Thomas Worsch’s research while affiliated with Karlsruhe Institute of Technology and other places

Fungal automata are a variation of the two-dimensional sandpile automaton of Bak et al. (Phys Rev Lett 59(4):381–384, 1987. https://doi.org/10.1103/PhysRevLett.59.381). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence HV. In particular we give a constructor that, given the description B of a circuit, computes the states of all cells in the finite support of the embedding configuration in O(logB)$O(\log \left| {B}\right| )$ space. As a consequence the prediction problem for fungal automata with update sequence HV is P$\textsf {P}$-complete. This solves an open problem of Goles et al. (Phys Lett A 384(22):126541, 2020. https://doi.org/10.1016/j.physleta.2020.126541).

Fungal automata are a variation of the two-dimensional sandpile automaton of Bak, Tang and Wiesenfeld (Phys. Rev. Lett., 1987). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence HV. In particular we give a constructor that, given the description B of a circuit, computes the states of all cells in the finite support of the embedding configuration in O(log|B|) space. As a consequence the prediction problem for fungal automata with update sequence HV is P-complete. This solves an open problem of Goles et al. (Phys. Lett. A, 2021).

Fungal automata are a variation of the two-dimensional sandpile automaton of Bak, Tang, and Wiesenfeld (Phys. Rev. Lett. 1987). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence HV. In particular we give a constructor that, given the description B of a circuit, computes the states of all cells in the finite support of the embedding configuration in $O(\log |B|)$ space. As a consequence the prediction problem for fungal automata with update sequence HV is $\mathsf{P}$-complete. This solves an open problem of Goles et al. (Phys. Lett. A, 2020).

In cellular automata with multiple speeds for each cell i there is a positive integer $p_i$ p i such that this cell updates its state still periodically but only at times which are a multiple of $p_i$ p i . Additionally there is a finite upper bound on all $p_i$ p i . Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.

We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single sweep of a bijective rule from left to right over an infinite tape. Such cellular automata are necessarily left-closing, and they move at least as much information to the left as they move information to the right.

In cellular automata with multiple speeds for each cell i there is a positive integer $p_i$ such that this cell updates its state still periodically but only at times which are a multiple of $p_i$. Additionally there is a finite upper bound on all $p_i$. Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.

We study the computational capacity of self-verifying iterative arrays (). A self-verifying device is a nondeterministic device whose nondeterminism is symmetric in the following sense. Each computation path can give one of the answers yes, no, or do not know. For every input word, at least one computation path must give either the answer yes or no, and the answers given must not be contradictory. It turns out that, for any time-computable time complexity, the family of languages accepted by s is a characterization of the so-called complementation kernel of nondeterministic iterative array languages, that is, languages accepted by such devices whose complementation is also accepted by such devices. s can be sped-up by any constant multiplicative factor as long as the result does not fall below realtime. We show that even realtime are as powerful as lineartime self-verifying cellular automata and vice versa. So they are strictly more powerful than the deterministic devices. Closure properties and various decidability problems are considered.

We study the computational capacity of self-verifying cellular automata with an emphasis on one-way information flow (). A self-verifying device is a nondeterministic device where each computation path can give one of the answers yes, no, or do not know. For every input word, at least one computation path must give either the answer yes or no, and the answers given must not be contradictory. Realtime are strictly more powerful than realtime deterministic one-way cellular automata. They can be sped-up from lineartime to realtime and are capable to simulate any lineartime computation of deterministic two-way CA. Closure and decidability properties are considered as well.

We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single left-to-right sweep of a bijective rule from left to right over an infinite tape.

We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single left-to-right sweep of a bijective rule from left to right over an infinite tape.