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.

On sets polynomially enumerable by iteration

Abstract: Sets whose members are enumerated by some Turing machine are called recursively enumerable. We define a set to be polynomially enumerable by iteration if its members are efficiently enumerated by iterated application of some Turing machine. We prove that many complex sets—including all exponential-time complete sets, all NP-complete sets yet obtained by direct construction, and the complements of all such sets—are polynomially enumerable by iteration. These results follow from more general results. In fact, we show that all recursively enumerable sets that are ⪯p1si-self-reducible are polynomially enumerable by iterations, and that all recursive sets that are p1si-self-reducible are bi-enumerable. We also show that when the ⪯p1si-self-reduction is via a function whose inverse is computable in polynomial time, then the above results hold with the polynomial enumeration given by a function whose inverse is computable in polynomial time. In the final section of the paper we show that no NP-complete set can be iteratively enumerated in lexicographically increasing order unless the polynomial time hierarchy collapses to NP. We also show that the sets that are monotonically bi-enumerable are “essentially” the same as the sets in parity polynomial time.

The degree hierarchy of undecidable problems of formal grammars

Abstract: After a brief discussion of historical matters in §1, twenty-seven predicates of formal grammers are introduced in §2. The next two sections discuss recursively enumerable predicates and nonrecursively enumerable predicates, respectively. These results show that the degree of unsolvability of a predicate is determined by its domain of definition. The paper concludes with a degree diagram and suggestions for further development. From a more comprehensive point of view these results complement the computational complexity classification of solvable problems and extend that classification to unsolvable problems based on their degree of unsolvability.

Simulating Turing Machines by P Systems with External Output

Abstract: In [3] a variant of the computation model introduced by Gh. Paun in [1] is considered: membrane systems with external output, which were proven to be universal, in the sense that they are able to generate all Parikh images of recursively enumerable languages. Here we give another proof of the universality of this model. The proof is carried out associating to each deterministic Turing machine a P system with external output that simulates its running. Thus, although we work with symbol-objects, we get strings as a result of computations, and in this way we generate directly all recursively enumerable languages, instead of their images through Parikh mapping, as it is done in [3].

Minimizing evolution-communication P systems and automata

Abstract: Evolution-communication P systems are a variant of P systems allowing both rewriting rules and symport/antiport rules, thus having separated the rewriting and the communication. The purpose of this paper is to solve an open problem stated in Reference [1], namely generating the family of Turing computable sets of vectors of natural numbers instead of the family of Turing computable sets of natural numbers. The same construction also reduces the 3-membrane non-cooperative case and the 2-membrane 1-catalyst case to the 2-membrane non-cooperative case. Also, EC P automata are introduced and it is proved that 2-membrane EC P automata with a promoter can accept all recursively enumerable languages. Finally, a definition of an extended system is given, and its universality is proved using the rules of more restricted types.