Sandwiches for promise constraint satisfaction

Promise Constraint Satisfaction Problems ( $$\mathrm{PCSP}$$ PCSP ) were proposed recently by Brakensiek and Guruswami as a framework to study approximations for Constraint Satisfaction Problems ( $$\mathrm{CSP}$$ CSP ). Informally a $$\mathrm{PCSP}$$ PCSP asks to distinguish between whether a given instance of a $$\mathrm{CSP}$$ CSP has a solution or not even a specified relaxation can be satisfied. All currently known tractable $$\mathrm{PCSP}$$ PCSP s can be reduced in a natural way to tractable $$\mathrm{CSP}$$ CSP s. In 2019 Barto presented an example of a $$\mathrm{PCSP}$$ PCSP over Boolean structures for which this reduction requires solving a $$\mathrm{CSP}$$ CSP over an infinite structure. We give a first example of a $$\mathrm{PCSP}$$ PCSP over Boolean structures which reduces to a tractable $$\mathrm{CSP}$$ CSP over a structure of size 3 but not smaller. Further we investigate properties of $$\mathrm{PCSP}$$ PCSP s that reduce to systems of linear equations or to $$\mathrm{CSP}$$ CSP s over structures with semilattice or majority polymorphism.