Die Problematik apriorischer Wahrscheinlichkeiten im System der induktiven Logik von Rudolf Carnap.
Die Unentscheidbarkeit der einstelligen unendlichwertigen Prädikatenlogik.
Die Vollständigkeit einer unverzweigten Variante des "analytischen" Entscheidungsverfahrens der klassischen Logik.
Difference of Function on Vector Space over F
In [11], the definitions of forward difference, backward difference, and central difference as difference operations for functions on R were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems [3], [10]. Differential cryptanalysis [4] that undertakes a general purpose attack against...
Differentiability of Polynomials over Reals
In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].
Differential Equations on Functions from R into Real Banach Space
In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real...
Differentiation in Normed Spaces
In this article we formalized the Fréchet differentiation. It is defined as a generalization of the differentiation of a real-valued function of a single real variable to more general functions whose domain and range are subsets of normed spaces [14].
Dijkstra's Interpretation of the Approach to Solving a Problem of Program Correctness
Diophantine geometry over groups I : Makanin-Razborov diagrams
This paper is the first in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the first paper we present the (canonical) Makanin-Razborov diagram that encodes the set of solutions of a system of equations. We continue by studying parametric families of sets of solutions, and associate with such a family a canonical graded Makanin-Razborov diagram, that encodes the collection...
Diophantine undecidability for addition and divisibility in polynomial rings
We prove that the positive-existential theory of addition and divisibility in a ring of polynomials in two variables A[t₁,t₂] over an integral domain A is undecidable and that the universal-existential theory of A[t₁] is undecidable.
Direct product decomposition of -algebras
Discourse on one way in which a quantum-mechanics language on the classical logical base can be built up
Discriminator varieties of Boolean algebras with residuated operators
The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral r-algebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated...
Discussion of the structure of uninorms
The paper deals with binary operations in the unit interval. We investigate connections between families of triangular norms, triangular conorms, uninorms and some decreasing functions. It is well known, that every uninorm is build by using some triangular norm and some triangular conorm. If we assume, that uninorm fulfils additional assumptions, then this triangular norm and this triangular conorm have to be ordinal sums. The intervals in ordinal sum are depending on the set of values of a decreasing...
Disjunction in modal description logics.
Distributive groupoids are symmetric-by-medial: An elementary proof
We present an elementary proof (purely in equational logic) that distributive groupoids are symmetric-by-medial.
Distributivity of ordinal sum implications over overlap and grouping functions
In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function...
Distributivity of strong implications over conjunctive and disjunctive uninorms
This paper deals with implications defined from disjunctive uninorms by the expression where is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a -norm or a -conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications...
Divergence measure between fuzzy sets using cardinality
In this paper we extend the concept of measuring difference between two fuzzy subsets defined on a finite universe. The first main section is devoted to the local divergence measures. We propose a divergence measure based on the scalar cardinalities of fuzzy sets with respect to the basic axioms. In the next step we introduce the divergence based on the generating function and the appropriate distances. The other approach to the divergence measure is motivated by class of the rational similarity...
Divisible ℤ-modules
In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].