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.

Degrees of Recursively Enumerable Topological Spaces

Abstract: In [5], Metakides and Nerode introduced the study of recursively enumerable (r.e.) substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff considered X , a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets of ω give rise to r.e. subsets of X . The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset of X ? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above. In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that given A any r.e. set and any r.e. open subset of X , there exists an r.e. open set ℋ which is a subset of and is dense in (in a topological sense) and in which A is coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree of A . We then go on and establish various results (both existential and universal) on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2.

Membrane division, restricted membrane creation and object complexity in P systems

Abstract: We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind. We show here that they generate all recursively enumerable languages, and that two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case. We then prove that 10+m symbols are sufficient to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting memb...

Well structured program equivalence is highly undecidable

Abstract: We show that strict deterministic propositional dynamic logic with intersection is highly undecidable, solving a problem in the Stanford Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We introduce the construction of program equivalence, which returns the value $\mathsf{T}$ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has $\Pi_1^1$-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program \texttt{skip}. We also show, in these cases, that the set of propositional statements valid over finite models is not recursively enumerable, so there is not even an axiomatisation for finitely valid propositions.

Complete Formal Systems for Equivalence Problems.

Abstract: We describe four complete and recursively enumerable formal systems S0,D0,H0,B0. Each one of them proves the decidability of some equivalence problem for some class of automata: namely the language equivalence problem for simple automata, the language equivalence problem for deterministic pushdown automata, the function equivalence problem for deterministic pushdown transducers with outputs in an abelian group, the bisimulation equivalence problem for loop-free pushdown automata.

A homomorphic characterization of time and space complexity classes of languages

Abstract: Recently, it has been shown that for each recursively enumerable language there exists an erasing homomorphism h 0 and homomorphisms h 1,h 2 such that L= h 0(e(h 1,h 2)) where (e(h 1,h 2)) is the set of minimal words on which h 1 and h 2 agree. Here we show that by restrictions on the erasing h 0 we obtain most time-complexity language classes, and by restrictions on the pair (h 1 h 2) we characterize all space complexity language classes.