Showing papers on "Undecidable problem published in 1976"

The decision problem for standard classes

Abstract: The standard classes of a first-order theory T are certain classes of prenex T -sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T , (1) the decision problem for any class of prenex T -sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K 1 , …, K n such that any undecidable standard class contains one of K 1 , …, K n . These results give direction to the study of the decision problem. Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

Undecidable properties of finite sets of equations

Abstract: Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical. Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive . Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

Recursively Enumerable Sets

Abstract: In this chapter we shall deal in some detail with the set Σ1 of relations (see 5.24). Such relations are called recursively enumerable for reasons which will shortly become clear. The study of recursively enumerable relations is one of the main branches of recursive function theory. They play a large role in logic. In fact, for most theories the set of Godel numbers of theorems is recursively enumerable. Thus many of the concepts introduced in this section will have applications in our discussion of decidable and undecidable theories in Part III. Unless otherwise stated, the functions in this chapter are unary.

The Complexity of Finite Memory Programs with Recursion

Abstract: In an earlier paper (JACM, 1976) we studied the computational complexity of a number of questions of both programming and theoretical interest (e.g. halting, looping, equivalence) concerning the behaviour of programs written in an extremely simple programming language. These finite memory programs or fmps model the behaviour of FORTRAN-like programs with a finite memory whose size can be determined by examination of the program itself. The present paper is a continuation in which we extend the analysis to include ALGOL-like programs (called fmp^(rec) s) with the finite memory augmented by an implicit pushdown stack used to support recursion. Our major results are the following. First, we show that at least deterministic exponential time is required to determine whether a program in the basic fmpr~C model accepts a nonempty set. Then we show that a model with a limited version of call-by-name requires exponential space to determine acceptance of a nonempty set, and that a more sophisticated model with rewritable conditional formal parametershas an undecidable halting problem. The same lower bounds apply to the equivalence problem, which in contrast to the situation for the basic fmp model is not known to be decidable (since it is not known whether equivalence of deterministic pushdown automata is decidable).

On the construction of simple first-order formulae without recursive models

Abstract: The paper is about solutions of the well known problems from Hilbert-Bernays whether every satisfiable closed formula of the restricted lower predicate calculus admits recursive models. Aanderaa's method is presented which associates formulae to register machine programs in such a way that the formula is the axiom of an essentially undecidable theory resp. satisfiable but without recursive models if the machine program enumerates two recursively inseparable sets. A simplification of Aanderaa's formulae brings their decision problems still closer to the corresponding stop problems of the machine programs and results in a more direct and technically less involved realization of the basic idea.[...]

On a certain basis of operators and predicates with an undecidable emptiness problem

Abstract: We shall consider algorithm schemata which are constructed from operators and predicates. Operators and predicates are assumed to be formal constructions which allow their interpretat ion as commands of an electronic computer. The class of schemata which we shall study are completely specified by the choice of a certain basis a set of operators and predicates from which the schemata are constructed. Problems of emptiness, inclusion and equivalence oriented on such a class of schemata are formulated as problems in the chosen basis.

Some Undecidable Theories

Abstract: Corresponding to our list in Chapter 13 of decidable theories we begin this section with a list of undecidable theories. As we have previously indicated, most undecidable theories satisfy one of the two stronger properties of inseparability or finite inseparability, and we shall indicate these properties in the table below.

General Theory of Undecidability

Abstract: In previous chapters we have introduced several concepts related to the notion of undecidable theories (complete theories, 11.9; theories, 11.29; decidable and undecidable theories, 13.1; syntactical and weak syntactical definability, 14.1; recursive axiomatizability, 14.4; spectral representability, 14.22). Our purpose in this chapter is to establish various relationships known to exist between these notions and related ones. These general theorems will be applied in the next chapter, in which numerous examples of undecidable theories are given. We proceed in this chapter from the simpler concepts to the more complicated ones.

Implicit Definability in Number Theories

Abstract: In this chapter we consider several ways in which number-theoretic functions and relations can be implicitly defined in number theories. We do not mean elementarily definable as in Chapter 11; the present notions of definability are expressed in terms of theories and not of structures. As we shall see, the notions lead to new equivalents of the notion of recursiveness; see 14.12, 14.20, and 14.26. They also form the basis for diagonalization procedures which produce many undecidable theories (see the next chapter). We shall be concerned with two types of implicit definability. The first, syntactic definability, follows; the second, spectral representability, is given in 14.22.

The Mind-Body Problem

Abstract: It often happens in the history of philosophy that problems which cannot be solved are dropped. This may come about when due to the advent of access to additional information a problem turns out to be not a genuine problem after all but only a pseudo-problem, or when interest shifts from a problem which proves to be either undecidable or of less importance than had been thought. In any case it means for some philosophical assumptions that they lead to trial runs which need not be made.