Topic
Undecidable problem
About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.
Papers published on a yearly basis
Papers
More filters
01 Apr 1991
TL;DR: It is proved that termination is undecidable for non-length-increasing string rewriting systems, using linear-bounded automata and the undecidability of confluence for terminating rewriting systems when terms begin by a fixed symbol is proved.
Abstract: We prove that termination is undecidable for non-length-increasing string rewriting systems, using linear-bounded automata. On the other hand, we prove the undecidability of confluence for terminating rewriting systems when terms begin by a fixed symbol. These two results illustrate that sometimes restriction of problem to recognizable domains modify decidability properties, sometimes it does not. (We only consider finite terms).
32 citations
01 Jan 2012
TL;DR: It is shown that the problem is (k+2)-EXPTIME complete for k-bounded combinatory logic with intersection types, for every fixed k (and hence non-elementary when k is a parameter), and the results give new insight into the expressive power of intersection types.
Abstract: In combinatory logic one usually assumes a fixed set of basic combinators (axiom schemes), usually K and S. In this setting the set of provable formulas (inhabited types) is PSPACE-complete in simple types and undecidable in intersection types. When arbitrary sets of axiom schemes are considered, the inhabitation problem is undecidable even in simple types (this is known as Linial-Post theorem).
k-bounded combinatory logic with intersection types arises from combinatory logic by imposing the bound k on the depth of types (formulae) which may be substituted for type variables in axiom schemes. We consider the inhabitation (provability) problem for k-bounded combinatory logic: Given an arbitrary set of typed combinators and a type tau, is there a combinatory term of type tau in k-bounded combinatory logic?
Our main result is that the problem is (k+2)-EXPTIME complete for k-bounded combinatory logic with intersection types, for every fixed k
(and hence non-elementary when k is a parameter). We also show that the problem is EXPTIME-complete for simple types, for all k.
Theoretically, our results give new insight into the expressive power of intersection types. From an application perspective, our results are useful as a foundation for composition synthesis based on combinatory logic.
32 citations
Journal Article•
TL;DR: In this paper, it was shown that the reachability question for some two-dimensional hybrid systems is undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as reachability problem for piecewise affine maps, that is a well known open problem.
Abstract: We revisited decidability of the reachability problem for low dimensional hybrid systems. Even though many attempts have been done to draw the boundary between decidable and undecidable hybrid systems there are still many open problems in between. In this paper we show that the reachability question for some two dimensional hybrid systems are undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as the reachability problem for Piecewise Affine Maps, that is a well known open problem.
31 citations
11 Dec 2013
TL;DR: This work shows that the problem is undecidable for architectures with information forks even for the fragment of LTL with temporal operators restricted to next and eventually, and establishes decidability (NEXPTIME-complete) for all architectures for a fragment that consists of a set of safety assumptions, and aSet of guarantees where each guarantee is a safety, reachability, or liveness condition.
Abstract: We consider the distributed synthesis problem for temporal logic specifications. Traditionally, the problem has been studied for LTL, and the previous results show that the problem is decidable iff there is no information fork in the architecture. We consider the problem for fragments of LTL and our main results are as follows: (1) We show that the problem is undecidable for architectures with information forks even for the fragment of LTL with temporal operators restricted to next and eventually. (2) For specifications restricted to globally along with non-nested next operators, we establish decidability (in EXPSPACE) for star architectures where the processes receive disjoint inputs, whereas we establish undecidability for architectures containing an information fork-meet structure. (3) Finally, we consider LTL without the next operator, and establish decidability (NEXPTIME-complete) for all architectures for a fragment that consists of a set of safety assumptions, and a set of guarantees where each guarantee is a safety, reachability, or liveness condition.
31 citations
25 Feb 1993
TL;DR: In this paper, the halting problem of any Turing machine in the form of a right-linear binary Horn clause is defined, and it is shown that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.
Abstract: This paper proposes a codification of the halting problem of any Turing machine in the form of only one right-linear binary Horn clause as follows: p(t) ← p(tt). where t (resp. tt) is any (resp. linear) term. Recursivity is well-known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.
31 citations