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Promise Constraint Satisfaction Problems (PCSP) were proposed recently by Brakensiek and Guruswami [2] 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 PCSPs can be reduced in a natural way to tractable CSPs. Barto [1] 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 PCSPs that reduce to systems of linear equations or to CSPs over structures with semilattice or majority polymorphism.

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