Infinite Argument Research Articles

ω-Groundedness of argumentation and completeness of grounded dialectical proof procedures

Dialectical proof procedures in assumption-based argumentation are in general sound but not complete with respect to both the credulous and skeptical semantics (due to non-terminating loops). This raises the question of whether we could describe exactly what such procedures compute. In a previous paper, we introduce infinite arguments to represent possibly non-terminating computations and present dialectical proof procedures that are both sound and complete with respect to the credulous semantics of assumption-based argumentation with infinite arguments. In this paper, we study whether and under what conditions dialectical proof procedures are both sound and complete with respect to the grounded semantics of assumption-based argumentation with infinite arguments. We introduce the class of ω-grounded and finitary-defensible argumentation frameworks and show that finitary assumption-based argumentation is ω-grounded and finitary-defensible. We then present dialectical procedures that are sound and complete wrt finitary assumption-based argumentation.

Riemann Zeta Based Surge Modelling of Continuous Real Functions in Electrical Circuits

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

Generic degrees are complemented

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let o be the set of natural numbers, i.e., (0, 1,2,3, . . .}. A string is a mapping from an initial segment of o into (0, l}. We identify a set A c w with its characteristic function. We now consider a set generic over the arithmetic sets. A set A E w is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every ,X:-set of strings S, there is a o cA such that o E S or (Vv Z= o)(v F# S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(=%) denote the set of degrees which are recursive in a. Before Cohen’s work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0’, i.e., the degree of a complete r.e. set, there is a degree a such that a’ = a U 0' = b. He actually proved this result by using the notion of forcing for 27 statements. In the construction of a real which satisfies some recursion-theoretic property, the notion of forcing makes the situation clear and it has become quite popular, see Lerman [8]. There is another important method in recursion theory, namely the priority method. Friedberg and Muchnik first independently invented the priority method to prove the existence of incomparable recursively enumerable degrees. The finite injury argument used there was improved by the infinite argument by Sacks, see [9]. Further, the 0”‘-priority argument was introduced by

Rational matrix structure

We bring together in this paper, in a unified way, certain results developed by various people regarding the pole/zero structure of a rational matrix and the structure of the vector-space generated by its columns. Pole and zero structure, at finite and infinite arguments, is compactly described by using elementary ideas from the language of valuation theory. The concept of column-reducedness of a rational matrix at some argument is introduced, and shown to determine when its pole/zero structure is simply that of its columns taken separately. We describe a procedure that operates on the Laurent expansion of a given rational matrix at the argument of interest in order to transform the matrix to one that is column-reduced at this argument but has the same pole/zero structure. The occurrence of such a "structure-extraction" algorithm in various contexts in system theory is pointed out. Special properties of rational bases (for a rational vector space) that are column-reduced at all arguments are noted, somewhat extending what is already well-known for minimal polynomial bases for such a space.