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.

∑2 Induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s Conjecture

Abstract: We prove that in everyB∑2 model (one satisfies ∑2 collection axioms but not ∑2 induction), every recursively enumerable (r.e.) set is either prompt or recursive. Consequently, over the base theory ∑2 collection, the existence of r.e. minimal pairs is equivalent to ∑2 induction. We also refute Shoenfield’s Conjecture inB∑2 models.

Minimal polynomial degrees of nonrecursive sets

Abstract: This is the second paper exploring a new link between recursion theory and computational complexity theory. From its inception part of the work in complexity theory has been guided by an analogy with recursion theory. However, here we explore a more concrete and direct connection. This connection stems from the notion of honest polynomial-time reducibi l i ty, ~ , defined in Homer [3]. ~ is a sl ight strengthening of the usual polynomial-time Turing reducibi l i ty. (Precise definitions and the main theorems from [3] are presented in the next section.) The particular problem studied is the existence of minimal sets for this reducibi l i ty. A set A is -minimal i f A ~ P and for any B, i f B ~ A then B~ P or A ~ B. In [3] two opposing results were proved. First i t was shown that no recursive set is of minimal ~ -degree. Second, that i f a set recursive in O" has minimal degree then P i NP. The method used to prove this theorem is recursion-theoretic in nature and is an adaption of the proof of the existence of minimal Turing degrees. The result gives a concrete connection between recursion theory and a fundamental problem of complexity theory. The results presented in this paper carry these methods and ideas further, There is clearly a gap between the f i r s t result, dealing with recursive sets and the second, dealing with sets recursive in 0". The intention here is to narrow that gap by looking at r.e. sets and nonrecursive sets with particular properties, but with respect to the same question of ~ -minimality. in section 3 we take another look at the second theorem mentioned above. A partial converse of this result is proved, l inking this problem to the existence of certain one-way functions. Sections 4 and 5 are devoted to trying to strengthen the f i r s t theorem mentioned above. Recursively enumerable sets are considered f i r s t . I t is shown that no recursively enumerable set which is P-immune can be ~ -minimal. (A set is P-immune i f i t contains no in f in i te polynomial time subset.) Arbitrary sets are next considered and i t is shown that i f a set and i ts complement are both P-immune then i t is not ~ -minimal. Topics for future work are discussed in the last section.

Strong noncomputability of random strings

Abstract: We prove that every infinite set of random strings is not recursively enumerable. In particular, the set of all random strings is not recursively enumerable. This property asserts that in a strong sense random strings are not constructable.

Context-dependent tree automata

Abstract: A new family of tree automata is studied\3-the “context-dependent tree automata≓ (CDTA). The CDTA are a generalization of the standard root-to-frontier tree automata studied over the past decade. The CDTA permit limited “communication≓ between different loci of the input tree. This communication mechanism permits CDTA to recognize many languages of trees not recognized by standard tree automata such as those defined by Thatcher, Doner, and Rounds. For example, nondeterministic CDTA recognize sets of trees whose frontiers are arbitrary recursively enumerable sets. However, there are simple recursive sets of trees which no nondeterministic CDTA can recognize. The CDTA recognize “highly transparent≓ derivations of many context-dependent languages. These derivation trees are not generated by any context-dependent grammar. The deterministic CDTA are not closed under any of the Boolean operations. The nondeterministic CDTA are closed only under union.

About time-varying distributed H systems

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 passing from a component to another one is specified in a cycle. It was proved by both authors (1999) that TVDH systems of degree 2 generate all recursively enumerable languages. It was made by modelling Turing machines and, in that modelling, every language is generated step by step or word by word. This solution is not a fully parallel one. A.Paun (1999) presented a complete parallel solution for TVDH systems of degree 4 by modelling type 0 formal grammars. Now we improved A.Paun's result by reducing the number of components of such TVDH systems down to 3. This question is open for 2 components, i.e. is it possible to construct TVDH systems of degree 2 which completely uses the parallel nature of molecular computations based on splicing operations (say model type 0 formal grammars)? We consider also original G.Paun's definition of TVDH systems and suggest a slightly different definition of TVDH systems based on the definition of H systems - extended time-varying distributed H systems (ETVDH systems). For this new definition we proved that ETVDH systems with one compdnent generate exactly the set of all regular languages and that with two components, they generate exactly the set of all recursively enumerable languages.