Two variable implicational calculi of prescribed many-one degrees of unsolvability

1976 ◽  
Vol 41 (1) ◽  
pp. 39-44 ◽  
Author(s):  
Charles E. Hughes
Keyword(s):  
Decision Problems ◽  
Constructive Proof ◽  
Distinct Variable ◽  
The Family ◽  
Propositional Calculi ◽  
Degrees Of Unsolvability

AbstractA constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

Many-one degrees associated with problems of tag

1973 ◽  
Vol 38 (1) ◽  
pp. 1-17 ◽  
Author(s):  
C. E. Hughes
Keyword(s):  
Formal Proof ◽  
Extensive Study ◽  
Decision Problems ◽  
Turing Machines ◽  
Natural Numbers ◽  
Halting Problem ◽  
The Family ◽  
Form Part ◽  
The Many ◽  
Degrees Of Unsolvability

Tag systems were defined by Post [9], [10] and have been studied by a number of researchers including Minsky [7], Maslov [6] and Aanderaa and Belsnes [1]. In their recent paper Aanderaa and Belsnes demonstrated that every r.e. many-one degree (exclusive of the degree of the empty set) is represented by the general halting problem for tag systems, that is, by the family of halting problems ranging over all tag systems. Their result depends upon an informal proof of this property for Turing machines but may be seen to be correct in light of a formal proof due to Overbeek [8]. Our aim is to extend their results to the general word problem for these systems. Specifically, we shall present an effective method which, when applied to an arbitrary r.e. set S, where S is neither empty nor the set of all natural numbers, produces a tag system R′ whose word and halting problems are both of the same many-one degree as the decision problem for S. The proof is realized by first constructing, from the description of an arbitrary Turing machine M, which machine has at least one mortal and one immortal configuration, a 5-register machine R, whose word and halting problems are both of the same many-one degree as the halting problem for M. From R we then construct the desired tag system R′. This construction combined with Overbeek's [8] shows that every r.e. many-one degree (exclusive of the degrees of the empty set and the set of all natural numbers) is represented by the general word and halting problems for tag systems. Moreover our results are seen to be best possible with regard to degrees of unsolvability in that it is not the case that every nonrecursive r.e. one-one degree is represented by either of the general decision problems for tag systems which are considered here. These results were first shown in the author's thesis [3] and were announced in [4], They form part of an extensive study into the many-one equivalence of general decision problems. An overview of the initial findings of this research project may be found in [5].

The undecidability theorem for the Horn-like fragment of linear logic (Revisited)

2016 ◽  
Vol 26 (5) ◽  
pp. 719-744 ◽  
Author(s):  
MAX KANOVICH
Keyword(s):  
Linear Logic ◽  
Decision Problems ◽  
Constructive Proof ◽  
Complexity Bounds ◽  
Wide Range ◽  
Image Position ◽  
Direct Correspondence

There has been an increased interest in the decision problems for linear logic and its fragments. Here, we give a fully self-contained, easy-to-follow, but fully detailed, direct and constructive proof of the undecidability of a very simple Horn-like fragment of linear logic, which is accessible to a wide range of people. Namely, we show that there is a direct correspondence between terminated computations of a Minsky machine M and cut-free linear logic derivations for a Horn-like sequent of the form \begin{equation*} \bang{\Phi_M},\ l_1 \vdash l_0, \end{equation*} where ΦM consists only of Horn-like implications of the following simple forms \begin{equation*} (l \llto l'),\ \ ((l\otimes r) \llto l'),\ \ (l\llto (r\otimes l')),\ \ and \ \ (l\llto (l'\oplus l'')), \end{equation*} where l1, l0, l, l′, l″ and r stand for literals.Neither negation, nor &, nor constants, nor embedded implications/bangs are used here.Furthermore, our particular correspondence constructed above provides decidability for each of the Horn-like fragments whenever we confine ourselves to any two forms of the above Horn-like implications, along with the complexity bounds that come from the proof.

A General Framework for Priority Arguments

1995 ◽  
Vol 1 (2) ◽  
pp. 189-201 ◽  
Author(s):  
Steffen Lempp ◽  
Manuel Lerman
Keyword(s):  
Information Content ◽  
Equivalence Relation ◽  
Decision Problem ◽  
General Framework ◽  
Equivalence Classes ◽  
Partial Ordering ◽  
Decision Problems ◽  
Natural Numbers ◽  
Relative Complexity ◽  
Degrees Of Unsolvability

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure theinformation contentof sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question is equivalent to: Given a computer with access to an oracle which will answer membership questions about a setA, can a program (allowing questions to the oracle) be written which will correctly compute the answers to all membership questions about a setB? If the answer is yes, then we say thatBisTuring reducibletoAand writeB≤TA. We say thatB≡TAifB≤TAandA≤TB. ≡Tis an equivalence relation, and ≤Tinduces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called thedegrees of unsolvability, and elements of this poset are calleddegrees.Post was particularly interested in computability from sets which are partially generated by a computer, namely, those for which the elements of the set can be enumerated by a computer.

scholarly journals A family of norms with applications in quantum information theory

2011 ◽  
Vol 11 (1&2) ◽  
pp. 104-123
Author(s):  
Nathaniel Johnston ◽  
David W. Kribs
Keyword(s):  
Semidefinite Program ◽  
Constructive Proof ◽  
Convex Mapping ◽  
Semidefinite Programs ◽  
Positive Partial Transpose ◽  
The Family ◽  
Partial Transpose ◽  
Arbitrary Convex ◽  
Matlab Implementation ◽  
Werner States

We consider the problem of computing the family of operator norms recently introduced. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory of semidefinite programming, including a new constructive proof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of the semidefinite program, including the case of Werner states and randomly generated states via the Bures measure, and approximate distributions of the norms are provided. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states.

REFLECTIONS ON MATHEMATICAL ECONOMICS IN THE ALGORITHMIC MODE

2012 ◽  
Vol 08 (01) ◽  
pp. 139-152
Author(s):  
K. VELA VELUPILLAI
Keyword(s):  
Axiom Of Choice ◽  
Decision Problems ◽  
Constructive Proof ◽  
Special Issue ◽  
Standard Analysis ◽  
Mathematical Economics ◽  
Existence Proofs ◽  
Skolem Theorem ◽  
The Axiom Of Choice ◽  
Non Standard Analysis

Non-standard analysis can be harnessed by the recursion theorist. But as a computable economist, the conundrums of the Löwenheim-Skolem theorem and the associated Skolem paradox, seem to pose insurmountable epistemological difficulties against the use of algorithmic non-standard analysis. Discontinuities can be tamed by recursive analysis. This particular kind of taming may be a way out of the formidable obstacles created by the difficulties of Diophantine Decision Problems. Methods of existence proofs, used by the "classical" mathematician — even if not invoking the axiom of choice — cannot be shown to be equivalent to the exhibition of an instance in the sense of a constructive proof. These issues were prompted by the fertile and critical contributions to this special issue.

SOME REMARKS ON THE HAIRPIN COMPLETION

2010 ◽  
Vol 21 (05) ◽  
pp. 859-872 ◽  
Author(s):  
FLORIN MANEA ◽  
VICTOR MITRANA ◽  
TAKASHI YOKOMORI
Keyword(s):  
Regular Language ◽  
Regular Languages ◽  
Decision Problems ◽  
Strict Sense ◽  
Closure Properties ◽  
New Words ◽  
The Family

We consider several problems regarding the iterated or non-iterated hairpin completion of some subclasses of regular languages. Thus we obtain a characterization of the class of regular languages as the weak-code images of the k-hairpin completion of center-disjoint k-locally testable languages in the strict sense. This result completes two results from [3] and [11]. Then we investigate some decision problems and closure properties of the family of the iterated hairpin completion of singleton languages. Finally, we discuss some algorithms regarding the possibility of computing the values of k such that the non-iterated or iterated k-hairpin completion of a given regular language does not produce new words.

scholarly journals Proper gap-labellings: on the edge and vertex variants

2019 ◽  
Author(s):  
Celso A. Weffort-Santos ◽  
Christiane N. Campos ◽  
Rafael C. S. Schouery
Keyword(s):  
Computational Complexity ◽  
Simple Graph ◽  
Decision Problems ◽  
Np Completeness ◽  
Polynomial Time Algorithms ◽  
The Family ◽  
Completeness Result ◽  
Np Complete ◽  
Proper Vertex ◽  
Vertex Colouring

Given a simple graph G, an ordered pair (π, cπ) is said to be a gap- [k]-edge-labelling (resp. gap-[k]-vertex-labelling) ofG ifπ is an edge-labelling (vertex-labelling) on the set {1, . . . , k}, and cπ is a proper vertex-colouring such that every vertex of degree at least two has its colour induced by the largest difference among the labels of its incident edges (neighbours). The decision problems associated with these labellings are NP-complete for k ≥ 3, and even when k = 2 for some classes of graphs. This thesis presents a study of the computational complexity of these problems, structural properties for certain families of graphs and several labelling algorithms and techniques. First, we present an NP-completeness result for the family of subcubic bipartite graphs. Second, we present polynomial-time algorithms for families ofgraphs. Third, we introduce a new parameter associated with gap-[k]-vertex-labellings ofgraphs.

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