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

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Publications (69)


Five transitions according to HVHVH
Compact representation of two cycles
Valid representations of and signals
A diagonal wire component with a signal moving south east
A diagonal wire for and signals

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Embedding Arbitrary Boolean Circuits into Fungal Automata
  • Article
  • Full-text available

March 2024

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15 Reads

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1 Citation

Algorithmica

Augusto Modanese

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Thomas Worsch

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(logB)O(\log \left| {B}\right| ) space. As a consequence the prediction problem for fungal automata with update sequence HV is PP\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).

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Embedding Arbitrary Boolean Circuits into Fungal Automata

October 2022

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11 Reads

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3 Citations

Lecture Notes in Computer Science

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).


Embedding arbitrary Boolean circuits into fungal automata

August 2022

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6 Reads

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(logB)O(\log |B|) space. As a consequence the prediction problem for fungal automata with update sequence HV is P\mathsf{P}-complete. This solves an open problem of Goles et al. (Phys. Lett. A, 2020).


Sketch of how a basic signal could move right in a MS-CAas fast as possible. The numbers in the top row are the periods of the cells. Time is going down. Triangles indicate active state transitions, i. e. when they are missing the state has to stay the same
Basic signal first moving right and bouncing back at the border with speed 1 although half of the cells has only period 2
Reflection of a “fast” signal at the right border. We use mc=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_c=6$$\end{document}, q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2$$\end{document}, hence bq=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_q=3$$\end{document}, h=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=1$$\end{document} and h+1=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1=2$$\end{document}. Therefore at the right end there are always h+1=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1=2$$\end{document} cells with period 2. We have introduced small spaces to make the boundaries of the last full block of cells with period 2 better visible. On the left hand side the last full block has 3 cells of period 2 and on the right hand side the last full block has 2 cells of period 3. For more details see the proof of Theorem 3
A faster algorithm for the Birthday Song Singers Synchronization Problem (FSSP) in one-dimensional CA with multiple speeds

August 2021

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60 Reads

Acta Informatica

In cellular automata with multiple speeds for each cell i there is a positive integer pip_i p i such that this cell updates its state still periodically but only at times which are a multiple of pip_i p i . Additionally there is a finite upper bound on all pip_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.


Sequentializing cellular automata

December 2020

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158 Reads

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1 Citation

Natural Computing

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.


A faster algorithm for the FSSP in one-dimensional CA with multiple speeds

March 2020

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7 Reads

In cellular automata with multiple speeds for each cell i there is a positive integer pip_i such that this cell updates its state still periodically but only at times which are a multiple of pip_i. Additionally there is a finite upper bound on all pip_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.


Iterative Arrays with Self-verifying Communication Cell

May 2019

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20 Reads

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1 Citation

Lecture Notes in Computer Science

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.


Self-verifying Cellular Automata: 13th International Conference on Cellular Automata for Research and Industry, ACRI 2018, Como, Italy, September 17–21, 2018, Proceedings

August 2018

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34 Reads

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1 Citation

Lecture Notes in Computer Science

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.


Sequentializing Cellular Automata

May 2018

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9 Reads

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4 Citations

Lecture Notes in Computer Science

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.



Citations (36)


... Again in 2024, Amir et al. [2] examine the Knuth-Morris-Pratt (KMP) automata and present a practical outcome that defies intuition. Also, Modanese and Worsch [3] in 2024, showed a fungal automaton with an update sequence horizontalvertical (HV) can be configured to include any Boolean circuit in its initial state. For texts with uniformly random symbols, the naive technique is thought to perform as well as the KMP algorithm. ...

Reference:

A Novel Approach to Construct Finite Automata Using Grid and Product Automata
Embedding Arbitrary Boolean Circuits into Fungal Automata

Algorithmica

... These gadgets typically consist of wires and logic gates (AND and OR). Nevertheless, when the support of the dynamics is symmetric and planar, it may be necessary to design more devices, such as diodes and crossover gates (Goles et al. , 2020Modanese and Worsch 2022). Crossover gates, in particular, hold a significant role as they often present the greatest challenge in terms of construction in a 2-dimensional context. ...

Embedding Arbitrary Boolean Circuits into Fungal Automata
  • Citing Chapter
  • October 2022

Lecture Notes in Computer Science

... In the one-dimensional setting, one possibility is that states are updated sequentially during a left-to-right (or right-toleft) sweep across the entire infinite line of cells. Such a setup is studied in [93] where the update performed once in each position is given by a reversible block rule A n −→ A n on n consecutive cells. The authors give a precise characterisation of the one-dimensional cellular automata that can be realised by such a sweep. ...

Sequentializing Cellular Automata
  • Citing Chapter
  • May 2018

Lecture Notes in Computer Science

... An interesting variant of cellular automata, introduced by Modanese and Worsch [12], falls between P and PSPACE in polynomial time. These are shrinking and expanding cellular automata, where a cell can not only update its state based on its neighbourhood, but also delete itself (shrinking); furthermore, new cells can be created between two existing ones (expanding). ...

Shrinking and Expanding Cellular Automata
  • Citing Conference Paper
  • June 2016

Lecture Notes in Computer Science

... The implementation of asynchrony directly affects the computational ability-for example, in chemical reaction networks, which are an asynchronous model, depending on the setting, their computational power varies between primitive recursive functions, Boolean circuits, and Turing machines [6]. Similarly, even though cellular automata are capable of universal computation, a certain threshold in terms of their size and state count needs to be met to allow for it: this question has been extensively investigated both for CAs [27,23,5,7,17,3,11,24,37] and ACAs [9,20,19,32,1]. Despite that, the limits of asynchronous universality are not known, and the smallest constructions still lag behind their synchronous counterparts, hinting at some foundational discrepancy between collective synchronous and asynchronous computation. ...

A 3-State Asynchronous CA for the Simulation of Delay-Insensitive Circuits

Lecture Notes in Computer Science

... In contrast to Turing machines it turned out that the family of languages accepted by reversible pushdown automata or reversible finite automata are proper subsets of the general families. Also reversible finite automata as well as reversible pushdown automata with states and lookahead have been considered [1,15]. ...

Degrees of reversibility for DFA and DPDA
  • Citing Conference Paper
  • July 2014

Lecture Notes in Computer Science

... In the case where one is allowed to choose freely the order of update of the cells, it was shown that there are some rules which allow one to return to the initial condition and some that do not always allow this [DSS12,SMD12]. With a different perspective, Worsch and Wacker examined how to construct an "inverse" rule, in the sense that its transition graph would be the "inverse" of the transition graph of the original inverse [WW13]. The design of asynchronous circuits with reversible gates is also a current important topic of research (see e.g. ...

On Completeness and Decidability of Phase Space Invertible Asynchronous Cellular Automata
  • Citing Article
  • January 2013

Fundamenta Informaticae

... This benchmarking approach is fundamentally different from above, but can lead to important outcomes that contribute to better application communication performance. The Special Karlsruhe MPI benchmark (SKaMPI) was created to benchmark MPI communications for supercomputer users and system administrators who want to tune their MPI libraries [12], evaluate and chose algorithms for collective communications [25,26], and ensure performance portability of the MPI library across platforms [17]. ...

Benchmarking Collective Operations with SKaMPI
  • Citing Article
  • January 2003

... These automata can be seen as finite automata whose state graphs are undirected. So, this notion is even stronger than the concept of time-symmetry studied in [3,11]. Time-symmetry appears in physics when a system can go back in time by applying the same transition function as for forward computations after a weak transformation of the phase-space. ...

Time-Symmetric Machines
  • Citing Conference Paper
  • July 2013

Lecture Notes in Computer Science