Showing papers on "Recursively enumerable language published in 1991"

Post's program and incomplete recursively enumerable sets.

Abstract: A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a first-order property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that (i) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete and (ii) there exists an r.e. set A satisfying Q(A). This resolves a long open question stemming from Post's program of 1944, and it sheds light on the fundamental problem of the relationship between the algebraic structure of an r.e. set A and the (Turing) degree of information that A encodes.

Inductive Inference of Recursive Functions: Complexity Bounds

Abstract: This survey includes principal results on complexity of inductive inference for recursively enumerable classes of total recursive functions Inductive inference is a process to find an algorithm from sample computations In the case when the given class of functions is recursively enumerable it is easy to define a natural complexity measure for the inductive inference, namely, the worst-case mindchange number for the first n functions in the given class Surely, the complexity depends not only on the class, but also on the numbering, ie which function is the first, which one is the second, etc It turns out that, if the result of inference is Goedel number, then complexity of inference may vary between log2n+o(log2n) and an arbitrarily slow recursive function If the result of the inference is an index in the numbering of the recursively enumerable class, then the complexity may go up to const·n Additionally, effects previously found in the Kolmogorov complexity theory are discovered in the complexity of inductive inference as well

Ordered rings over which output sets are recursively enumerable sets

Abstract: In a recent paper [BSS], L. Blum, M. Shub, and S. Smale developed a theory of computation over the reals and over commutative ordered rings ; in 9 of [BSS] they showed that over the reals (and over any real closed field) the class of recursively enumerable sets and the class of output sets are the same ; it is question (Problem 9.1 in [BSS] to characterize ordered rings with this property (abbreviated by O=R.E. here). In this paper we prove essentially that in the class of (linearly) ordered rings of infinite transcendence degree over Q, that are dense (for the order) in their oral closures, only real closed fields have property O=R.E

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.

Jump embeddings in the turing degrees

Abstract: Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and

Some properties of the syntactic p -recursion categories generated by consistent, recursively enumerable extensions of peano arithmetic

Abstract: The dominical categories were introduced by Di Paola and Heller, as a first step toward a category-theoretic treatment of the generalized first Godel incompleteness theorem [1]. In his Ph.D. dissertation [7], Rosolini subsequently defined the closely related p-categories, which should prove pertinent to category-theoretic representations of incompleteness for intuitionistic systems. The precise relationship between these two concepts is as follows: every dominical category is a pointed p-category, but there are p-categories, indeed pointed p-isotypes (all pairs of objects being isomorphic) with a Turing morphism that are not dominical. The first of these assertions is an easy consequence of the fact that in a dominical category C by definition the near product functor when restricted to the subcategory Ct, of total morphisms of C (as “total” is defined in [1]) constitutes a bona fide product such that the derived associativity and commutativity isomorphisms are natural on C × C × C and C × C, respectively, as noted in [7]. As to the second, p-recursion categories (that is, pointed p-isotypes having a Turing morphism) that are not dominical were defined and studied by Montagna in [6], the syntactic p-categories ST and S′T associated with consistent, recursively enumerable extensions of Peano arithmetic, PA. These merit detailed investigation on several counts.

The power of alternating one-reversal counters and stacks

Abstract: The relation between reversals and alternation is studied in two simple models of computation: the 2-counter machine with a one-way input tape whose counters make only one reversal (1-reversal 2CM) and the one-way pushdown automaton whose pushdown store makes only one reversal (1-reversal PDA). The following is shown: (a) alternating 1-reversal 2CM’s accept all recursively enumerable languages; (b) alternating 1-reversal PDA’s accept exactly the languages accepted by exponential time-bounded deterministic TM’s. The first improves on the known result that alternating 1-reversal 4CM’s accept all recursively enumerable languages. The second improves an earlier result that alternating PDA’s with no restrictions on reversals accept exactly the exponential-time languages.

A diophantine definition of integers in the rings of rational numbers

Abstract: The author considers rings of rational numbers which are integral at all the primes except, possibly, primes contained in a finite set. In such rings a Diophantine definition of ℤ is constructed to show that all the recursively enumerable subsets of the ring are Diophantine.

Finite automata over a direct product of free semigroups and groups and languages

Abstract: A finite-automaton-algebraic model of formal languages is described. It is shown how particular cases of this model characterize the classes of recursively enumerable and context-free languages.

A universal partial recursive function

Abstract: The present article is devoted to the study of the properties of universal functions. It contains a number of assertions regarding the algebraic properties of the orbits of universal functions. It is proved that the union of all cycles of fixed length of a universal function possesses the property of being creative. A description is obtained of recursively enumerable sets that are splinters and infinite cycles of universal functions. Several sufficiency conditions for the universality of functions of the form ~8 (~ a universal function) are presented.