On Immortal Configurations in Turing Machines

Abstract

In this paper we investigate the behaviour of Turing machines seen as dynamical systems. In this context, a Turing machine does not start from a specified initial state and tape, but from any configuration. In this context, we call a configuration immortal if the machine runs forever starting from it. The seminal article of Hooper [8] proves that we cannot decide whether a Turing machine has an immortal configuration (is immortal). However, the result does not say anything about what the immortal configurations look like. In fact, in the construction by Hooper, if the Turing machine has an immortal configuration, then it has one where the tape is almost completely empty.A few results give some structure on the set of immortal configurations: Kurka [13] asked whether there always exists a (temporally) periodic configuration, and was refuted by Blondel et al. [1]. Delvenne and Blondel [6] proved that the set of immortal configurations, if nonempty, always contains quasi-periodic configurations.In this article, we go further, and give some news results on the set of immortal configurations. We prove in particular in Theorem 2 that there exists a Turing machine for which the set of immortal configurations is nonempty and contains no computable configurations.The main ingredient is a new proof of Hooper's theorem by Kari and Ollinger [11] combined with an encoding into subshifts of an effective set with no recursive points. We first define all relevant vocabulary in the next section, then proceed to the proof.

Tiling Problems on Baumslag-Solitar groups.

“…Following the ideas in [9] and [10], we exhibit a weakly aperiodic tile set (see Section 3), and we prove that the domino problem is undecidable on these groups (see Section 4). Analogously to [7], we conclude that there exist arecursive tile sets on the Baumslag-Solitar groups, that is, tile sets that admit valid tilings but all valid tilings are non-recursive.…”

“…Combining this two facts, we can conclude that for any Baumslag-Solitar group, it is possible to encode any Turing machine inside a tiling on this group. It was shown in [7] that there exists a Turing machine M that halts on every recursive configuration but does not halt on some non-recursive configuration. Executing our construction on M one obtains a tile set on BS(m, n) that admits a valid tiling, and all valid tilings are non recursive.…”

On relations between properties in transitive Turing machines

“…To define a dynamical system, we have chosen to restrict our scope to complete machines (nonetheless, incomplete machines will be used in the construction of other complete machines). Note that other authors define dynamical systems over incomplete machines (for example, [15]), but here, we exclude this case for simplicity. With these considerations and providing X = Q × Z × Σ Z with the product topology, that is, taking the discrete metric for Q and Z and the already defined metric for Σ Z , X is not a compact set.…”

“…These two different topologies are not equivalent; therefore, a given machine may have different properties depending on the considered topological model, as Kůrka established. The seminal work of Kůrka inspired a large line of research that considers properties such as immortality [14,15,17], entropy [13,16,18,21], equicontinuity [9], periodicity [5,6,17,18], transitivity and minimality [6,12].…”

Optimization-Based Approach to Tiling of Finite Areas With Arbitrary Sets of Wang Tiles

“…All tilings of finite-sized areas are restricted to be subsets of the infinite plane, which is not required in the finite domain, and it is almost impossible to introduce boundary conditions. On top of that, there provably exist aperiodic tile sets that admit only non-recursive tiling [28,29], forbidding design of any problem-specific algorithm to tile the infinite plane.…”