Showing papers on "Undecidable problem published in 1972"

Decision problems in computational models

Abstract: Some of the assertions about programs which we might be interested in proving are concerned with correctness, equivalence, accessibility of subroutines and guarantees of termination. We should like to develop techniques for determining such properties efficiently and intelligently wherever possible. Though theory tells us that for a realistic programming language almost any interesting property of the behaviour is effectively undecidable, this situation may not be intolerable in practice. An unsolvability result just gives us warning that we may not be able to solve all of the problems we are presented with, and that some of the ones we can solve will be very hard.In such circumstances it is very reasonable to try and determine necessary or sufficient conditions on programs for our techniques to be assured of success; however, in this paper we shall discuss a more qualitative, indirect, approach. We consider a range of more or less simplified computer models, chosen judiciously to exemplify some particular feature or features of computation. A demonstration of unsolvability in such a model reveals more accurately those sources which can contribute to unsolvability in a more complicated structure. On the other hand a decision procedure may illustrate a technique of practical use. It is our thesis that this kind of strategy of exploration can and will yield insight and practical advances in the theory of computation. Provided that the model retains some practical relevance, the dividends are the greater the nearer the decision problem lies to the frontier between solvability and unsolvability.

Sufficient Conditions for the Undecidability of Intuitionistic Theories with Applications

Abstract: ?0. Statement of results. Let A be a set of axioms of a theory T<(A) of classical predicate calculus (CPC); A may also be considered as a set of axioms of a theory TH(A) of Heyting's predicate calculus (HPC). Our aim is to investigate the decision problem of TH(A) in HPC for various known theories A of CPC. Theorem I(a) of ?1 states that if A is a finitely axiomatizable and undecidable theory of CPC then TH(A) is undecidable in HPC. Furthermore, the relations between theorems of HPC are more complicated and so two CPC-equivalent axiomatizations of the same theory may give rise to two different HPC theories, in fact, one decidable and the other not. Semantically, the Kripke models (for which HPC is complete) are partially ordered families of classical models. Thus a formula expresses a property of a family of classical models (i.e. of a Kripke model). A theory 0 expresses a set of such properties. It may happen that a class of Kripke models defined by a set of formulas2? is also definable in CPC (in a possibly richer language) by a CPC-theory 0'! This establishes a connection between the decision problem of 0 in HPC and that of i' in CPC. In particular if ' is undecidable, so is 0. Theorems II and III of ?1 give sufficient conditions on ? to be such that the corresponding 0' is undecidable. 0' is shown undecidable by interpreting the CPC theory of a reflexive and symmetric relation in 0'. (In the proof the passage to 0' is implicit and a direct syntactical translation into 0 is given.) As a corollary we obtain that the following theories among others are undecidable in HPC: (a) The monadic theory with one monadic letter. (b) The theory of unary functions with a decidable equality. (c) The theory of abelian groups with decidable equality. (d) The theory of a decidable linear order. (e) The theory of a symmetric and transitive relation. For the precise formulation of the theories and for the proofs see ?2. The undecidability of two monadic letters in both the intuitionistic and modal logic was first proved by Kripke [3], [4]. Maslov-Minc-Orevkov [1] gave a syntactical proof of the undecidability of one letter in intuitionistic logic. Slomson [2] and

Some undecidable problems in group theory

Abstract: In this paper we obtain general results for undecidable first order decision problems about groups (that is, problems about elements in a particular group, such as the word and conjugacy problems). We shall describe a class Q of such decision problems and a construction A such that if P is a problem in Q, then A(P) will be a finitely presented group in which P is recursively undecidable. This work then yields an analog of the Adjan-Rabin theorem for quotient-closed properties. In the past 20 years there has been a rash of undecidability results for finitely presented groups. Some of these are the conjugacy problem [12], the word problem ([4,] [13]), the isomorphism problem ([1], [14]), the center problem [2], and many others (see [2] for a collection of other examples). With the single exception of the Adjan-Rabin theorem, which shows the undecidability of the isomorphism problem, each of these results is obtained by providing a construction for the particular problem in question, and then concluding the desired unsolvability from peculiarities of the construction. The Adjan-Rabin theorem, on the other hand, gives a general construction which can be applied to any Markov property P of finitely presented groups, and from which one can conclude the impossibility of deciding which presentations present groups enjoying P. In this paper we obtain general results for undecidable first order decision problems (that is, problems about elements in a particular group, such as the word and conjugacy problems). We shall describe a class Q of such decision problems and a construction A such that if P is a problem in Q, then A(P) will be a finitely presented group in which P is recursively undecidable. The following list tabulates some problems in Q (the (?x)( ) notation Received by the editors November 4, 1971. AMS 1970 suibject classifications. Primary 02F47, 02H15, 20A10, 20E30, 20F10. 1 This research conducted at the University of Illinois, while the author held an NSF Traineeship. ( American Mathematical Society 1972

Subrecursive program schemata I & II(I. Undecidable equivalence problems, II. Decidable equivalence problems)

Abstract: The study of program schemata and the study of subrecursive programming languages are both concerned with limiting program structure in order to permit a more complete analysis of algorithms while retaining sufficiently rich computing power to allow interesting algorithms. In this paper we combine these approaches by defining classes of subrecursive program schemata and investigating their equivalence problems. Since the languages are all subrecursive, any scheme written in any one of them must halt (as long as we assume the basic functions and predicates are all total). Hence equivalence of schemes is the first question of interest we can ask about these languages. We consider schematic versions of various subrecursive programming languages similar to the Loop language. We distinguish between Pre-Loop and Post-Loop languages on the basis of whether the exit condition in an iteration loop is tested before iteration, as in Algol (Pre-), or after iteration, as in FORTRAN (Post-). We show that at the program level all these languages have the same computing power (the primitive recursive functions) and all have unsolvable equivalence problems (of arithmetic degree p01). But at the level of schemes, Pre-Loop has an unsolvable equivalence problem, while at least one formulation of Post-Loop has a solvable equivalence problem.

Decision problems in computational models.

Abstract: Some of the assertions about programs which we might be interested in proving are concerned with correctness, equivalence, accessibility of subroutines and guarantees of termination. We should like to develop techniques for determining such properties efficiently and intelligently wherever possible. Though theory tells us that for a realistic programming language almost any interesting property of the behaviour is effectively undecidable, this situation may not be intolerable in practice. An unsolvability result just gives us warning that we may not be able to solve all of the problems we are presented with, and that some of the ones we can solve will be very hard.In such circumstances it is very reasonable to try and determine necessary or sufficient conditions on programs for our techniques to be assured of success; however, in this paper we shall discuss a more qualitative, indirect, approach. We consider a range of more or less simplified computer models, chosen judiciously to exemplify some particular feature or features of computation. A demonstration of unsolvability in such a model reveals more accurately those sources which can contribute to unsolvability in a more complicated structure. On the other hand a decision procedure may illustrate a technique of practical use. It is our thesis that this kind of strategy of exploration can and will yield insight and practical advances in the theory of computation. Provided that the model retains some practical relevance, the dividends are the greater the nearer the decision problem lies to the frontier between solvability and unsolvability.