Showing papers on "Undecidable problem published in 1977"

Undecidable questions about the maximum invariant set

Abstract: This paper shows that the class of all unilateral cellular spaces has the computation universality of the second kind which is quite different from that of the ordinary kind. Applying that in various forms, it shows many undecidable questions about the maximum invariant set of a csm system. The main result is the following: Given an arbitrary csm, it is recursively unsolvable to determine whether the maximum invariant set is regular or not.

Talking about free groups in naturally enriched languages

Abstract: The elementary theory of a non-commutative free group with a predicate for equality of length is proved hereditarily undecidable. This theory is shown to be ‘finitely axiomatizable’ modulo the second order stipulation that some centralizer be free cyclic. The theory of free groups in a weaker language using only a predecessor concept is investigated in the hope of shedding some light on Tarski's problems.

Decidability (undecidability) of equivalence of Minsky machines with components consisting of at most seven (eight) instructions

Abstract: S. Ginsburg and H. Spanier ~I] investigated semilinear sets, i. e. the sets which are first order definable in the monoid (N, +) of nonnegative integers. The relation of semilinear /partial/ functions defined below to the semilinear sets is analogous to the relation of /partial/ recursive functions to th~recursive sets. Semilinear /partial/ functions will be characterized by the so called semilinear Minsky machines. We sAall find a rather large recursive class of these machines and its very simple subclasses sufficient for computing of all semilinear /partial/ functions. ~insky machines are sets of instructions but we shall also consider them as oriented graphs; roughly speaking, an arc from A to B means that the instruction B can innnediately follow the instruction A in a computation. Hence machimes can be partitioned into (strongly connected) components. We shall show that all machines with components consisting of at most seven instructions are semilinear; then we shall prove their equivalence is decidable. On the other hand, ~ii~sky machines with components consisting of at most eight insZructions compute all polynomials with integer coefficients. We shall reduce the undecidable problem of solvability of Diophantine equations to the equivalence problem of these machines. In this way we shall show that the last problem is undecidable. It is interesting that the halting problem for these machines can be shown to be decidable.

Verification Decidability of

Abstract: A program annotated with inductive assertions is said to be verification decidable if ail of the verification conditions generated from the program and assertions are formulas in a decidable theory. We define a theory, which we call Presburger array theory, containing two logical sorts: integer and array-of-integer . Addition, subtraction, and comparisons are permitted for integers. We allow array contents and assign functions, and, since the elements of the arrays are integers, array accesses may be nested. The first result is that the validity of unquantified formulas in Presburger array theory is decidable, yet quantified formulas in general are undecidable. We also show that, with certain restrictions, we can add a new predicate Perm(M,N) ~ meaning array M is a permutation of array N — to the assertion language and still have a solvable decision problem for verification conditions generated from unquantified assertions. The significance of this result is that almost all known sorting programs, when annotated with inductive assertions for proving that the output is a permutation of the input, are verification decidable.