Recursively enumerable classes and their application to recursive sequences of formal theories

1965 ◽  
Vol 8 (3-4) ◽  
pp. 104-121 ◽  
Author(s):  
Marian Boykan Pour-El ◽  
Hilary Putnam
Keyword(s):  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Formal Theories

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

Is Rationality Normative?

2017 ◽  
Author(s):  
Ralph Wedgwood
Keyword(s):  
Rational Choice ◽  
Rational Belief ◽  
False Beliefs ◽  
Ought Implies Can ◽  
Normative Concept ◽  
Original Meaning ◽  
Formal Theories

In its original meaning, the word ‘rational’ referred to the faculty of reason—the capacity for reasoning. It is undeniable that the word later came also to express a normative concept—the concept of the proper use of this faculty. Does it express a normative concept when it is used in formal theories of rational belief or rational choice? Reasons are given for concluding that it does express a normative concept in these contexts. But this conclusion seems to imply that we ought always to think rationally. Four objections can be raised. (1) What about cases where thinking rationally has disastrous consequences? (2) What about cases where we have rational false beliefs about what we ought to do? (3) ‘Ought’ implies ‘can’—but is it true that we can always think rationally? (4) Rationality requires nothing more than coherence—but why does coherence matter?

scholarly journals DIOPHANTINE SETS OF POLYNOMIALS OVER ALGEBRAIC EXTENSIONS OF THE RATIONALS

2014 ◽  
Vol 79 (3) ◽  
pp. 733-747
Author(s):  
CLAUDIA DEGROOTE ◽  
JEROEN DEMEYER
Keyword(s):  
Minimal Polynomial ◽  
Finite Extension ◽  
Algebraic Extension ◽  
Recursively Enumerable

AbstractLet L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

scholarly journals Log-concavity of P-recursive sequences

2021 ◽  
Vol 107 ◽  
pp. 251-268
Author(s):  
Qing-hu Hou ◽  
Guojie Li
Keyword(s):  
Recursive Sequences

scholarly journals When catalytic P systems with one catalyst can be computationally complete

2021 ◽  
Author(s):  
Artiom Alhazov ◽  
Rudolf Freund ◽  
Sergiu Ivanov
Keyword(s):  
Computational Complexity ◽  
P Systems ◽  
Membrane Systems ◽  
Computational Completeness ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

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