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For 3-regular graphs, there are fewer equivalence classes than would otherwise exist over all possible graphs. Since a dichotomy theorem only needs to prove the complexity of one constrain function in each equivalence class, this leads to a shorter proof. I see the authors use Buchberger's algorithm as a subroutine, and exploit the structure of their problem to prove that the running time is polynomially bounded. Grant Passmore and others write about them in the context of SMT solvers.

I'm not an expert in Groebner bases nor in SMT solvers, so it's hard for me to evaluate how well this reference answers your question.

Gröbner Bases, Coding, and Cryptography - CERN Document Server

There are cases in which this proof system outperforms Resolution exponentially but it does not seem to me that there is a real performance improvement for general instances. The exceptions usually are built for the characteristic of the underlying field.

Yet Polynomial Calculus has not been studied as much as Resolution, thus well tested heuristics are not available. See also this this for application in cryptanalysyis I don't know very much about that.

Gröbner Bases, Coding, And Cryptography

But this joins the use of grobner basis in refutating CNFS. Home Questions Tags Users Unanswered. Ask Question.

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Workshop D1: Gröbner Bases in Cryptography, Coding Theory, and Algebraic Combinatorics

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