Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer’s framework in which the Fourier transform is used to represent constraints by multilinear polynomials. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We completely characterize the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps, and also give several additional applications.
This work was initiated when all three authors were in residence at the Simons Institute for the Theory of Computing. Albert Atserias partly funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR), and by MINECO through TIN2013-48031-C4-1-P (TASSAT2); Simone Severini partly funded by The Royal Society, Engineering and Physical Sciences Research Council (EPSRC), National Natural Science Foundation of China (NSFC).
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