The Group of Reversible Turing Machines

Abstract

Part 2: Regular PapersInternational audienceWe consider Turing machines as actions over configurations in <math xmlns="http://www.w3.org/1998/Math/MathML"> <msup> <mi>&#x03A3;<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </mrow> </msup></math>which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two

Groups and Monoids of Cellular Automata

“…right-angled Artin groups) [35], and contains a group not satisfying the Tits alternative [49] (we give another proof in Proposition 5). In the journal version of [3] we prove that there is an f.g. subgroup with undecidable torsion problem. Since the constructions of the present paper are constructive, Theorem 1 combined with [33] provides a new proof of this.…”

“…When one alphabet is fixed, the group of all synchronous automata transductions is not finitely generated, as it has infinite abelianization (consider the signs of permutations performed on different levels or the tree). A large finitely-generated subgroup of "elementary Turing machines" is constructed in the journal version of [3], but the author does not know whether it is f.g.-universal.…”

“…A sofic shift is a subshift defined by a regular language of forbidden patterns; in particular all full shifts A Z are trivially sofic. 1 Though the journal version of [3] is not submitted or available online, it precedes the results of this paper and uses different methods -there the work of producing a "generating set" is done in the group of Turing machines, while here it is done in the group of RCA. Theorem 5.…”

No Tits alternative for cellular automata

“…It turns out that Lemma 2 generalizes to such actions, and any infinite subshift can be used in place of this (almost) periodic subshift. The generalization is based on a commutator trick from [4,43,1,8,3]. The trick to adapting the construction to cellular automata on N is to consider 'actions of the trace of another cellular automaton'.…”

Transitive action on finite points of a full shift and a finitary Ryan’s theorem

“…See Section 7 for the (easy) proofs, and [11] for a reference for Thompson's V . As an auxiliary result, in Lemma 5.4 we also prove that a certain natural group of homeomorphisms on the full shift A Z called G 0 , which corresponds to the reversible one-state oblivious Turing machines of [1], acts ∞-transitively on finite points.…”