Showing papers on "Undecidable problem published in 1983"

Recurring Dominoes: Making the Highly Undecidable Highly Understandable (Preliminary Report)

Abstract: In recent years many diverse logical systems for reasoning about programs have been shown to posses a highly undecidable, viz Π 1 1 -complete, validity problem. All such known results are reproved in this paper in a uniform and transparent manner by reductions from recurring domino problems. These are simple variants of the classical unbounded domino (or tiling) problems introduced by Wang and the bounded versions defined by Lewis. While the former are (weakly) undecidable and the latter complete in various complexity classes, the problems in the new class are Σ 1 1 -complete.

Variants of Robinson's essentially undecidable theoryR

Abstract: Cobham has observed that Raphael Robinson's well known essentially undecidable theoryR remains essentially undecidable if the fifth axiom scheme\(\left( {x \leqq \bar n \vee \bar n \leqq x} \right)\) is omitted. We note that whether the resulting system is in a sense “minimal essentially undecidable” depends on what the basic constants are taken to be. We give an essentially undecidable theory based on three axiom schemes involving only multiplication and less than or equals.

On some variants of Post's Correspondence Problem

Abstract: A variant of Post's Correspondence Problem is considered where two different index words are allowed provided that one of them can be obtained from the other by permuting a fixed number of subwords. It is shown that this variant is undecidable. Post's Correspondence Problem is also extended to circular words, doubly infinite words and doubly infinite powers of words, and shown to be undecidable in all these extensions.

Independent instances for some undecidable problems

Abstract: — In this note we prove the existence of effectively independent instances (with respect to an arbitrary recursively axiomatizable, consistent, intuitively true and sufficiently rich theory) for some well-known undecidable problems including the Emptiness Problem, the Finiteness Problem, the Totality Problem, the Halting Problem and the Post Correspondence Problem. Applications in the Theory of Diophantine Equations and in the Formai Language Theory will be analyzed. Résumé. — Dans cette note nous prouvons Vexistence, pour quelques problèmes indècidables bien connus, d'instances indépendants (relativement à une théorie récursivement axiomatisable, consistante, intuitivement vraie et suffisamment riche). Les problèmes traités comprennent le problème de la vacuité, de lafinitude, de la totalité, de l'arrêt et le problème de correspondance de Post. On analyse les applications à la théorie des équations diophantiennes et des langages formels.

Remarks on the development of computability

Abstract: The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. It is suggested in Section 3 that a central item was the problem of generalizing Godel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of...

Reducibilities among decision problems for HNN groups, vector addition systems and subsystems of Peano arithmetic

Abstract: Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show that although these problems are not primitive recursively decidable, they do admit decision procedures which are primitive recursive in the Ackermann function. By the class of vector groups VA we understand the HNN groups G( P 1, ql, . p4, qk\.) given by (1) (al...,akl b; a'bPI'a= b '...,ak = b where the exponent pairs p,, q, occurring in (I) are positive and relatively prime. (For concepts and results of a group-theoretic nature not explicitly discussed here the reader should consult Lyndon and Schupp [5].) Let G be a vector group, m a positive conjugate power of I in G when b' xb'x -, where x in G is given by a positive word in the generators al,. . . ,ak, b of G (i.e. one which involves no negative exponents). The set of positive conjugate powers of I in G, or positive conjugate power set is denoted PCP(l, G). By the equality problem for positive conjugate power sets, we mean the question of determining for any integers 1, 12 and vector groups G1, G2, whether PCP(I1, G1) = PCP(12, G2). The special equality problem is to decide the equality problem in those cases where l1 = 12 and G2 arises from G1 by removing a particular generating symbol a, and its corresponding defining relation a'bP a = bq, from the presentation of G1 as in (I). The finite special equality problem is to decide the special equality problem in those cases where PCP(11, G) is finite. We identify a decision problem with the set of Godel numbers of its instances and use this identification to discuss the complexity of the problem. A function g is primitive recursive in a function f iff g is in the class obtained by primitive recursion and composition from f together with the usual initial functions. It is shown in Anshel [1] that the special equality problem for vector groups is undecidable. In contrast, we will prove THEOREM 1. The finite special equality problem for vector groups is (i) decidable but not primitive recursive, (ii) primitive recursive in the Ackermann function. Received by the editors June 27, 1981. 1980 Mathematics Subject Classification. Primary 20F10, 03D40, 03F30; Secondary 68C99. 01983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page

Decision problem for separated distributive lattices

Abstract: It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable. Introduction. A distributive lattice with 0 is separated if it satisfies the following separation property: for every x1, x2 there are Yi < x1 and Y2 < x2 such that Yil Y2 are disjoint (i.e. Yi A Y2 = 0) and Yi V Y2 x1 V x2. Alistair Lachlan introduced separated distributive lattices in [La] in connection with his study of the first-order theory of the lattice of recursively enumerable sets. He mentioned to me a question whether the first-order theory of separated distributive lattices is decidable. The answer is negative: in ?2 a known undecidable theory is interpreted in the firstorder theory of separated distributive lattices. The known undecidable theory is the first-order theory of the following structures: a Boolean algebra with a distinguished subalgebra. About undecidability of it see [Ru]. Actually the first version of the undecidability proof used the closure algebra CACD of Cantor Discontinuum, i.e. the Boolean algebra of subsets of Cantor Discontinuum with the closure operation. CACD is easily interpretable in the separated distributive lattice of functions f from Cantor Discontinuum into {0, 1, 2} such thatf1(2) is clopen. By [GS1] a finitely axiomatizable essentially undecidable arithmetic reduces to the first-order theory of CACD, hence to the first-order theory of the mentioned separated distributive lattice of functions, hence to the first-order theory of separated distributive lattices. The last step is somewhat complicated by the fact that [GS1] does not interpret the standard model N of arithmetic in CACD. (Even though [GS2] reduces the second-order theory of N to the first-order theory of CACD, [GS3] proves that N cannot be interpreted in CACD.) However the Boolean algebra of subsets of Cantor Discontinuum with a distinguished subalgebra of clopen (closed and open) sets is easily interpretable in CACD. This way I came to use Rubin's result which made the undecidability proof very simple. From the other side the cited result of [GS1] can be used to reprove Rubin's theorem and Received October 12 1980; revised August 30, 1981. 'The results were obtained and the paper was written during the Logic Year in the Institute for Advanced Studies of Hebrew University. ? 1983, Association for Symbolic Logic 0022-4812/83/4801-0020/$01.40

Behavioural Categoricity of Abstract Data Type Specifications

Abstract: In this note we want to present the concept of behavioural categoricity of an abstract data type specification. Intuitively, a specification is behaviourally categoric if it captures the external views the user can have on the data type. More specifically, using this specification, it is possible to prove that two objects are equal if and only if they behave the same, or informally speaking, if and only if they implement the same black box. Providing a general algorithm for proving the behavioural categoricity of any specification is impossible because that algorithm could also decide whether a finite presentation of a group presents the trivial group or not, which Rabin proved to be undecidable. We show by an example of a specification of circular lists that the proof of the categoricity must be done carefully. In this note we want to present the concept of behavioural categoricity of an abstract type specification. Intuitively, a specification is behaviourally categoric if it captures all the external views the user can have on the data type. More specifically, using this specification, it is possible to prove that two objects are equal if and only if they behave the same, or informally speaking, if and only if they implement the same black box. Providing a general algorithm for proving behavioural categoricity of any specification is impossible, because that algorithm could also decide whether a finite presentation of a group presents the trivial group or not, which Rabin proved to be undecidable. We show by an example of a specification of circular lists, that the proof of the categoricity must be done carefully.

The tau-Theory for Free Groups Is Undecidable.

Abstract: In this paper we give short proofs to the undecidability of the τ -theory for free groups and other relevant theories.

Descriptional complexity measures of context-free languages

Abstract: The properties of several new descriptional complexity measures of context-free languages are discussed. Though these measures seem to be very simple the basic algorithmic problems remain to be undecidable.

An algorithmically unsolvable problem in analysis

Abstract: The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all 4-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group. Algorithmically unsolvable problems abound in logic, algebra, combinatorial topology, and diophantine equations (for a review and references see, for instance, [1]). In this note we point out an algorithmically unsolvable decision problem whose natural context is analysis on Riemannian manifolds. Let M be a compact differentiable manifold, 7T1(M) its fundamental group, M its universal cover. If g is a Riemannian metric on M (and so also on M), let Xo(M, g) denote the infimum of the spectrum of the Laplace-Beltrami operator -/ acting in L2(M, g). Whether XO(M, g) = 0 or XO(M, g) > 0 holds is independent of g and is, therefore, a diffeomorphism invariant of M. The following question then arises: Is there an algorithm for deciding which of the two cases holds? To make this question meaningful one considers a recursively enumerable class C of compact differentiable manifolds, i.e. one that can be put into one-to-one correspondence with a recursively enumerable set of positive integers which serve as code names for the manifolds (for this notion see [2, ?1.1]). An algorithm is then understood to be a Turing machine which, when presented with the code number of any M in C, computes in a finite number of steps which of the above alternatives holds. THEOREM. If C is such that for every finitely presented group G there is an M in C with g1(M) = G, then there is no algorithm for the above decision problem. The proof depends on a recent theorem of Brooks [3] which establishes a connection between the condition XA(M) = 0 and g1(M), namely, the condition holds if and only if gl1(M) is an amenable group. Thus, to prove our theorem, it is only necessary to show that amenability is an algorithmically undecidable property in the class of finitely presented groups. Amenability may be defined in various ways (see [4]), but from our point of view the essential fact is that amenability is a so-called Markov property. A property P of finitely presented groups is called a Markov property whenever (A) there is at least one finitely presented group G1 with the Received by the editors June 9, 1982. 1980 Mathematics Subject Classification. Primary 03D35; Secondary 31 C 12, 58C40.

The Meaning of the Conjecture P NP for Mathematical Logic

Abstract: I think that the conjecture P + NP is not as widely taught in courses of mathematical logic as it should be, in view of its capital importance for the foundation of mathematics. Therefore I am writing this note in the hope that all logicians will always include it in the introductory courses of their subject although it does not appear yet in the appropriate books. The original paper of S. Cook [1], where the conjecture was formulated, was indeed written from the point of view of logic but it became the domain of computer scientists (see [2]), particularly because of a paper of Karp [3] where the combinatorial or computational aspects of the conjecture were developed in a very suggestive way. Let us assume that the teacher has already presented the concept of a first order theory and the concept of a Turing machine (neither the concepts of a decidable theory nor that of a recursive function are needed). Then he may proceed as follows: By a normal theory we shall mean a theory which is formalized with a finite alphabet in first order logic with equality and is axiomatizable by a finite set of axioms and axiom schemata in which one can prove 3xy [ x + y ]. (In [4] it is proved that every theory which is recursively axiomatizable and contains a minimal amount of arithmetic or set theory is normal.) By a proof in a normal theory we mean a Hilbert style proof from the axioms. Let E be a finite alphabet and E* the set of all words, i.e., finite sequences of elements of E. For any E*, I denotes the length of (. Now we introduce a more abstract concept of a theory which we will call a T7T-theory. A T7T-theory is a set of pairs T c E* x E* such that there exists a polynomial P(x, y) and a Turing machine M such that, for any (T, 1) E E* x E*, M can decide in time < P(ITI, I7TI) if (T, 7T) E T. If (i, 7T) E T, thenT is called a theorem of T and 7 is called a proof of i-in T. Every normal theory defines a Tg-theory since the time necessary to check the correctness of a Hilbert style proof in a normal theory can be estimated from above by a polynomial in the length of that proof. Now, a T7n-theory T will be called amenable (to automatization) iff there exists another polynomial Po(x, y) and another Turing machine Mo such that, given any word XE * and any positive integer n, MO can decide in time < Po(ITi-, n) if there exists a X E =* with Ig I < n and such that (i, s) E T. (Notice that if we replaced the condition < Po(iTI, n) by the condition < Po(ITI, Cn) where c = card E, then the concept would trivialize since every T7T-theory would be amenable. In fact, given a time Po(lTI, cn), the machine can form all sequences of symbols of length n and find out if any of them is a proof of i.) It is clear that, after Godel's 1931 discovery that all sufficiently strong theories are undecidable,