Showing papers on "Undecidable problem published in 1975"

The monadic theory of order

Abstract: We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are modeltheoretic, and we do not use automaton theory.

A note on undecidable extensions of monadic second order successor arithmetic.

Abstract: Let SC (sequential calculus) be the interpreted system of monadic second order arithmetic with 0, ' as nonlogical constants for zero and successor function and with first order and monadic second order quantifiers. Btichi [1] proved that the theory given by SC is decidable. In this note we show by a generalization of a method presented in Elgot-Rabin [2] that the theory resulting from SC by adjoining a unary function with an infinite inverse image for infinitely many numbers is undecidable (Theorem 1). This serves to prove a conjecture of Siefkes [4] concerning the undecidability of an extended theory SC +h , where h is a unary function of a certain growth (Theorem 2). We adopt the notation of Siefkes [4].

Even simple programs are hard to analyze

Abstract: It has long been known that most questions of interest about the behavior of programs are recursively undecidable. These questions include whether a program will halt, whether two programs are equivalent, whether one is an optimized form of another, and so on. On the other hand, it is possible to make some or all of these questions decidable by suitably restricting the computational ability of the programming language under consideration. The Loop language of Meyer and Ritchie [MR], for example, has a decidable halting problem, but undecidable equivalence. Restricting the computational ability still further, virtually all of these questions are decidable for finite automata and generalized sequential machines (except that Griffiths [Gri] has shown equivalence undecidable for nondeterministic gsms).A natural question to ask is how hard it is to solve these problems for programming languages for which they are decidable, and it is with this area that we are concerned in this paper. In particular we describe a programming language modeled on current higher-level languages which has exactly the computational power of deterministic finite state transducers with final states, and analyze the space and time required to decide various questions of programming interest about the language. We find that questions about halting, equivalence, and optimization are already intractable for this very simple language. We also study extensions to the language such as simple arithmetic capabilities, arrays, and recursive subroutines with both call-by-value and call-by-name parameter passing mechanisms, some of which extend the capabilities of the language and/or increase the complexity of its decidable problems. In one case, that of recursion with call-by-name, the previously decidable questions are seen to become undecidable.

A theorem on shortening the length of proof in formal systems of arithmetic

Abstract: This note is concerned with an aspect of the length of proof of formulas in recursively enumerable theories T adequate for recursive arithmetic. In particular, we consider the relative length of proof of formulas in the theories T and T ( S ), where F represents an r.e. set A in T and T ( S ) is the theory obtained from T by adjunction, as a new axiom, of a sentence S undecidable in T . Throughout the sequel T is a consistent, r.e. theory with standard formalization [7] in which all recursive functions of one variable are definable, and in which there is a binary formula x ≤ satisfying the well-known conditions [7]: Here is the constant term corresponding to the natural number n . W n is the n th r.e. set in a standard enumeration of the r.e. sets. Also, we assume an a priori Godel numbering of our formalism satisfying the usual conditions, so that all formulas are numbers ab initio. In the more common applications of the theorem below, if F is a k -ary formula of T , is a natural number that measures in some way the length of the shortest proof of in T .

The decision problem for recursively enumerable degrees

Abstract: If I have any message today for mathematicians in general, it is that consideration of difficult problems can be useful even when the problem is at present beyond solution. The problem I will discuss is unlikely to be solved in the near future, but I hope to show how the study of it leads to many more accessible problems. In order to state the problem, we need some definitions. To save words, we agree that number means natural number (nonnegative integer) and set means set of numbers. A set A is recursive if there is an algorithm for determining whether any given number is in A. A set A is recursive in a set B if there is an algorithm by which we can decide whether any given number x is in A, provided we are supplied with answers to all questions we choose to ask of the form Ts y in BT. As an example, let A ={2x : x e B}. Then B is recursive in A ; for x e B iff 2x € A. Also A is recursive in B ; for x e A iff x is even and \\x e B. (All this is independent of the choice of B.) Writing A^RB for A is recursive in B, we easily see that (1) A ^ R A ,

On models of protection in operating systems

Abstract: In [ii], a model of protection systems was introduced. Much of that paper is devoted to an explanation of the model. There are examples of the use of the model in capturing aspects of real systems. A formulation of the safety question is presented and it is shown that the question of whether a given general protection system is safe or not is undecidable. That paper has provoked a number of questions as to why certain features were or were not included in the model. The present paper attempts to answer these questions as well as giving some new results.