Showing papers on "Undecidable problem published in 1971"

Negative Solution of the Decision Problem for Sentences True in Every Subalgebra of

Abstract: It was shown by Taiclin [6], and independently announced by Tarski [7], that the elementary theory of commutative cancellation semigroups is hereditarily undecidable. In his proof Tarski exhibited a subsemigroup of 〈 N , ·〉, the natural numbers with multiplication, whose theory is both hereditarily and essentially undecidable. (The details of his construction were published by V. H. Dyson [1].) In connection with these results, Tarski suggested to the author that it would be of interest to solve the decision problem for the theory K which consists of all elementary sentences which are true in every subalgebra (i.e. every subsemigroup) of 〈 N , +〉. The object of this note is to prove that the theory K is hereditarily undecidable.

The decision problem for mathematical structures of quantum theory

Abstract: The purpose of this letter is to enunciate some results concerning the ~!ecision problem relative to a well-known mathematical structure of quantum theory. Such a structure, originally proposed by BIRKHOFF and yon NEU~ANN (1) and later developed by PI toN (3) and EMCH (3) together with JAVCH (4,5), is essentially the structure of the <( propositions ~ (*) on a given quantum system and turns out to be the same of a complete, atomic, orthocomplemented, weakly modular, satisfying-the-covering-law lattice ((( system of propositions ~, in the nomenclature introduced by PIRON). Let us denote with TQ~ the theory of the above systems of propositions ; we will prove that TQM is undecidable, in a sense that will be specified below. To begin with, it is useful to recall some definitions which are of common use in mathematical logic; they arc essentially due to TARSKI (6). The theories discussed in the following are all formalized within the lower predicate calculus, and they have the same logical constants (i.e. quantifiers, connectives, symbol of equivalence) and rules of inference. Let us call T anyone of these theories; we assume further tha t T is specified by its nonlogical constants (i.e. symbols of operations, and so on) and nonlogical axioms (i.e. relations). We suppose known to the reader the definitions of a formula, a sentence (i.e. a formula without free variables), a provable sentence (in terms of nonlogical constants and nonlogical axioms), together with the definitions of consistency, completeness, and finite axiomatizabili ty of a theory T. Moreover, we suppose also known the concept of a general recursive set.

The Generalized Continuum Hypothesis is Equivalent to the Generalized Maximization Principle

Abstract: In spite of the work of Godel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis (GCH) from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth (or falsity). Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search for “other (hitherto unknown) axioms of set theory which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts” (see Godel [7, p. 265]).

The Immortality Problem for Non-Erasing Turing Machines

Abstract: Publisher Summary In this chapter M is considered a Turing machine (TM). An instantaneous description (ID) of M is a triple 〈q,X,n〉 where q∈K, X∈ ∑∞ and n≥1. describes that M is in state q with the read-write head scanning square no. n and that the tape T contains X. M is to stop if M tries to go off the tape at the left end. M is called a non-writing TM if it contains no write-instructions. The immortality problem (IP) associated with a set of TM-s is the problem of deciding, for a given TM in the set, whether or not there exists an immortal ID. It is shown that IP for non-erasing TM-s is decidable, if the tape is allowed to contain ultimately periodic words. If, however, the tape is restricted to contain only a finite number of non-blanks, then the IP for the set of nonerasing TM-s is recursively undecidable (of degree 0").