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Undecidable problem
About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.
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TL;DR: This paper shows that the system of Kreisel described in [2] is creative, which depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.
Abstract: In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Godel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable. In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A . (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest. Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.
28 citations
12 Sep 2011
TL;DR: This work has identified several new minimal undecidable logics amongst the fragments of Halpern-Shoham logic HS, including the logic of the overlaps relation, over the classes of all and finite linear orders, as well as the Logic of the meet and subinterval relations,over the class of dense linear orders.
Abstract: Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undesirability. Identifying minimal undecidable interval logics is thus a natural and important issue in the research agenda in the area. The decidability status of a logic often depends on the class of models (in our case, the class of interval structures)in which it is interpreted. In this paper, we have identified several new minimal undecidable logics amongst the fragments of Halpern-Shoham logic HS, including the logic of the overlaps relation, over the classes of all and finite linear orders, as well as the logic of the meet and subinterval relations, over the class of dense linear orders. Together with previous undecid ability results, this work contributes to delineate the border of the dark side of interval temporal logics quite sharply.
28 citations
01 Jan 2002
TL;DR: It is shown that in the case where all constructors are either unary or nullary, the first-order theory is decidable for both structural and non-structural subtyping, and the decidability results are shown by reduction to a decision problem on tree automata.
Abstract: We investigate the first-order of subtyping constraints. We show that the first-order theory of non-structural subtyping is undecidable, and we show that in the case where all constructors are either unary or nullary, the first-order theory is decidable for both structural and non-structural subtyping. The decidability results are shown by reduction to a decision problem on tree automata. This work is a step towards resolving long-standing open problems of the decidability of entailment for non-structural subtyping.
28 citations
TL;DR: It is proved that the origin constrained domino problem is undecidable for any group of the form G × Z subject to a technical condition on G and a simulation theorem which is valid in any finitely generated group is presented.
28 citations
27 Jul 1996
TL;DR: It is proved decidability results for fragments of monadic simultaneous rigid E-unification are found and the connections between this notion and some algorithmic problems of logic and computer science are shown.
Abstract: The notion of simultaneous rigid E-unification was introduced in 1987 in the area of automated theorem proving with equality in sequent-based methods, for example the connection method or the tableau method. Recently, simultaneous rigid E-unification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid E-unification and show the connections between this notion and some algorithmic problems of logic and computer science.
28 citations