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A decomposition of the ab-controlled 3-cycle (xs xt ys) into a composition of four 2-controlled swaps.
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It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of $\{0,1\}^n$ can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every...
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It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of \(\{0,1\}^n\) can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform ever...
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Citations
... In this section, we show that OBpZ d , n, kq is finitely generated. Our proof is based on the existence of strongly universal reversible gates for permutations of A m , which can be found in [1,38] for the binary alphabet case, and generalized to other alphabets in [7]. We need a finite generating set for permutations of QˆΣ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n ‰ k). ...
... Our proof is based on the existence of strongly universal reversible gates for permutations of A m , which can be found in [1,38] for the binary alphabet case, and generalized to other alphabets in [7]. We need a finite generating set for permutations of QˆΣ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n ‰ k). ...
... The following result was proved in [7] (Lemmas 3 and 5): It permutes the positions of the m tape symbols according to π. Now we can conjugate the prefix applicationf using a rewiring to getf π " r´1 π˝f˝r π , we callf π an application of f in the coordinates Based on the decomposition in Figure 3, we first conclude that any controlled 3-cycle f of QˆΣ m is a composition of four applications of controlled swaps of QˆΣ m´2 . ...
We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; 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; and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines. We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely-generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb{Z}^d$-shift on a non-trivial alphabet if and only if $d \geq 2$.
... We solve this question by showing that universality on Z implies universality on Z/nZ for all large enough n in Sect. 4. We include copypasteable implementations of standard gates in terms of shifts of asynchronous ECA 57 in the appendices. ...
... The following is essentially folklore in the theory of reversible circuits. It is known that every even permutation on {0, 1} n can be written as a finite composition of applications of the NOT gate and Toffoli gate when we are allowed to apply these gates to arbitrary 3-tuples of wires (this is the folklore part, see e.g., [4,14]). The swaps in S allow arbitrary permutations of the coordinates, so we can conjugate any 3-tuple of coordinates to the support {0, 1, 2} of the Toffoli gate c 2 n , and similarly we can apply the NOT gate in any coordinate. ...
... There are obvious generalizations of G which seem interesting. It is a standard direction of generalization in the theory of reversible gates to change the binary alphabet to a higher-arity one, and sometimes the qualitative properties change, e.g., the parity of the finitely generated part of the group of reversible gates depends on the parity of the alphabet [3,4,11]. Here, the fact we have a geometry for the arrangement of the wires allows us to do much more: we can change the geometry to an arbitrary group, and instead of just increasing the size of the alphabet, we can replace the set of legal configurations by a subshift. ...
We give some optimal size generating sets for the group generated by shifts and local permutations on the binary full shift. We show that a single generator, namely the fully asynchronous application of the elementary cellular automaton 57 (or, by symmetry, ECA 99), suffices in addition to the shift. In the terminology of logical gates, we have a single reversible gate whose shifts generate all (finitary) reversible gates on infinitely many binary-valued wires that lie in a row and cannot (a priori) be rearranged. We classify pairs of words u , v such that the gate swapping these two words, together with the shift and the bit flip, generates all local permutations. As a corollary, we obtain analogous results in the case where the wires are arranged on a cycle, confirming a conjecture of Macauley-McCammond-Mortveit and Vielhaber.
... There are finite reversible gate sets generating the even permutations of Σ n for any finite alphabet Σ and n ∈ N, in the sense that there exists k ∈ N such that every even permutation of Σ n for n ≥ k is a finite composition of bijections φ : Σ k → Σ k applied to length-k subsequences of the coordinates. See [27,2,40] for formal definitions and proofs in the case when |Σ| is odd (in which case also the odd permutations are finitely generated) and [3] for the case of even |Σ|. In [1], this observation is used to show that a certain group of reversible Turing machines is finitely generated. ...
... , n − 1}, where n is such that reversible gates with at most n inputs and outputs generate all even permutations of Σ m for all m. From the result of [3] it follows that n = 4 suffices in general, and [2,40] show that n = 2 suffices in the case of odd |Σ|. We will show G ′ = G 0 . ...
... Many properties of interest are local, for example amenability, residual finiteness and soficness. Topological full groups and automorphism groups of subshifts on a group G are direct limits of the corresponding groups on the finitely generated subgroups of G due to the existence of local rules.3 More precisely, the finite-index subgroup of the automorphism group that stabilizes the point 0 Z -or alternatively the automorphism group of the corresponding 0-pointed subshift. ...
We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is Z (the shift group), giving a finitary version of Ryan's theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing SFTs. We show that any such set of automorphisms must generate an infinite group. We ask many related questions and mention some easy transitivity results in topological full groups and Thompson's V.
... Appendix B: OB is finitely generated Next we prove that OB(Z d , n, k) is finitely generated. This is based on the existence of strongly universal reversible gates for permutations of A m , recently proved for the binary alphabet A = {0, 1} in [2,33], and generalized to other alphabets in [7]. We need a finite generating set for permutations of Q × Σ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n = k). ...
... This is based on the existence of strongly universal reversible gates for permutations of A m , recently proved for the binary alphabet A = {0, 1} in [2,33], and generalized to other alphabets in [7]. We need a finite generating set for permutations of Q × Σ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n = k). ...
... The following lemma was proved in [7] (Lemmas 3 and 5): Let H = (V, E) be the graph with vertices V = Q × Σ m and edges {s, t} that connect elements s and t having Hamming distance one. This graph is clearly connected, so we get from Lemma 4(b) that the controlled 3-cycles generate all its even permutations: ...
We consider Turing machines as actions over configurations in \(\varSigma ^{\mathbb {Z}^d}\) 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.
... Appendix B: OB is finitely generated Next we prove that OB(Z d , n, k) is finitely generated. This is based on the existence of strongly universal reversible gates for permutations of A m , recently proved for the binary alphabet A = {0, 1} in [2,33], and generalized to other alphabets in [7]. We need a finite generating set for permutations of Q × Σ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n = k). ...
... This is based on the existence of strongly universal reversible gates for permutations of A m , recently proved for the binary alphabet A = {0, 1} in [2,33], and generalized to other alphabets in [7]. We need a finite generating set for permutations of Q × Σ m , and hence the proof in [7] has to be adjusted to account for non-homogeneous alphabet sizes (that is, due to possibly having n = k). ...
... The following lemma was proved in [7] (Lemmas 3 and 5): Let H = (V, E) be the graph with vertices V = Q × Σ m and edges {s, t} that connect elements s and t having Hamming distance one. This graph is clearly connected, so we get from Lemma 4(b) that the controlled 3-cycles generate all its even permutations: ...
We consider Turing machines as actions over configurations in $\Sigma^{\mathbb{Z}^d}$ 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.
It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of $\{0,1\}^n$ can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented. We generalize these results to non-binary logic: For any finite set $A$, a finite gate set can generate all even permutations of $A^n$ for all $n$, without any auxiliary symbols. This directly implies the previously published result that a finite gate set can generate all permutations of $A^n$ when the cardinality of $A$ is odd, and that one auxiliary symbol is necessary and sufficient to obtain all permutations when the cardinality of $A$ is even. We also consider the conservative case, that is, those permutations of $A^n$ that preserve the weight of the input word. The weight is the vector that records how many times each symbol occurs in the word or, more generally, the image of the word under a fixed monoid homomorphism from $A^*$ to a commutative monoid. It turns out that no finite conservative gate set can, for all $n$, implement all conservative even permutations of $A^n$ without auxiliary bits. But we provide a finite gate set that can implement all those conservative permutations that are even within each weight class of $A^n$.
We present a collection of results concerning the structure of reversible gate classes over non-binary alphabets, including (1) a reversible gate class over non-binary alphabets that is not finitely generated (2) an explicit set of generators for the class of all gates, the class of all conservative gates, and a class of generalizations of the two (3) an embedding of the poset of reversible gate classes over an alphabet of size $k$ into that of an alphabet of size $k+1$ (4) a classification of gate classes containing the class of $(k-1,1)$-conservative gates, meaning gates that preserve the number of occurrences of a certain element in the alphabet.
