Showing papers on "Idempotence published in 1988"
TL;DR: Embedding sets as a datastructure into resolution-based deduction requires a unification algorithm respecting associativity, commutativity and idempotence of the constructor function union, which is presented.
32 citations
13 citations
01 Jan 1988
TL;DR: In this article, the complexity of the membership problem of a set of generators is shown to be NP-hard for monoids of threshold 2 or more, and NP-complete in commutative, J-and R-trivial monoids.
Abstract: Given a finite set X of states, a finite set of transformations of X (generators), and another transformation f of X, we analyze the complexity of the membership problem, which consists in deciding whether f can be obtained by composition of the generators. This problem is studied for various classes (pseudovarieties) of monoids. It is shown that the complexity is NP-hard for monoids of threshold 2 or more, and NP-complete in commutative, J- and R-trivial monoids. For idempotent monoids (aperiodic of threshold one), the problem is NP-complete in the general case; subcases are analyzed, and a largest class of aperiodic monoids is identified for which the problem is in FL, as well as a largest class for which the problem is not NP-hard.
The problem which consists in characterizing an idempotent monoid is also addressed: given a set of transformations, it can be decided in NC$\sp2$ whether the monoid they generate is idempotent. Similar tests are given for three subclasses of idempotent monoids: R$\sb1$, L$\sb1$, and N$\sb3$; in all three cases, the complexity is NC$\sp1$.
A sequential upper bound is also given for each of the parallel complexities given above.
10 citations
TL;DR: In this paper, it was proved that if A is a nonassociative algebra that verifies A 2 = A and has an idempotent, then A and its duplicate have isomorphic automorphism groups and isomorphic derivation algebras.
6 citations
TL;DR: In this paper, the authors generalize this result to the case of the ton-groupoids and show that the idempotent and medial laws form a basis for the equational theory of the then-groupoid if and only if α 1, α 2,..., αn−1 are algebraically independent.
Abstract: Fajtlowicz and Mycielski ([2]) showed that if we define an operationx·y=px+(1-p)x on the real numbersR for somep e R, then the idempotent and medial laws form a basis for the equational theory of the groupoid (R,) if and only ifp is a transcendental number. In this paper, we generalize this ton-groupoids. Namely, if we define ann-ary operation [x
1x2h.xn]=α1x1 + α2x2+h.+(1-α1-h.αn−1) onR for some α1, α2, h., αn−1 in R, then the idempotent and medial laws form a basis for the equational theory of then-groupoid (R, [ ]) if and only if α1, α2,..., αn−1 are algebraically independent.
4 citations
06 Oct 1988
TL;DR: This paper considers queries consisting of a single clause containing a single predicate symbol, which are a notational variant of the full, untagged tableau queries with recursive semantics and generalizes these results to obtain syntactic, polynomial-time computable characterizations of idempotence for certain classes of untyped queries.
Abstract: Previous work has addressed the issues of idempotence and boundedness for various restricted classes of Horn-clause queries. In this paper, we consider queries consisting of a single clause containing a single predicate symbol. As such, these queries are a notational variant of the full, untagged tableau queries with recursive semantics. The study of the idempotence and boundedness for single-clause, single-predicate queries has previously been restricted to the typed case. We generalize these results to obtain syntactic, polynomial-time computable characterizations of idempotence for certain classes of untyped queries.
2 citations