Ville Salo

Block maps between primitive uniform and Pisot substitutions

In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift.

Solving the Induced Subgraph Problem in the Randomized Multiparty Simultaneous Messages Model

We study the message size complexity of recognizing, under the broadcast congested clique model, whether a fixed graph H appears in a given graph G as a minor, as a subgraph or as an induced subgraph. The n nodes of the input graph G are the players, and each player only knows the identities of its immediate neighbors. We are mostly interested in the one-round, simultaneous setup where each player sends a message of size

Distributed Testing of Excluded Subgraphs

{\mathcal O}\log n

Groups and Monoids of Cellular Automata

to a referee that should be able then to determine whether H appears in G. We consider randomized protocols where the players have access to a common random sequence. We completely characterize which graphs H admit such a protocol. For the particular case where H is the path of 4 nodes, we present a new notion called twin ordering, which may be of independent interest.

The Group of Reversible Turing Machines

We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangle-freeness. Hence, in particular, testing whether a graph is \(K_4\)-free, and testing whether a graph is \(C_4\)-free can be done in a constant number of rounds (where \(K_k\) denotes the k-node clique, and \(C_k\) denotes the k-node cycle). On the other hand, we show that testing \(K_k\)-freeness and \(C_k\)-freeness for \(k\ge 5\) appear to be much harder. Specifically, we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test \(K_k\)-freeness and \(C_k\)-freeness in a constant number of rounds for \(k\ge 5\).

Strongly Universal Reversible Gate Sets

We discuss groups and monoids defined by cellular automata on full shifts, sofic shifts, minimal subshifts, countable subshifts and coded and synchronized systems. Both purely group-theoretic properties and issues of decidability are considered.

Group-Walking Automata

We consider Turing machines as actions over configurations in

PSPACE-Completeness of Majority Automata Networks

\Sigma^{\mathbb{Z}^d}

Independent finite automata on Cayley graphs

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.

Playing with Subshifts

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