Showing papers on "Recursively enumerable language published in 1996"

Test Tube Distributed Systems Based on Splicing

Abstract: We define a symbol processing mechanism with the components (test tubes) working as splicing schemes in the sense of T. Head and communicating by redistributing the contents of tubes (in a similar way to the separate operation of Lipton-Adleman). (These systems are similar to the distributed generative mechanisms called Parallel Communicating Grammar Systems.) Systems with finite initial contents of tubes and finite sets of splicing rules associated to each component are computationally complete, they characterize the family of recursively enumerable languages. The existence of universal test tube distributed systems is obtained on this basis, hence the theoretical proof of the possibility to design universal programmable computers with the structure of such a system.

DNA computing based on splicing: universality results.

Abstract: The paper extends some of the most recently obtained results on the computational universality of specific variants of H systems (e.g. with regular sets of rules) and proves that we can construct universal computers based on various types of H systems with a finite set of splicing rules as well as a finite set of axioms, i.e. we show the theoretical possibility to design programmable universal DNA computers based on the splicing operation. For H systems working in the multiset style (where the numbers of copies of all available strings are counted) we elaborate how a Turing machine computing a partial recursive function can be simulated by an equivalent H system computing the same function; in that way, from a universal Turning machine we obtain a universal H system. Considering H systems as language generating devices we have to add various simple control mechanisms (checking the presence/absence of certain symbols in the spliced strings) to systems with a finite set of splicing rules as well as with a finite set of axioms in order to obtain the full computational power, i.e. to get a characterization of the family of recursively enumerable languages. We also introduce test tube systems, where several H systems work in parallel in their tubes and from time to time the contents of each tube are redistributed to all tubes according to certain separation conditions. By the construction of universal test tube systems we show that also such systems could serve as the theoretical basis for the development of biological (DNA) computers.

Undecidable fragments of elementary theories

Abstract: We introduce a general framework to prove undecidability of fragments. This is applied to fragments of theories arising in algebra and recursion theory. For instance, the V3V-theories of the class of finite distributive lattices and of the p.o. of recursively enumerable many-one degrees are shown to be undecidable.

Splicing Grammar Systems

Abstract: The aim of this paper is to bring together two new and powerful tools: on the one hand, the splicing operation as a basic operation on DNA sequences and, on the other hand, the parallelism and communication features in grammar systems. As expected, the result of the above combination is a very powerful mechanism, leading to a new characterization of recursively enumerable languages.

Embeddings of Heyting algebras

Abstract: In this paper we study embeddings of Heyting Algebras. It is pointed out that such embeddings are naturally connected with Derived Rules. We compare the Heyting Algebras embeddable in the Heyting Algebra of the Intuitionistic Propositional Calculus (IPC), i.e. the free Heyting Algebra on countably infinitely many generators, and those embeddable in the Heyting Algebra of Heyting's Arithmetic (HA). A partial result is obtained. We show that every recursively enumerable prime Heyting Algebra is embeddable -in the Heyting Algebra of HA*, a ‘natural’ extension of HA.

Definability in the Recursively Enumerable Degrees

Abstract: §1. Introduction. Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets) always seem to be either actually computable (recursive) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K? Let be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl)with least element 0, the degree (equivalence class) of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of . The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved: Theorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into . Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degree a there are r.e. degrees b, c < a such that b ∨ c = a. Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degrees a < b there is an r.e. degree c such that a < c < b.

Bimodal logics for extensions of arithmetical theories

Abstract: We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ0 + EXP, PRA); (PRA, IΣn); (IΣm, IΣn) for 1 ≤ m < n; (PA, ACA0); (ZFC, ZFC + CH); (ZFC, ZFC + ¬CH) etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

Lattice embeddings below a nonlow2 recursively enumerable degree

Abstract: We introduce techniques that allow us to embed below an arbitary nonlow2 recursively enumerable degree any lattice currently known to be embeddable into the recursively enumerable degrees.

On a quantitative notion of uniformity

Abstract: One topic arising in recent research on “Bounded Query Classes” is to consider quantitative aspects of recursion theory, and in particular various notions of parameterized recursive approximations of sets. An important question is, for which values of the parameters - depending on the type of approximation - the approximated set is necessarily recursive. Beigel's Nonspeedup Theorem, Rummer's Cardinality Theorem and Trakhtenbrot's Theorem provide answers using nonuniform constructions. This paper investigates to which extend these constructions can be made uniform. Beigel's Nonspeedup Theorem is equivalent to the statement that every branch of a recursively enumerable tree of bounded width is recursive. There is no algorithm which computes a branch from the index of the tree, but there are nontrivial positive results by weakening the requirements as follows: For some fixed number k, an algorithm is wanted which, given an index of a tree, outputs a list of k programs such that at least one of them computes a branch of the tree up to finitely many errors. What is the least k for which this works? In this paper it is shown that, for recursively enumerable trees of width at most n, the least possible k is 2n−1. For trees arising from Trakhtenbrot's Theorem with parameters m, n, the optimal value is k = n − m + 1. In addition, several other, related classes of trees are investigated.

The Sacks density theorem and Σ 2 -bounding

Abstract: The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P− + BΣ2. The proof has two components: a lemma that in any model of P− + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

Accepting Multi-Agent Systems

Abstract: We consider cooperating distributed (CD) grammar systems and variants thereof as language acceptors. If the CD grammar systems work in the modes ..., then their generating capacity equals their accepting capacity. Contrary to this, we obtain a new characterization of the context-sensitive languages by accepting CD grammar systems (with or without l-productions) working in t-mode. Moreover, accepting hybrid CD (HCD) grammar systems with l-productions characterize the recursively enumerable languages.

Uniform representation of recursively enumerable sets with simultaneous rigid E-unification

Abstract: Recently it was proved that the problem of simultaneous rigid E­unification (SREU) is undecidable. Here we perform an in­depth investigation of this matter and obtain that one can use SREU to uniformly represent any recursively enumerable set. From the exact form of this representation follows that SREU is undecidable already for 6 rigid equations with ground left hand sides and 2 variables. There is a close correspondence between solvability of SREU problems and provability of the corresponding formulas in intuitionistic first order logic with equality. Due to this correspondence we obtain a new (uniform) representation of the recursively enumerable sets in intuitionistic first order logic with equality with one binary function symbol and a countable set of constants. From this result follows the undecidability of the EE­fragment of intuitionistic logic with equality. This is an improvement of a recent result regarding the undecidability of the E*­fragment in general.

DNA splicing systems and post systems.

Abstract: This paper concerns the formal study on the generative powers of extended splicing (H) systems. First, using a classical result by Post which characterizes the recursively enumerable languages in terms of his Post Normal systems, we establish several new characterizations of extended H systems which not only allow us to have very simple alternative proof methods for the previous results mentioned above, but also give a new insight into the relationships between families of extended H systems. We show a kind of normal form for extended H systems exactly characterizing the class of regular languages. We also show a new representation result for the family of context-free languages in terms of extended H systems.

Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices

Abstract: We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint.We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21].

An introduction to formal languages and machine computation

Abstract: This book provides an elementary introduction to formal languages and machine computation. The materials covered include computation-oriented mathematics, finite automata and regular languages, push-down automata and context-free languages, Turing machines and recursively enumerable languages, and computability and complexity. As integers are important in mathematics and computer science, the book also contains a chapter on number-theoretic computation. The book is intended for university computing and mathematics students and computing professionals.

Translating the hypergame paradox: Remarks on the set of founded elements of a relation

Abstract: In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct an undecidable formula.

A comment on the joint embedding property

Abstract: The paper presents an alternative proof of the known result that no recursively enumerable number theory has the joint embedding property.

Conditional Tabled Eco-Grammar Systems versus (E)TOL Systems

Abstract: We investigate the generative capacity of the so-called conditional tabled eco-grammar systems (CTEG). They are a variant of eco-grammar systems, generative mechanisms recently introduced as models of the interplay between environment and agents in eco-systems. In particular, we compare the power of CTEG systems with that of programmed and of random context T0L systems and with that of ET0L systems. CTEG systems with one agent only (and without extended symbols) are found to be surprisingly powerful (they can generate non-ETOL languages). Representation theo­rems for ET0L and for recursively enumerable languages in terms of CTEG languages are also presented.

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.

Kins of context-free languages

Abstract: We study languages which can be described as limits of fast converging infinite sequences of context-free languages. Such a sequence \(L_0 \subseteq L_1 \subseteq L_2 \subseteq\) ... is fast converging if each string w of its limit language belongs to an Li which has a grammatical description very concise in comparison with the length of w . We prove that these languages are closely related to context-free languages in several properties: pumping lemma, interchange lemma, regularity of unary languages, full AFL properties. The languages can differ, however, in their computational complexity: we construct languages of arbitrarily high complexity, even languages which are not recursively enumerable, but have fast context-free approximations.