122277498
Example of an independent system of words which cannot be included in a finite complete system
122087792
Normal algorithms connected with the computation of Boolean functions
The article deals with lower and upper bounds for the complexity of normal algorithms related to the computation of Boolean functions.
118293516
An Approach to Constructive Mathematical Logic
Publisher Summary In constructive mathematical logic, the chapter explains the logical connectives applied to propositions in terms of the logical connectives applied to actions. This presents no difficulties for conjunction, disjunction, and existential quantifiers, because the applications of these connectives to actions are almost immediately clear. Difficulties of a different kind arise in connection with the generality quantifier. The natural explanation of this quantifier includes the requirement of some unique general method of proving every instance of the general proposition in question. This leads to the acceptation of some, possibly transfinite, hierarchy of Carnap's inference rules. If at the same time implication is introduced as deducibility, then one shall possibly have a transfinite hierarchy of implications.
31097901
On the Inversion Complexity of a System of Functions
1. Let n be a natural number different from zero. We will call the alphabet {0, 1, xl , . . , xn, &, V, l, ( , )} an alphabet of a formula in n variables. We will denote this alphabet by ~ . We will call the letters Xl, • • • , x. of the alphabet • ~ variables and the letters 0, 1 constants. We will denote by formulas in n variables words in the alphabet ,I)~ which are defined by the following generating rules. F1. Single-lettered words
17070359
Insolubility of the Problem of Homeomorphy ∗
1. We consider, from the general problem of homeomorphy, the problem of finding an algorithm that determines whether two given polyhedra are homeomorphic. In this case, polyhedra are combinatorially given through their triangulation and we must understand the term “algorithm” in the precise sense what the it offers i.e., e.g., as a “classifying algorithm”. In addition to the general problem of the Homeomorphy, there are, of course, different subproblems which themselves refer to polyhedra or the those resulting classes. One may, for example, set up the problem of homeomorphy for polyhedra of degree no higher than n, a fixed natural number. One may, in exactly the same way, set up the problem of homeomorphy for the n-manifolds, if one could clearly decide what a “manifold” is. Another natural restriction that can be made to the polyhedra to be matched is fixing one of them. In this case, the problem of the homeomorphy of a given polyhedron A consists of finding an algorithm which, for any polyhedron, determines whether it is homeomorphic to the polyhedron A. One of these problems has been solved for a long time, i.e. the problem of homeomorphy for 2-manifolds or the problem of the homeomorphy of a given 2-manifold. However, we have found the following results:
115545888
On a Semantical Language Hierarchy in a Constructive Mathematical Logic
The subject of this paper is a constructive mathematical logic, i.e. a logic suitable for constructive mathematics. Constructive mathematics differs radically from classical mathematics in the understanding of existence and disjunction. We say in constructive mathematics that there exists an object satisfying some condition iff we possess a construction of such an object. We say that the disjunction ‘A or B’ takes place (A and B are propositions) iff we possess a construction of a true proposition which coincides either with A or with B. Whatever might be the classical understanding of existence and disjunction, it differs from the constructive one inasmuch as such constructions are not needed according to the classicist’s point of view.
122621540
Numerical modeling of three-dimensional viscous flows using a space-marching method with pressure iterations
A method of calculating steady three-dimensional compressible and incompressible flows on the basis of simplified Navier-Stokes equations is examined. Some calculation results are presented. These relate to the problem of supersonic flow past blunt bodies and to incompressible channel flows.