  Concurrent Constraint Programming (CCP) is a simple and powerful model for
concurrency where agents interact by telling and asking constraints. Since
their inception, CCP-languages have been designed for having a strong
connection to logic. In fact, the underlying constraint system can be built
from a suitable fragment of intuitionistic (linear) logic --ILL-- and processes
can be interpreted as formulas in ILL. Constraints as ILL formulas fail to
represent accurately situations where "preferences" (called soft constraints)
such as probabilities, uncertainty or fuzziness are present. In order to
circumvent this problem, c-semirings have been proposed as algebraic structures
for defining constraint systems where agents are allowed to tell and ask soft
constraints. Nevertheless, in this case, the tight connection to logic and
proof theory is lost. In this work, we give a proof theoretical interpretation
to soft constraints: they can be defined as formulas in a suitable fragment of
ILL with subexponentials (SELL) where subexponentials, ordered in a c-semiring
structure, are interpreted as preferences. We hence achieve two goals: (1)
obtain a CCP language where agents can tell and ask soft constraints and (2)
prove that the language in (1) has a strong connection with logic. Hence we
keep a declarative reading of processes as formulas while providing a logical
framework for soft-CCP based systems. An interesting side effect of (1) is that
one is also able to handle probabilities (and other modalities) in SELL, by
restricting the use of the promotion rule for non-idempotent c-semirings.This
finer way of controlling subexponentials allows for considering more
interesting spaces and restrictions, and it opens the possibility of specifying
more challenging computational systems.

Logic in Computer Science