Recursively enumerable language

About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.

Can we Compute the Similarity Between Surfaces

Abstract: A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Frechet distance. Whereas efficient algorithms are known for computing the Frechet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all. Here, using a discrete approximation we show that it is {\em upper semi-computable}, i.e., there is a non-halting Turing machine which produces a monotone decreasing sequence of rationals converging to the result. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some specified value, is recursively enumerable. Furthermore, we show that a relaxed version of the problem, the computation of the {\em weak Frechet distance} can be solved in polynomial time. For this, we give a computable characterization of the weak Frechet distance in a geometric data structure called the {\em free space diagram}.

The Infinite Injury Priority Method

Abstract: One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {R,},ncw called requirements. If n s be undone for the sake of R, thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often. Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity. The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own. The reader need have no prior knowledge of priority arguments for in ?1 we review the finite injury method using a version invented by Sacks for his Splitting Theorem [15]. In ?2 we discuss the two principal obstacles in extending the strategy to the infinite injury case. We show how the obvious and well-known solution to the first obstacle has automatically solved the second and more fundamental one. We then prove the two main lemmas upon which all of the theorems depend, and from these we prove the Thickness Lemma of Shoenfield [21, p. 83]. In ?3 we apply the method to derive the Yates Index Set Theorem, and results of

TeMP : A Temporal Monodic Prover

Abstract: First-Order Temporal Logic, FOTL, is an extension of classical first-order logic by temporal operators for a discrete linear model of time (isomorphic to ℕ, that is, the most commonly used model of time). Formulae of this logic are interpreted over structures that associate with each element n of ℕ, representing a moment in time, a first-order structure (D n ,I n ) with its own non-empty domain D n . In this paper we make the expanding domain assumption, that is, D n ⊆ D m if n

Recursively Enumerable Reals and Chaitin Omega Numbers

Abstract: A real α is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real α dominates an r.e. real β if from a good approximation of a from below one can compute a good approximation of β from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23] Ω-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability of a universal self-delimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number is Ω-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.