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.

Monogenic Post Normal Systems of Arbitrary Degree

Abstract: From an arbitrary Turing machine, Z, a monogenic Post normal system, v(Z), is constructed. It is then shown not only that the halting problem for Z is reducible to that of v(Z) but also that the halting problem for v(Z) is reducible to that of Z. Since these two halting problems are of the same degree, the halting problem for the normal system can have an arbitrary (recursively enumerable) degree of undecidability.

Syntactic complexity of context-free grammars over word monoids

Abstract: The syntactic complexity of context-free grammars defined over word monoids is investigated. It is demonstrated that every recursively enumerable language can be defined by a ten-nonterminal context-free grammar over a word monoid generated by an alphabet and six words of length two. Open problems are formulated.

A finitely presented group with almost solvable conjugacy problem

Abstract: The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd(m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

On the learnability of vector spaces

Abstract: The central topic of the paper is the learnability of the recursively enumerable subspaces of V∞/V, where V∞ is the standard recursive vector space over the rationals with countably infinite dimension, and V is a given recursively enumerable subspace of V∞. It is shown that certain types of vector spaces can be characterized in terms of learnability properties: V∞/V is behaviourally correct learnable from text iff V is finitely dimensional, V∞/V is behaviourally correct learnable from switching type of information iff V is finite-dimensional, 0-thin, or 1-thin. On the other hand, learnability from an informant does not correspond to similar algebraic properties of a given space. There are 0-thin spaces W 1 and W 2 such that W 1 is not explanatorily learnable from informant and the infinite product (W 1 )∞ is not behaviourally correct learnable, while W 2 and the infinite product (W 2 )∞ are both explanatorily learnable from informant.