ersatz
A monad for expressing SAT or QSAT problems using observable sharing.
http://github.com/ekmett/ersatz
| LTS Haskell 22.40: | 0.5@rev:2 |
| Stackage Nightly 2024-11-06: | 0.5@rev:2 |
| Latest on Hackage: | 0.5@rev:2 |
ersatz-0.5@sha256:8ef8abed9dd9847344d6f8b1793ecf1f5dc24fe152095c6d093ca2a47b65a67c,9425Module documentation for 0.5
Ersatz
Ersatz is a library for generating QSAT (CNF/QBF) problems using a monad. It takes care of generating the normal form, encoding your problem, marshaling the data to an external solver, and parsing and interpreting the result into Haskell types.
What differentiates Ersatz is the use of observable sharing in the API.
For instance to define a full adder:
full_adder :: Bit -> Bit -> Bit -> (Bit, Bit)
full_adder a b cin = (s2, c1 || c2)
where (s1,c1) = half_adder a b
(s2,c2) = half_adder s1 cin
half_adder :: Bit -> Bit -> (Bit, Bit)
half_adder a b = (a `xor` b, a && b)
as opposed to the following code in satchmo:
full_adder :: Boolean -> Boolean -> Boolean
-> SAT ( Boolean, Boolean )
full_adder a b c = do
let s x y z = sum $ map fromEnum [x,y,z]
r <- fun3 ( \ x y z -> odd $ s x y z ) a b c
d <- fun3 ( \ x y z -> 1 < s x y z ) a b c
return ( r, d )
half_adder :: Boolean -> Boolean
-> SAT ( Boolean, Boolean )
half_adder a b = do
let s x y = sum $ map fromEnum [x,y]
r <- fun2 ( \ x y -> odd $ s x y ) a b
d <- fun2 ( \ x y -> 1 < s x y ) a b
return ( r, d )
This enables you to use the a much richer subset of Haskell than the purely monadic meta-language, and it becomes much easier to see that the resulting encoding is correct.
To allocate fresh existentially or universally quantified variables or to assert that a Bit is true and add the attendant circuit with sharing to the current problem you use the SAT monad.
verify_currying :: (MonadQSAT s m) => m ()
verify_currying = do
(x::Bit, y::Bit, z::Bit) <- forall_
assert $ ((x && y) ==> z) === (x ==> y ==> z)
We can then hand that off to a SAT solver, and get back an answer:
main = solveWith depqbf verify_currying >>= print
Support is offered for decoding various Haskell datatypes from the solution provided by the SAT solver.
Examples
Included are a couple of examples included with the distribution.
Neither are as fast as a dedicated solver for their respective
domains, but they showcase how you can solve real world problems
involving 10s or 100s of thousands of variables and constraints
with ersatz.
sudoku
% time ersatz-sudoku
Problem:
┌───────┬───────┬───────┐
│ 5 3 │ 7 │ │
│ 6 │ 1 9 5 │ │
│ 9 8 │ │ 6 │
├───────┼───────┼───────┤
│ 8 │ 6 │ 3 │
│ 4 │ 8 3 │ 1 │
│ 7 │ 2 │ 6 │
├───────┼───────┼───────┤
│ 6 │ │ 2 8 │
│ │ 4 1 9 │ 5 │
│ │ 8 │ 7 9 │
└───────┴───────┴───────┘
Solution:
┌───────┬───────┬───────┐
│ 5 3 4 │ 6 7 8 │ 9 1 2 │
│ 6 7 2 │ 1 9 5 │ 3 4 8 │
│ 1 9 8 │ 3 4 2 │ 5 6 7 │
├───────┼───────┼───────┤
│ 8 5 9 │ 7 6 1 │ 4 2 3 │
│ 4 2 6 │ 8 5 3 │ 7 9 1 │
│ 7 1 3 │ 9 2 4 │ 8 5 6 │
├───────┼───────┼───────┤
│ 9 6 1 │ 5 3 7 │ 2 8 4 │
│ 2 8 7 │ 4 1 9 │ 6 3 5 │
│ 3 4 5 │ 2 8 6 │ 1 7 9 │
└───────┴───────┴───────┘
ersatz-sudoku 1,13s user 0,04s system 99% cpu 1,179 total
regexp-grid
This solves the regular crossword puzzle from the MIT mystery hunt.
% time ersatz-regexp-grid
ersatz-regexp-grid 2,45s user 0,05s system 99% cpu 2,502 total
Contact Information
Contributions and bug reports are welcome!
Please feel free to contact me through github or on the #haskell IRC channel on irc.freenode.net.
-Edward Kmett