Sandwiches for promise constraint satisfaction

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