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.

Equality of streams is a Π02-complete problem

Abstract: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: Π 0 2 . Since the Π 0 2 class includes properly both the recursively enumerable and the co-recursively enumerable classes, this result implies that neither the set of pairs of equal streams nor the set of pairs of unequal streams is recursively enumerable. Consequently, one can find no algorithm for determining equality of streams, as well as no algorithm for determining inequality of streams. In particular, there is no complete proof system for equality of streams and no complete system for inequality of streams.

Time-Varying Distributed H Systems of Degree 2 Can Carry Out Parallel Computations

Abstract: A time-varying distributed H system (TVDH system) is a splicing system which has the following feature: at different moments one uses different sets of splicing rules (these sets are called components of TVDH system). The number of components is called the degree of the TVDH system. The passage from a component to another one is specified in a cycle. It was proved by the first two authors (2001) that TVDH systems of degree one generate all recursively enumerable languages. The proof was made by a sequential modelling of Turing machines. This solution is not a fully parallel one. A. P?un (1999) presented a complete parallel solution for TVDH systems of degree four by modelling type-0 formal grammars. His result was improved by the first two authors by reducing the number of components of such TVDH systems down to three (2000). In this paper we improve the last result by reducing the number of components of such TVDH systems down to two. This question remains open for one component, i.e. is it possible to construct TVDH systems of degree one which completely uses the parallel nature of molecular computations based on splicing operations (say model type-0 formal grammars).

Recursive assertions are not enough - or are they?☆

Abstract: Call a set of assertions A complete (with respect to a class of programs S ) if for any p , q∈ A and S∈ S , wherever { p } S { q } holds, then all intermediate assertions can be chosen from A . This paper is devoted to the study of the problem which sets of assertions are complete in the above sense. We prove that any set of recursive assertions containing true and false is not complete. We prove the completeness for while programs of some more powerful assertions, e.g. the set of recursively enumerable assertions. Finally, we show that by allowing the use of an ‘auxilliary’ coordinate, the set of recursive assertions is complete for while programs.

Geometric and Higher Order Logic in terms of Abstract Stone Duality

Abstract: The contravariant powerset and its generalisations X to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers satisfy the Euclidean principle that ^F ( ), ^F (>). Conversely, when the adjunction ( ) a ( ) is monadic, this equation implies that classies some class of monos and the Frobenius law 9x:( (x)^ ), (9x: (x))^ for the existential quantier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural denitions of discrete and Hausdor spaces correspond to equality and inequality, whilst the quantiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdor spaces and proper maps. The category of overt discrete spaces forms a pretopos. The paper concludes with a converse of Par e’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantiers, making no reference to subsets.