % v)
    if ans and ans.lower()[0] == 'y':
        return QueryResult(True, None)
    else:
        print("Please input a vertex on the shortest path between"
              " '%s' and the target. The graph is: " % v)
        for w in G.incident_edges:
            print("%s: %s" % (w, G.incident_edges[w]))

        target = None
        while target not in G.vertices:
            target = input("Input neighboring vertex of '%s': " % v)

    return QueryResult(
        False,
        G.edge(v, target)
    )

output = binary_search(G, simple_query)
print("Found target: %s" % output)

The query function just prints out a reminder of the graph and asks the user to answer the query with a yes/no and a relevant edge if the answer is no.

An example run:

is 'g' the target? [y/N] n
Please input a vertex on the shortest path between 'g' and the target. The graph is:
e: [Edge(source='e', target='d', weight=1)]
i: [Edge(source='i', target='h', weight=1), Edge(source='i', target='j', weight=1), Edge(source='i', target='l', weight=1)]
g: [Edge(source='g', target='f', weight=1), Edge(source='g', target='h', weight=1)]
l: [Edge(source='l', target='i', weight=1), Edge(source='l', target='m', weight=1)]
k: [Edge(source='k', target='j', weight=1)]
j: [Edge(source='j', target='i', weight=1), Edge(source='j', target='k', weight=1)]
c: [Edge(source='c', target='b', weight=1), Edge(source='c', target='d', weight=1), Edge(source='c', target='f', weight=1)]
f: [Edge(source='f', target='c', weight=1), Edge(source='f', target='g', weight=1)]
m: [Edge(source='m', target='l', weight=1)]
d: [Edge(source='d', target='c', weight=1), Edge(source='d', target='e', weight=1)]
h: [Edge(source='h', target='g', weight=1), Edge(source='h', target='i', weight=1)]
b: [Edge(source='b', target='a', weight=1), Edge(source='b', target='c', weight=1)]
a: [Edge(source='a', target='b', weight=1)]
Input neighboring vertex of 'g': f
is 'c' the target? [y/N] n
Please input a vertex on the shortest path between 'c' and the target. The graph is:
[...]
Input neighboring vertex of 'c': d
is 'd' the target? [y/N] n
Please input a vertex on the shortest path between 'd' and the target. The graph is:
[...]
Input neighboring vertex of 'd': e
Found target: e

A likely story

The binary search we implemented in this post is pretty minimal. In fact, the more interesting part of the work of Emamjomeh-Zadeh et al. is the part where the response to the query can be wrong with some unknown probability.

In this case, there can be many shortest paths that are valid responses to a query, in addition to all the invalid responses. In particular, this rules out the strategy of asking the same query multiple times and taking the majority response. If the error rate is 1/3, and there are two shortest paths to the target, you can get into a situation in which you see three responses equally often and can’t choose which one is the liar.

Instead, the technique Emamjomeh-Zadeh et al. use is based on the Multiplicative Weights Update Algorithm (it strikes again!). Each query gives a multiplicative increase (or decrease) on the set of nodes that are consistent targets under the assumption that query response is correct. There are a few extra details and some postprocessing to avoid unlikely outcomes, but that’s the basic idea. Implementing it would be an excellent exercise for readers interested in diving deeper into a recent research paper (or to flex their math muscles).

But even deeper, this model of “query and get advice on how to improve” is a classic  learning model first formally studied by Dana Angluin (my academic grand-advisor). In her model, one wants to design an algorithm to learn a classifier. The allowed queries are membership and equivalence queries. A membership is essentially, “What’s its label of this element?” and an equivalence query has the form, “Is this the right classifier?” If the answer is no, a mislabeled example is provided.

This is different from the usual machine learning assumption, because the learning algorithm gets to construct an example it wants to get more information about, instead of simply relying on a randomly generated subset of data. The goal is to minimize the number of queries before the target hypothesis is learned exactly. And indeed, as we saw in this post, if you have a little extra time to analyze the problem space, you can craft queries that extract quite a lot of information.

Indeed, the model we presented here for binary search on graphs is the natural analogue of an equivalence query for a search problem: instead of a mislabeled counterexample, you get a nudge in the right direction toward the target. Pretty neat!

There are a few directions we could take from here: (1) implement the Multiplicative Weights version of the algorithm, (2) apply this technique to a problem like ranking or clustering, or (3) cover theoretical learning models like membership and equivalence queries in more detail. What interests you?

Until next time!