During the night, Mayra’s health monitor started to detect a new bacterial infection. This implanted device, continuously monitoring her bloodstream, warns Mayra that she is probably infected with the bacteria burkholderia pseudomallei. This bacteria, categorized as tropical even decades ago, is now relatively common in the United States.
Because of its dangerousness, Mayra is informed that she has to go to a Center for Diseases without delay. (Otherwise, she could have just gone to the drugstore: over-the-counter phage cocktails are generally used for more trivial infections).
At the Center, her blood is drawn to confirm the infection. The computer, after having extracted and sequenced the genome of the bacteria, identifies phage φ192e1-a as the perfect candidate to fight this infection. (The -a postfix indicates that this phage is not originally found in nature. It has been discovered by artificial intelligence, successfully tested, and FDA approved.)
Synthetic bacteriophages are now the preferred method to fight bacterial and fungal infections.
Three main reasons for this.
First, the prevalence of antibiotic resistance has dramatically increased during the last decades. This situation was long expected, and the regulatory landscape slowly but surely adapted accordingly in favor of phage therapy. Second, one of the least expected effects of global warming has been the emergence of human diseases caused by fungi. Phages have revealed themselves as more powerful than classic fungicides to fight these complex infections. Third, continuous monitoring and synthetic phage production have made phage therapy more convenient and less expensive.
There is no longer the need to keep specimens in phage banks: they are now generated on demand. The whole DNA sequence of φ192e1-a is recorded in a database, and these phages are synthesized. Once produced, they are orally administered.
Soon, Mayra is healed before developing any symptoms.
Puzzles can be defined as one-player constraint-logic games (Hearn, Robert A., and Erik D. Demaine. Games, Puzzles, and Computation. AK Peters, 2009, p. 55). Generally speaking, these constraints are “rules defining legal moves” (Hearn, Robert A. Games, Puzzles, and Computation. Ph.D. dissertation, Massachusetts Institute of Technology, 2006, p. 18) toward a solution.
Hearn and Demaine use this terminology in the context of constraint graphs. For our demonstration, we place it in the general context of constraint satisfaction. On this basis, it is an apparent fact that, the more a puzzle is constrained, the more it is solvable efficiently: each constraint diminishes the size of the search space.
The core rules of a puzzle are what I would like to call “first-order constraints.” They are defining the game itself.
Let’s take a 9x9 Sudoku square as an example. Its rules are well known (Knuth, Donald Ervin. The Art of Computer Programming. Addison-Wesley, 2020, vol. 4, fascicle 5, p. 72):
By definition, these rules constitute first-order constraints: they are used for solving the puzzle (by pruning the search space); they validate that a puzzle has been solved.
TetraVex, an edge-matching puzzle, is characterized by the following first-order constraints:
First-order constraints are enough to verify whether a puzzle has been solved or not.
When the size and complexity of a game are computationally tractable, simple backtracking solvers can solve them in a reasonable time. They are using first-order constraints not only at the verification stage (to verify that the puzzle has been solved) but also to reduce the size of the search tree during the solving process (e.g., for a TetraVex solver: for the next node, only select tiles which are edge-matching the current one).
To be more efficient, solvers are generally optimized. They can notably:
Solvers optimized that way are generally very efficient. However, they fail to solve puzzles that are deemed intractable due to their size. Over a certain threshold, they cannot solve in a reasonable time.
An advanced solving technique is the translation (or encoding, or conversion, …) of a puzzle into another class of problem. It offers the advantage of the possibility to use solvers dedicated to solving the latter. In other words, the general idea is to benefit from these highly optimized solvers. This is an indirect optimization: instead of optimizing the solver itself, the solving process is delegated to another tool or algorithm.
A Sudoku can be translated into a propositional formula (typically into the CNF format) to be solved by a SAT-solver (e.g., zChaf). It can also be encoded into a binary matrix solvable by algorithm X/dancing links. A Sudoku can additionally be solved by being translated into a graph. (Another example can be given with Instant Insanity, easily solvable after having been translated into a graph).
In other words and respectively, a Sudoku can be translated into a propositional satisfiability problem, an exact cover problem, or a graph coloring problem, then solved accordingly.
Intellectually stimulating for modest puzzles, this approach has its limits. Larger puzzles are translated into large objects: they produce large formulas, matrices, graphs, etc. The translation does not reduce per se the complexity of the original representation of a puzzle.
It is just another representation of an NP-complete problem. You had one problem. Now, you have two.
First-order constraints, even supplemented with direct optimization and/or translation into other problems, are insufficient to solve some puzzles because of the combinatorial explosion.
Eternity 2, a puzzle launched in 2007, is a perfect illustration: this 256-piece edge-matching puzzle has still not been solved. It is intractable.
Its first-order constraints are:
There is also a rule that distinguishes it from TetraVex puzzles: its tiles can be rotated. This rule can be formulated as a constraint:
Considerable computing and brain power have been thrown into this NP-complete problem. Sophisticated solvers have been created, greatly optimized for parallelism, allowing some mismatches between the edges to reach an optimal overall score, etc. But none of them has been able to solve it.
Some people have suggested translating Eternity 2 into a SAT problem, a binary matrix, a graph, etc., à la Sudoku. In our opinion, merely translating this puzzle and using solvers dedicated to these classes of problem just translates this intractable edge-matching problem into an intractable SAT/exact cover/graph problem.
We are optimistic that this puzzle can be solved but differently, by discovering and using constraints complementary to the first-order ones.
We call them “second-order constraints.”
Second-order constraints can be defined as non-obvious constraints that are emerging from the first-order ones. They are as fundamental yet can go unnoticed. They are implicit but necessary rules.
The following table compares these two classes of constraint.
First-Order Constraints | Second-Order Constraints |
---|---|
Defined by human will (puzzle creator) | Emanating from first-order rules |
Explicitly defined | Unnoticed until discovered |
Required to find a solution | Helpful to find a solution |
Required to validate a solution | Not required to validate a solution |
Sudokus give an example of a second-order constraint: the Phistomefel Ring. In a solved Sudoku, the numbers in the central ring are always the same as in the four corners, as illustrated by this image (in this instance, both corners and ring contain the numbers { 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9 }
):
This singular characteristic is demonstrated in a video.
The Phistomefel Ring second-order constraint, inherent to all Sudokus, is so non-obvious that it has been discovered in 2020 while modern Sudokus exist since 1979. It has always been there, but nobody noticed it for several decades.
The Phistomefel Ring has been discovered by someone searching for patterns. According to its discoverer: “I was pondering a bit about the geometry of the Sudoku grid the other day and came across an interesting find.” (Source in German).
This interesting find is the observation of the existence of an invariant—the equality between two sets (ring = corners) of numbers. In other words: observationally, second-order constraints are non-obvious invariants in solved instances of a puzzle.
The existence of second-order constraints inherent to puzzles deemed intractable as Eternity 2 is an open question.
If they exist, they have not been found yet. Or, if some of them have been found, they are not critical enough to solve the puzzle.
A pessimistic view would conclude that this puzzle does not contain any second-order constraint. A more optimistic stance (ours) believes that Eternity 2 contains several of them, but they remain latent.
To reveal them, one approach could be to purposely study solved Eternity 2-like puzzles. Edge-matching puzzles similar to Eternity 2 could be randomly generated; then, an algorithm could try to detect invariants in these instances; finally, a puzzle would be solved using both first-order and second-order constraints.
The nature of this algorithm (basically: a second-order constraints detector) is unknown, though. At this stage, it is purely speculative. But it is probably worth investigating. In particular—and before tackling the gargantuan Eternity 2 puzzle—it would be interesting to test this hypothesis by constructing an algorithm able to detect by itself the Phistomefel Ring (alongside potential other second-order constraints) in Sudokus.
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