  Intensional sets, i.e., sets given by a property rather than by enumerating
elements, are widely recognized as a key feature to describe complex problems
(see, e.g., specification languages such as B and Z). Notwithstanding, very few
tools exist supporting high-level automated reasoning on general formulas
involving intensional sets. In this paper we present a decision procedure for a
first-order logic language offering both extensional and (a restricted form of)
intensional sets (RIS). RIS are introduced as first-class citizens of the
language and set-theoretical operators on RIS are dealt with as constraints.
Syntactic restrictions on RIS guarantee that the denoted sets are finite,
though unbounded. The language of RIS, called L_RIS , is parametric with
respect to any first-order theory X providing at least equality and a decision
procedure for X-formulas. In particular, we consider the instance of L_RIS when
X is the theory of hereditarily finite sets and binary relations. We also
present a working implementation of this instance as part of the {log} tool and
we show through a number of examples and two case studies that, although RIS
are a subclass of general intensional sets, they are still sufficiently
expressive as to encode and solve many interesting problems. Finally, an
extensive empirical evaluation provides evidence that the tool can be used in
practice.

Logic in Computer Science