Formal language theory is the field of study of non-commutative objects so-called “languages” (sets of words). While many problems on languages are difficult to solve due to the non-commutativity, the situation could be drastically changed if we consider a commutative invariant of a language. This paper explains some commutative invariants of languages with concrete examples.

The halting property as such is not guaranteed for limitimg recursion, since its evaluation process requires infinitely many steps. Nevertheless, as limiting recursive functions are useful in the study of computational aspects of mathematics on the continuum, it is desirable that some extended notion of computation be bestowed upon limiting recursion. We will here present a framework which general recursion and limiting recursion can share in characterizing their functions, and in which the halting property of the former and the identification in the limit of the latter can be expressed in terms of a formula stating a kind of compactness of a sequence of sets. We regard this fact as claiming that limiting recursion is conjunctive to general recursion, and hence can be viewed as a computation in an extended sense.