Topic
Idempotence
About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.
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TL;DR: In this paper, a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check, is presented, and results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.
Abstract: It is important that practical data-flow analyzers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the soundness of Sharing assume the occurs-check. This paper provides a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check. Results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.
4 citations
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TL;DR: In this article, a new approach to the Hutchinson-Barnsley theory for idempotent measures is presented, which is based on the embedding of the space of the IDM space to the fuzzy set.
Abstract: We provide a new approach to the Hutchinson-Barnsley theory for idempotent measures first presented in N. Mazurenko, M. Zarichnyi, Invariant idempotent measures, Carpathian Math. Publ., 10 (2018), 1, 172--178. The main feature developed here is a metrization of the space of idempotent measures using the embedding of the space of idempotent measures to the space of fuzzy sets. The metric obtained induces a topology stronger than the canonical pointwise convergence topology. A key result is the existence of a bijection between idempotent measures and fuzzy sets and a conjugation between the Markov operator of an IFS on idempotent measures and the fuzzy fractal operator of the associated Fuzzy IFS. This allows to prove that the Markov operator for idempotent measures is a contraction w.r.t. the induced metric and, from this, to obtain a convergence theorem and algorithms that draw pictures of invariant measures as greyscale images.
4 citations
TL;DR: In this paper, it was shown that AI-matching, which is solving matching word equations in free idempotent semigroups, is NP-complete, and that the problem is solvable in polynomial time.
Abstract: We show that AI-matching (AI denotes the theory of an associative and idempotent function symbol), which is solving matching word equations in free idempotent semigroups, is NP-complete. Note: this is a full version of the paper [9] and a revision of [8].
4 citations
TL;DR: The probability that the product of l square matrices of size n over a finite field with q elements will be nilpotent is shown to be 1-[( q n -1)/ q n ] l.
4 citations
TL;DR: This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite, and establishes a generalisation to inverse semigroups of Anisimov's Theorem for groups.
Abstract: The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This paper proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalization to inverse semigroups of Anisimov's Theorem for groups.
4 citations