Topic
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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26 Jun 1994
TL;DR: This work shows that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary.
Abstract: Second-order unification is undecidable in general. Miller showed that unification of so-called higher-order patterns is decidable and unitary. We show that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two second-order terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.
27 citations
TL;DR: For every Turing machine, there is an automaton group that simulates it as discussed by the authors, which has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-iterated commutator is the identity or not.
27 citations
TL;DR: In this article, a tight connection between infinite paths in recursive trees and Hamiltonian path in recursive graphs is established, and a corollary is that determining Hamiltonicity is highly undecidable.
Abstract: A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ
1
1
-complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case.
27 citations
01 Jan 2005
TL;DR: A graphical, user-friendly and very simple language as inter-agent specification language: Live Sequence Charts (LSC) is presented and its properties are investigated: it is highly succinct, but inexpressive.
Abstract: The problem of relating inter-agent and intra-agent behavioral specifications is investigated. These two views are complimentary, in that the former is closer to scenario-based user requirements whereas the latter is design-oriented. We use a graphical, user-friendly and very simple language as inter-agent specification language: Live Sequence Charts (LSC). LSC is presented and its properties are investigated: it is highly succinct, but inexpressive. There are essentially two ways to relate inter-agent and intra-agent specifications: (i) by checking that an intra-agent specification is correct with respect to some LSC specification and (ii) by automatically building an intra-agent specification from an LSC specification. Several variants of these problems exist: closed/open systems and centralized/distributed systems. We give inefficient but optimal algorithms solving all problems, besides synthesis of open distributed systems, which we show is undecidable. All the problems considered are difficult, even for a very restricted subset of LSCs, without alternatives, interleaving, conditions nor loops. We investigate the cost of extending the language with control flow constructs, conditions, real-time and symbolic instances. An implementation of the algorithms is proposed. The applicability of the language is illustrated on a real-world case study.
27 citations
TL;DR: It is proved that there exists no algorithm to decide whether the language generated by a context-free grammar is dense with respect to the lexicographic ordering, and it is shown that it is undecidable whether the order is dense.
26 citations