A SAT Attack on Tarski's High School Algebra Problem
arxiv.orgI love SAT solver papers, always interesting to see auxiliary variable techniques, since those aren't really listed anywhere central
here for example, instead of saying {f(x,y,z)==g(x,y,z)}, they instead make variable a_w:=(f(x,y,z)=w||g(x,y,z)=w), and then apply "at most 1" to it. Can't be unequal if both functions only can have 1 result in total
this adds an index to iterate over, but separates internal subexpressions of f() and g(), removing 2 indixes (in this problem) and thus dropping whole power of n of clauses
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what I don't get is that they aren't searching Tarski's problem per se, but for one specific solution to it (one identity that isn't resulting from given). I'd totally look for arithmetic models that violate expectations in other ways than Wilkie
Why is subtraction not part of the algebra? It’s certainly familiar to every high school math student. This omission allows the counterexample, so the reveal is a bit of a disappointment IMHO.
Subtraction is not closed over positive integers, which is untidy. The point of Tarski’s conjecture was to propose a minimal number of axioms and operations, AFAICS they define the standard semiring of positive integers (with the natural definition of exponentiation added).
(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)
Well, yes, but negative numbers are also well known to every high school math student.
Sure. But "High School Algebra (Excluding Subtraction) Problem" isn’t as catchy a name.
They subtracted the subtraction exclusion in the name of simplicity?
I'm not sure, but maybe it is due to that the expression a - b can be replaced as a + (-b)?
Similarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...
We’re in the semiring of positive integers, so there are no additive (or multiplicative) inverses.
Because subtraction is not a total operation on positive integers. Negative numbers leave the domain.
The subtraction point is interesting but I don't think it makes the result disappointing. The whole point of Tarski's problem is what follows from that very restricted set of elementary identities so finding the exact minimum countermodel under those rules still seems like a pretty satisfying result.