SATisfying Solutions to Difficult Problems(vaibhavsagar.com) |
SATisfying Solutions to Difficult Problems(vaibhavsagar.com) |
In practice that are a bunch of problems that seem to be SAT, but they are either SAT at a scale where the solver still can't find a solution in any reasonable time or they turn out to not really be SAT because there is that one extra constraint that is quite difficult to encode in simple logic.
And it is unrewarding experimenting because it ends up being a day or so remembering how to use a SAT solver, then rediscovering how horrible raw sat solver interfaces are and trying to find a library that builds SAT problems from anything other than raw boolean algebra (the intros really undersell how bad the experience of using SAT solvers directly is, the DIMACS sat file format makes me think of the year 1973), then discovering that a SAT solver can't actually solve the alleged-SAT problem. Although if anyone is ever looking for a Clojure library I can recommend rolling-stones [0] as having a pleasant API to work with.
PS: Yes, I develop a propositional SAT solver that used to be SOTA [1]. I nowadays develop a propositional model counter (for computing probabilities), that is currently SOTA [2]
[1] https://github.com/msoos/cryptominisat/ [2] https://github.com/meelgroup/ganak/
And if not that, then they are used as the core component of a custom solver that speaks some higher level domain language.
Encoding things in CNF is a pain (even with higher level APIs like PySAT). But it's not usually something you should be doing unless you are developing a solver of some sort yourself.
This is really not true.
SAT solvers are really good these days, and many (exact) algorithms (for NP-hard problems) simply use some SOTA SAT solvers and are automatically competitive.
"MiniZinc" is the name of the Pythonic-ish-like syntax targeting (ie on-the-fly translating to) numerous different solvers (somewhere around half-a-dozen to a dozen or so, don't recall exactly =)
"""Create a MiniZinc script to find the optimal E12-series resistor values for the base ($R_B$) and collector ($R_C$) of a PN2222A transistor switch.
Specifications:
Collector Voltage Supply ($V_{CC}$): 12V DC
Base (Input) Voltage ($V_{IN}$): 3.3V DC
Target Collector Current ($I_C$): Maximize, but do not exceed, 1W (1 watt).
The script must correctly model the circuit to ensure the transistor is in saturation and must execute without Gecode solver errors like 'Float::linear: Number out of limits"""
After a few more try-paste exception loops, that generated a lovely and readable MiniZinc script with variables I can adjust for other circuits. It was exciting to see that basically every constraint problem I learned in school ("at what angle should you swim across the river..." is just a matter of encoding the problem and running a solver (I still think people should learn the basics of constraint problems, but after a certain point the problems are just tricky enough that it makes more sense to teach people how solvers work, and how to encode problems for them...")
MILPs (Mixed Integer Linear Programs) are basically sets of linear constraints, along with a linear optimization functions, where your variables can either be reals or ints.
Notably, you can easily encode any SAT problem as a MILP, but it's much easier to encode optimization problems, or problems with "county" constraints as MILPs.
HiGHS didn't exist, SCIP was "non-commercial", CBC was ok but they've been having development struggles for awhile, GLPK was never even remotely close to commercial offerings.
I think if something like Gurobi or Hexaly were open source, you'd see a lot more use since both their capabilities and performance are way ahead of the open source solutions, but it was the commercial revenue that made them possible in the first place.
Using CP-SAT from Google OR-Tools as a fake MILP solver by scaling the real variables is pretty funny though and works unreasonably well (specially if the problem is highly combinatorial since there's a SAT solver powering the whole thing)
If you get sick of MILPs, maybe you could use a representation of your finite field instead of the field itself? That way you could do everything in C^n, and preserve differentiability to use SGD or something like it.
> This is interesting, but techniques like CDCL seem to only ever find any one valuation that makes a proposition true.
So, when you find the solution, you can add a (long!) clause that blocks that solution from appearing again and then run solver again.
Most, if not all, SAT solvers have a command line option for that enumeration done behind the scenes, for picosat it is "picosat --all".
A more clever approach is to emulate depth-first search using a stack of assumption literals. The solver still retains learned conflict clauses so it's more efficient than naive DPLL.
Also Grobner basis algorithm over polynomials in Z_2 can be used to solve SAT. A SAT problem can be encoded as a set of quadratic polynomials, and if the generated ideal is all polynomials, the system is unsatisfiable (that's Nullstellenansatz). I don't understand how we can get high degree polynomials when running Grobner basis algorithm that specifically prefers low degree polynomials. It intuitively doesn't make sense.
Sudokus that can be solved with reasoning alone, can be solved in polynomial time.
I recently discovered that solving exact covers, and probably also with SAT, using a generic strategy, does not always result in the most efficient way for finding solutions. There are problems that have 10^50 solutions, yer finding a solution can take a long time.
Is 'reasoning alone' actually well defined in the context of Sudoku?
Sudoku's are finite, so I can solve all of them with just a lookup table in constant time..
CryptoMiniSAT has native support for Gaussian Elimination but it has to put a lot of effort into recovering XORs from the CNF.
A different format (XORSAT + 2SAT) plus an efficient algorithm to exchange information from the two sides of the problem would be interesting.
(a | b | c) = ((a ^ ~x) | b) & (x | c)
where x is a variable that can be chosen. The LHS is satisfiable iff RHS is satisfiable. This gives you extra variable per clause, but you can eliminate some of them using GE. (Probably not an ideal approach to easy problems, but IMHO worth trying for the hard ones.)
Then you get a set of clauses that have what I call "generalized literals" - essentially a linear combination of literals - in clauses that only have 1 OR (I call this 2-XORSAT). These can be trivially transformed to intersection of XORSAT and 2SAT.
Another thing that DIMACS is missing, it's not very modular. So you cannot represent a bunch of conditions using already pre-solved XORSAT portion, and apply that to a new problem. (For example, we could encode a generic multiplier in SAT, relating inputs and outputs. Then we could presolve the XORSAT portion of the multiplier. Then we want to factorize a number. Well, just add a bunch of constants specifying output that needs to be satisfied and you don't need to solve the XORSAT for the multiplier again.)
Or, you can convert the 2-XORSAT into quadratic polynomials over Z_2 that all need to be equal to 0. Then you can use the Grobner basis algorithm to find whether these polynomials can be linearly combined (using polynomials as coefficients) to give 1 (i.e. they generate an ideal which contains the whole ring of polynomials), meaning the contradictory equation 1=0 has to be satisfied, and so the problem is unsatisfiable.
What I would really like to understand, how it can happen, if you are running the Grobner basis algorithm and keep an eye on the degree, that you build the degree up from low degree polynomials. It seems to me, if the problem is unsatisfiable, you should be always able to get by with polynomials of low degree, because why carry extra variables if you're gonna eliminate the terms anyway? (If low degree polynomials are sufficient for the Grobner basis, it would imply P=NP by the way, because there is only polynomial number of polynomials of a given maximum degree.)
But yeah CryptoMiniSAT looks somewhat promising.
Just my 2 cents.
These are the types of questions that can easily be answered by simply trying.
Unfortunately, putting the algorithms for XORSAT and 2SAT together is not trivial at all, they are quite different (but Grobner bases over GF(2) seem very promising in that).
But I agree that the fact that both XORSAT and 2SAT have polynomial algorithms is quite a strong indicator that full SAT has a one too. :-) (On the other hand, there is IMHO only very little actual evidence for P!=NP.)
From the syzygies, surely?
See my problem? Somehow, I feel these syzygies need to be constructed from the degree 2 polynomials (the initial set), but then we can probably only work with the polynomials of the low degree instead, from which those syzygies are constructed.
To me, this is a very interesting question. Because if we can find the Grobner basis (even for just those ideals that coincide with the whole ring) in GF(2) in a way that we only need to construct polynomials of bounded degree, then there is a polynomial algorithm for SAT. Since most people believe there isn't such algorithm, I would really like to see a counterexample to this situation.
I like the higher level CSP more as an interface but those are _probably_ best solved by compilation to SAT. SMT also worth a look.
I have only very quickly skimmed it, but I wouldn't be surprised if the theorem D.21 in Kaufman's thesis (https://danielakaufmann.at/wp-content/uploads/2020/11/Kaufma...) turned out to be true for all the unsatisfiability PAC proofs, not just the circuits she is looking at. (As I commented below. If you're proving contradiction, looking for element 1 in the ideal using Grobner basis, then it seems somewhat unreasonable to require degree of basis polynomials larger than 3, if you start from all polynomials with degree less or equal than 2. If you look what e.g. 2SAT algorithm is doing algebraically, it only needs degree 3 polynomials as well, although the monomial of degree 3 is immediately eliminated. So if the Grobner basis algorithm needs to build large degree polynomials, because no small degree will help you, it's likely your system is already satisfiable. Would really like to see a counterexample.)
The problem with this kind of constraints is that they are not local and really disturb CDCL based SAT solvers because you cannot remove the constraint unless you have fixed all its variables. Now, cryptominisat has a specific focus on this kind of constraints and implements ad-hoc routines to exploit this routine. cryptominisat is hence focused toward applications where this kind of constraint naturally appears, be it circuit synthesis or cryptography.
Now, CDCL solver is another beast. Gaussian elimination / Grobner basis are not implemented in these solvers simply because it is "too costly". One way of understanding why is just to reframe what one call SAT solver. This is not really a tool to solve SAT (hell, they can't even deal with the pigeon hole principle without preprocessing), the purpose of a SAT solver is to quickly solve instances coming from natural encoding of constraint systems. In this framework, unit propagation and clause learning are basically extremely efficient because they uncover hidden structure in the way, us human, encode constraint problems. Hence (CDCL) SAT solvers are highly efficient in many applications. But it is easy to kill one with a few variables by simply going out of this framework. For example, they are killers in solving packages dependencies because the structure of these problems matches this approach, CDCL solvers are then just an extremely refined way of bruteforcing your way through the search space.
Now, if your application heavily contains XOR-constraints, then it is likely a (CDCL) SAT solver is not the right approach. For circuit synthesis, some people use BDD based approach (where parity can be efficiently encoded) but sometimes you will simply have to develop your dedicated solver or use other solvers for other NP-complete problem that will have a different area of specialization (as already observed in many comments).
Even simple goal-directed block's world-like robotic planning problems where the number of moves and the items to pickup/putdown are variable are much easier formulated and solved using Prolog.
You can only encode decision problems in NP into a SAT instance of polynomially-balanced size. Sure, that's a lot of things, but there are things provably not in this set.
A basic question, before searching these: are they "input compatible"? I mean, can a problem be formulated once and then be solved by a variety of solvers? Or does each one of them use its own input language?
Nearly every solver supports the MPS format, but that's a really old format straight from the era of punchcards, it sucks.
Many solvers support the nl format, which is a low level format spat out by the AMPL tool (commercial software for mathematical modeling).
Many solvers support the CPLEX lp format, which is a nice human readable and writable format.
Google OR-Tools includes an API for mathematical modeling that supports the relevant open source MIP solvers plus Gurobi I think and its own CP solver. There are Python and Julia packages that try to do the same (rather than calling the solver APIs directly they usually spit out a problem in nl format though).
MiniZinc supports various open source MILP solvers plus various CP solvers. Very nice language, very high level.
For MINLP the only standard I know of is OSiL but support for it is spotty, mostly supported by open source tools I think.
def objective(model: WtaModel) -> SumExpression:
return sum(
model.target_values[t_j]
* prod(
model.survival_rates[w_i][t_j]
** model.target_selected[w_i, t_j]
for w_i in model.weapon_types
)
for t_j in model.targets
)[1] https://jump.dev/ [2] https://jump.dev/JuMP.jl/stable/installation/#Supported-solv...