Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

The first-order theory of linear one-step rewriting is undecidable

Abstract: The theory of one-step rewriting for a given rewrite system R and signature σ is the first-order theory of the following structure: its universe consists of all σ-ground terms, and its only predicate is the relation “x rewrites to y in one step by R”. The structure contains no function symbols and no equality. We show that there is no algorithm deciding the ∃ ∗ ∀ ∗ - fragment of this theory for an arbitrary finite, linear and non-erasing term-rewriting system. With the same technique we prove that the theory of encompassment plus one-step rewriting by the rule f(x) → g(x) and the modal theory of one-step rewriting are undecidable.

Hamiltonian paths in infinite graphs

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.

The Hunt for a Red Spider: Conjunctive Query Determinacy Is Undecidable

Abstract: We solve a well known, long-standing open problem in relational databases theory, showing that the conjunctive query determinacy problem (in its "unrestricted" version) is undecidable.

Decision Problems for Additive Regular Functions

Abstract: Additive Cost Register Automata (ACRA) map strings to integers using a finite set of registers that are updated using assignments of the form "x := y + c" at every step. The corresponding class of additive regular functions has multiple equivalent characterizations, appealing closure properties, and a decidable equivalence problem. In this paper, we solve two decision problems for this model. First, we define the register complexity of an additive regular function to be the minimum number of registers that an ACRA needs to compute it. We characterize the register complexity by a necessary and sufficient condition regarding the largest subset of registers whose values can be made far apart from one another. We then use this condition to design a PSPACE algorithm to compute the register complexity of a given ACRA, and establish a matching lower bound. Our results also lead to a machine-independent characterization of the register complexity of additive regular functions. Second, we consider two-player games over ACRAs, where the objective of one of the players is to reach a target set while minimizing the cost. We show the corresponding decision problem to be EXPTIME-complete when costs are non-negative integers, but undecidable when costs are integers.