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

Brakensiek和Guruswami最近提出了承诺约束满足问题（\（\ mathrm {PCSP} \））作为研究约束满足问题（\（\ mathrm {CSP} \））的近似框架。非正式地，\（\ mathrm {PCSP} \）要求区分给定的\（\ mathrm {CSP} \）实例是否具有解，甚至不能满足指定的松弛度。所有当前已知的易处理\（\ mathrm {PCSP} \）可以自然地减少为易处理\（\ mathrm {CSP} \） s。在2019年，Barto展示了布尔结构上的\（\ mathrm {PCSP} \）的示例，为此，归约需要解决\（\ mathrm {CSP} \）在无限的结构上。我们给出布尔结构上的\（\ mathrm {PCSP} \）的第一个示例，该布尔结构在大小为3但不小于3的结构上减小为易于处理的\（\ mathrm {CSP} \）。进一步，我们研究了\（\ mathrm {PCSP} \）的性质，该性质在具有半晶格或多数多态性的结构上可简化为线性方程组或\（\ mathrm {CSP} \）。