Suppose we would like to check whether a given array of real numbers is sorted (say, in nondecreasing order). Performing this task exactly requires reading the entire array. Here we consider the approximate version of the problem: testing whether an array is sorted or “far” from sorted. We consider two natural definitions of the distance of a given array from a sorted array. Intuitively, we wou...

Let R be a non empty poset and let A, B be many sorted sets indexed by R. One can check that every many sorted relation between A and B which is os-compatible is also order-sorted. Let R be a non empty poset and let A be a many sorted set indexed by the carrier of R. An order sorted relation of A is an order sorted relation of A, A. Let S be an order sorted signature and let U1 be an order sort...

In an algebraic framework, where equational, membership and existence formulas can be expressed, decorated terms and rewriting provide operational semantics and decision procedures for these formulas. We focus in this work on testing sort inheritance, an undecidable property of speciications, needed for uniication in this context. A test and three speciic processes, based on completion of a set...

We investigate parametrization for order-sorted algebraic specifications. As a prerequisite we study free constructions for order-sorted algebras and relate the various approaches to order-sorting. Then we analyse parameter passing, the result being that the notion of ordersorted specification has to be restricted in order to establish our main result, namely, that parameter passing satisfies t...

We propose a theory for reasoning about actions based on order-sorted predicate logic where one can consider an elaborate taxonomy of objects. We are interested in the projection problem: whether a statement is true after executing a sequence of actions. To solve it we design a regression operator that takes advantage of well-sorted unification between terms. We show that answering projection q...

We extend order-sorted unification by permitting regular expression sorts for variables and in the domains of function symbols. The obtained signature corresponds to a finite bottom-up unranked tree automaton. We prove that regular expression order-sorted (REOS) unification is of type infinitary and decidable. The unification problem presented by us generalizes some known problems, such as, e.g...

Generalization, also called anti-unification, is the dual of unification. Given terms t and t′, a generalization is a term t′′ of which t and t′ are substitution instances. The dual of a most general unifier (mgu) is that of least general generalization (lgg). In this work, we extend the known untyped generalization algorithm to, first, an order-sorted typed setting with sorts, subsorts, and su...

(1) Let S be an order sorted signature, U0 be a strict non-empty order sorted algebra of S, and A be a subset of U0. Then A is an order sorted generator set of U0 if and only if for every OSSubset O of U0 such that O = OSClA holds OSGenO = U0. Let us consider S, let U0 be a monotone order sorted algebra of S, and let I1 be an order sorted generator set of U0. We say that I1 is osfree if and onl...

Let S be an order sorted signature and let U0 be an order sorted algebra of S. A many sorted subset indexed by the sorts of U0 is said to be an OSSubset of U0 if: (Def. 2) It is an order sorted set of S. Let S be an order sorted signature. One can verify that there exists an order sorted algebra of S which is monotone, strict, and non-empty. Let S be an order sorted signature and let U0 be a no...