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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.


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TL;DR: The set-theoretical predicate Taut(x), "x is a tautology," taken in the infinitary sense is considered, to completely settle the question of ?
Abstract: Methods are being developed for treating questions of decidability in fewer than Q steps, Q being a regular2 nondenumerable cardinal. In this paper we consider the set-theoretical predicate Taut(x), "x is a tautology," taken in the infinitary sense. In case x is hereditarily finite there is no question that it is decidable in finitely many steps. But what if x is infinite? We are not assuming that x is in any way constructive or even that the propositional formulas can be well-ordered, so it is not appropriate to treat this predicate as one of natural numbers or even as one of ordinals. There is available however the notion of X1-predicate of sets, studied by Levy in [4], which reduces to a notion equivalent to recursively enumerable predicate of natural numbers when the universe is restricted to the hereditarily finite sets. The El-predicates retain their original meaning when the universe is restricted to Ha, the set of sets hereditarily of power less than Q2 (Theorem 1.4). Call a predicate on Ho a Zn-predicate if it is the restriction to Ha of a E1-predicate, a HIe-predicate if its complement with respect to H. is a V-predicate. There is little doubt that a subset C of H0 whose elements are generable in fewer than n steps by some effective procedure, is EQ. This is because it always seems to be possible to attach to an effective procedure a Zermelo-Fraenkel formula W(v0, vl) in such a way that if if is satisfied by (x0, x1) in a transitive set, then x1 is the output for input x0, and conversely, if x1 is the output for input x0, then there is a transitive set in which (x0, x1) satisfies W. Thus x1 is the output for x0 iff (3t)(Trans(t) A x0, x1 E t A }= 'e(xo, xj)). But the satisfaction predicate relative to a set is II 1 r) . (See Levy [4]. The calculus of Appendix I suffices for this.) Thus the condition, ''x1 is the output for x0," is El. But then the predicate x1 E C for x1 E H0 is equivalent to (3x0) (xl is the output for x0), a 1,-predicate. By Theorem 1.4, C is El. Returning to the predicate Taut(x) and its restriction Q-Taut(x) to x Ec Ha, the results in this paper specify the Q for which Q-Taut is El (assuming the continuum hypothesis (CH)) and specify also the Q for which Q-Taut is En, given a parameter in H0 (assuming the generalized continuum hypothesis (GCH)). These results plus the completeness theorems in [2] completely settle the question of ?-effective axiomatizability for ?2-propositional calculi assuming GCH, for it is precisely in the positive cases, the cases where Q-Taut is En given a parameter in Ho, that

7 citations

Journal ArticleDOI
TL;DR: It is shown that for all α which are regular cardinals of L (ℵ 1 L is, of course, such an α), there are simple α-r.e. sets with different 1-types, and another theorem of (ω) points the way.
Abstract: Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*- finite if it is α-finite and has ordertype less than α* (the Σ 1 projectum of α). An a-r.e. set S is simple if (the complement of S ) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ 1 L ( L is Godel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ 1 L is, of course, such an α), there are simple α-r.e. sets with different 1-types. The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “ S is a maximal α-r.e. set”, “ S is an r -maximal α-r.e. set”, or “ S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L , there are no maximal, r -maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

7 citations

Journal ArticleDOI
TL;DR: A terminal weighted grammar is defined, where the terminal generated at any step of a derivation is defined as a function of time, and it is seen that terminal weighted regular grammars generate exactly the class of recursively enumerable sets.
Abstract: Motivated by the idea of describing parquet deformations using grammars and also of describing an infinite number of terminals starting with only a finite set, this paper defines a terminal weighted grammar, where the terminal generated at any step of a derivation is defined as a function of time It is seen that terminal weighted regular grammars generate exactly the class of recursively enumerable sets Terminal weighted matrix grammars are used to describe parquet deformations The extension of terminal weights to array grammars is also discussed

7 citations

Journal ArticleDOI
TL;DR: It turns out that going through all propagating morphismsf and g, the family of maximal solutions obtained equals the famdy of complements of recursively enumerable languages after intersecting with regular languages and mapping with propagating Morphisms.
Abstract: After extending two word morphismsfand g to languages, an equationf(X) = g(X) can be written and ItS language soluUons investigated. An elementary characterization of the famdy of all solutions of the equation is ~ven, and it is used to mvesttgate the maximal solution which is the mum subject of this paper. It turns out that going through all propagating morphismsf and g, the family of maximal solutions obtained equals the famdy of complements of recurslvely enumerable languages after intersecting with regular languages and mapping with propagating morphisms. In the general case (of arbitrary morphismsf and g) the corresponding family is larger and includes the full-AFL closure of the family of complements of recursively enumerable languages.

7 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202220
202127
202022
201918
201823