Using cabal-install's dependency solver as a SAT solver!?

Dependency resolution in Haskell ecosystem is a hard computational problem. While I'm unsure how hard problem is to picking individual package versions without any additional features, selecting assignment of automatic package flags seems to be very hard: it seems we encode arbitrary boolean satisfiablity problems, SAT, into automatic package flag selection problem.

Real world flag selection problems are easy. Yet, I wanted to try how well cabal-install's solver copes with problems it haven't been tuned for.

#Boolean satisfiability problems

From Wikipedia:

In logic and computer science, the Boolean satisfiability problem is the problem of determining if there exists an interpretation that satisfies a given Boolean formula.

The problems are given to solvers in conjunctive normal form:

(x₁ ∨ x₂) ∧ (x₃ ∨ x₄)

and solvers job is to find an assignment making the formula true. In the example above there are many solutions, e.g. setting all variables to true.

#Sudoku

One of go to examples of what you can do with a SAT solvers is solving sudoku puzzles.

Our running example will be a very simple 2×2 sudoku puzzle.

┌─────┬─────┐
│   1 │ 4   │
│   3 │   2 │
├─────┼─────┤
│     │ 3   │
│     │     │
└─────┴─────┘

Problem encoding is somewhat of art form, but for sudoku its quite simple.

The problem variables are numbers in cells i, j. We can encode each number using four variables, x(i,j,k), and requiring that exactly one is true. We could also use only two "bits" to encode four options (so called binary encoding), but using one-bit per option makes easier to encode the sudoku rules.

Recall the sudoku rules: each number have to occur exactly once in each row, column and subsquare.

With our number encoding the puzzle rules are easy to encode. For example for each row i and number k we require that exactly one literal in x(i,j,k), j <- [1..4] is true And similarly for columns and subsquares.

For what it's worth, sudoku can be very neatly encoded using Applicatives and Traversables. See the StackOverflow answer by Conor McBride.

SAT solvers consume a DIMACS format which looks like:

p cnf 64 453
-60 -64 0
-48 -64 0
-48 -60 0
-44 -64 0
-44 -60 0
-44 -48 0
44 48 60 64 0
-59 -63 0
-47 -63 0
-47 -59 0
-43 -63 0
-43 -59 0
-43 -47 0
43 47 59 63 0
-58 -62 0
...

Borrowing the DIMACS format explanation from varisat's documentation:

A DIMACS file begins with a header line of the form p cnf <variables> <clauses>. Where <variables> and <clauses> are replaced with decimal numbers indicating the number of variables and clauses in the formula. Following the header line are the clauses of the formula. The clauses are encoded as a sequence of decimal numbers separated by spaces and newlines. For each clause the contained literals are listed followed by a 0.

The above is beginning of encoding of our sudoku problem. There are 4 × 4 cells and each number uses 4 variables, so in total there are 64 variables.

The exactly once encoding I used is done using naive (binomial) at most one encoding. You can see a pattern:

-60 -64 0
-48 -64 0
-48 -60 0
-44 -64 0
-44 -60 0
-44 -48 0
44 48 60 64 0

The last line is requiring that at least one of four variables (44, 48, 60, 64) is true. The first 6 lines are pairwise requirements that at most one of the variables is true: n over 2 i.e. 6 pairs. All of sudoku rules are such exactly one of four constraints, which we have 64 in total: 16 for digits, 4 rows, columns and subsquares with four numbers, That is 7 × 64 = 448 clauses.

The final 5 clauses are initial value constraints. As we know 5 numbers, we tell that the following bits must be true.

The sudoku.cnf file indeed ends with five unit clauses:

...
43 0
30 0
23 0
12 0
5 0

When we run the SAT solver, e.g. z3 -dimacs sudoku.cnf it will immediately give a solution which looks something like

sat
-1 2 -3 -4 5 -6 -7 -8 -9 -10 -11 12 -13 -14 15 -16
-17 -18 -19 20 -21 -22 23 -24 25 -26 -27 -28 -29 30 -31 -32
33 -34 -35 -36 -37 38 -39 -40 -41 -42 43 -44 -45 -46 -47 48
-49 -50 51 -52 -53 -54 -55 56 -57 58 -59 -60 61 -62 -63 -64 

For each of 64 variables it prints whether the satisfying assignment for that variable is true (positive) or false (negative).

When we decode the solution we'll get a solution to our sudoku puzzle:

┌─────┬─────┐
│ 2 1 │ 4 3 │
│ 4 3 │ 1 2 │
├─────┼─────┤
│ 1 2 │ 3 4 │
│ 3 4 │ 2 1 │
└─────┴─────┘

#Encoding as cabal automatic flags

So how can we encode a SAT problem as flag selection one?

It is hopefully obvious that each variable will be represented by an automatic flag, i.e. a flag which solver can (should) choose an assignment for:

flag 1
  manual: False
  default: False

(Yes, flag names can be "numbers", they are still treated as strings).

The default value shouldn't matter, but it's probably better to pick False as most variables in sudoku problem are indeed false.

Let's next think how to encode clauses. When the CNF is satisfiable each clause should evaluate to true. When CNF is unsatisfiable it's enough that any clause evaluate to false. Recall clauses are disjunctions of literals:

x₁ ∨ ¬ x₂ ∨ ¬ x₃ ∨ x₄

Then we can encode such clause as a conditional in a component stanza of .cabal file. There shouldn't be an install plan if a clause value if false:

if !(flag(1) || !flag(2) || !flag(3) || flag(4))
  build-depends: unsatisfiable <0

or equivalently:

if !flag(1) && flag(2) && flag(3) && !flag(4))
  build-depends: unsatisfiable <0

The library stanze in the resulting sudoku.cabal file looks like

library
  if flag(60) && flag(64)
    build-depends: unsatisfiable <0

  if flag(48) && flag(64)
    build-depends: unsatisfiable <0

  if flag(48) && flag(60)
    build-depends: unsatisfiable <0

  if flag(44) && flag(64)
    build-depends: unsatisfiable <0

  if flag(44) && flag(60)
    build-depends: unsatisfiable <0

  if flag(44) && flag(48)
    build-depends: unsatisfiable <0

  if !flag(44) && !flag(48) && !flag(60) && !flag(64)
    build-depends: unsatisfiable <0

...

  if !flag(43)
    build-depends: unsatisfiable <0

  if !flag(30)
    build-depends: unsatisfiable <0

  if !flag(23)
    build-depends: unsatisfiable <0

  if !flag(12)
    build-depends: unsatisfiable <0

  if !flag(5)
    build-depends: unsatisfiable <0
...

And we can ask cabal-install to construct an install plan with

cabal build --dry-run

On my machine it took 17 seconds to complete. (I actually didn't know what to expect, I'd say 17 seconds is not bad).

cabal-install writes out plan.json file which contains the build plan. It's a JSON file which can be inspected directly or queried with cabal-plan utility.

cabal-plan topo --show-flags

shows

sudoku-0 -1 -10 -11 +12 -13 -14 +15 -16 -17 -18 -19 +2 +20 -21 -22 +23
  -24 +25 -26 -27 -28 -29 -3 +30 -31 -32 +33 -34 -35 -36 -37 +38 -39 -4
  -40 -41 -42 +43 -44 -45 -46 -47 +48 -49 +5 -50 +51 -52 -53 -54 -55
  +56 -57 +58 -59 -6 -60 +61 -62 -63 -64 -7 -8 -9

there is only one package in the install plan, and we asked cabal-plan to also show the flag assignment, which it does. The output is almost the same as from z3!

If we decode this solution, we get the same answer:

┌─────┬─────┐
│ 2 1 │ 4 3 │
│ 4 3 │ 1 2 │
├─────┼─────┤
│ 1 2 │ 3 4 │
│ 3 4 │ 2 1 │
└─────┴─────┘

#Conclusion

We successfully used cabal-install dependency solver as a SAT solver. It is terribly slow, but it's probably still faster at solving 2×2 sudoku puzzle than myself. The code is available on GitHub if you want to play with it.

However, it is not unheard that we need to encode some logical constraints in cabal file.

For example transformers-compat encodes which transformers version it depends on using kind of unary encoding: each bucket is encoded using single flag: Removing some unrelated bits:

...

  if flag(four)
    build-depends: transformers >= 0.4.1 && < 0.5
  else
    build-depends: transformers < 0.4 || >= 0.5

  if flag(three)
    build-depends: transformers >= 0.3 && < 0.4
  else
    build-depends: transformers < 0.3 || >= 0.4

...

The choice of transformers versions forces assignments to the automatic flags (four, three, ...) and then we can alter build info of a package based on that.

That is an indirect way of writing (encoding!) something like

  if build-depends(transformers >= 0.4.1 && <0.5)
    ...

  if build-depends(transformers >= 0.3 && <0.4)
    ...

A common example in the past was adding old-locale dependency when the old version of time library was picked:

  if flag(old-locale)
    build-depends:
        old-locale  >=1.0.0.2 && <1.1
      , time        >=1.4     && <1.5

  else
    build-depends: time >=1.5 && <1.7

which could be written as

  build-depends: time >=1.4 && <1.7
  if build-depends(time < 1.5)
    build-depends: old-locale >=1.0.0.2 && <1.1

Another example is functor-classes-compat which also encodes transformers and base version subsets, but it is using binary encoding of four options. There the implied constraints are also (hopefully) disjoint making flag assignment deterministic.

I think that automatic flags are good feature to have. It is a basic building block, but is a "low-level" feature. On the other hand if build-depends(...) construct is more difficult to use wrong, and probably covers 99% of the use cases for automatic flags. If you are mindful that you encoding if build-depends (...) constraint, then you'll probably use cabal's automatic flags correctly. Conversely, if you are using automatic flags to encode something else, like solving general SAT problems, most likely you are doing something wrong.

• 2024-04-21 – A note about coercions
• 2024-04-18 – What makes a good compiler warning?
• 2024-04-12 – Core Inspection
• 2024-04-01 – Implicit arguments
• 2024-03-17 – ST with an early exit

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