  Weighted Logic is a powerful tool for the specification of calculations over
semirings that depend on qualitative information. Using a novel combination of
Weighted Logic and Here-and-There (HT) Logic, in which this dependence is based
on intuitionistic grounds, we introduce Answer Set Programming with Algebraic
Constraints (ASP(AC)), where rules may contain constraints that compare
semiring values to weighted formula evaluations. Such constraints provide
streamlined access to a manifold of constructs available in ASP, like
aggregates, choice constraints, and arithmetic operators. They extend some of
them and provide a generic framework for defining programs with algebraic
computation, which can be fruitfully used e.g. for provenance semantics of
datalog programs. While undecidable in general, expressive fragments of ASP(AC)
can be exploited for effective problem-solving in a rich framework. This work
is under consideration for acceptance in Theory and Practice of Logic
Programming.
