  A value of a CSP instance is typically defined as a fraction of constraints
that can be simultaneously met. We propose an alternative definition of a value
of an instance and show that, for purely combinatorial reasons, a value of an
unsolvable instance is bounded away from one; we call this fact a gap theorem.
  We show that the gap theorem implies NP-hardness of a gap version of the
Layered Label Cover Problem. The same result can be derived from the PCP
Theorem, but a full, self-contained proof of our reduction is quite short and
the result can still provide PCP-free NP-hardness proofs for numerous problems.
The simplicity of our reasoning also suggests that weaker versions of
Unique-Games-type conjectures, e.g., the d-to-1 conjecture, might be accessible
and serve as an intermediate step for proving these conjectures in their full
strength.
  As the second, main application we provide a sufficient condition under which
a fixed template Promise Constraint Satisfaction Problem (PCSP) reduces to
another PCSP. The correctness of the reduction hinges on the gap theorem, but
the reduction itself is very simple. As a consequence, we obtain that every CSP
can be canonically reduced to most of the known NP-hard PCSPs, such as the
approximate hypergraph coloring problem.
