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**Published in:** LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)

We analyze the computational complexity of basic reconfiguration problems for the recently introduced surface Chemical Reaction Networks (sCRNs), where ordered pairs of adjacent species nondeterministically transform into a different ordered pair of species according to a predefined set of allowed transition rules (chemical reactions). In particular, two questions that are fundamental to the simulation of sCRNs are whether a given configuration of molecules can ever transform into another given configuration, and whether a given cell can ever contain a given species, given a set of transition rules. We show that these problems can be solved in polynomial time, are NP-complete, or are PSPACE-complete in a variety of different settings, including when adjacent species just swap instead of arbitrary transformation (swap sCRNs), and when cells can change species a limited number of times (k-burnout). Most problems turn out to be at least NP-hard except with very few distinct species (2 or 3).

Robert M. Alaniz, Josh Brunner, Michael Coulombe, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Elise Grizzell, Ryan Knobel, Jayson Lynch, Andrew Rodriguez, Robert Schweller, and Tim Wylie. Complexity of Reconfiguration in Surface Chemical Reaction Networks. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alaniz_et_al:LIPIcs.DNA.29.10, author = {Alaniz, Robert M. and Brunner, Josh and Coulombe, Michael and Demaine, Erik D. and Diomidova, Jenny and Gomez, Timothy and Grizzell, Elise and Knobel, Ryan and Lynch, Jayson and Rodriguez, Andrew and Schweller, Robert and Wylie, Tim}, title = {{Complexity of Reconfiguration in Surface Chemical Reaction Networks}}, booktitle = {29th International Conference on DNA Computing and Molecular Programming (DNA 29)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-297-6}, ISSN = {1868-8969}, year = {2023}, volume = {276}, editor = {Chen, Ho-Lin and Evans, Constantine G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.10}, URN = {urn:nbn:de:0030-drops-187936}, doi = {10.4230/LIPIcs.DNA.29.10}, annote = {Keywords: Chemical Reaction Networks, reconfiguration, hardness} }

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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.

Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, and Jayson Lynch. Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ani_et_al:LIPIcs.SAND.2023.5, author = {Ani, Joshua and Coulombe, Michael and Demaine, Erik D. and Diomidov, Yevhenii and Gomez, Timothy and Hendrickson, Dylan and Lynch, Jayson}, title = {{Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {5:1--5:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.5}, URN = {urn:nbn:de:0030-drops-179414}, doi = {10.4230/LIPIcs.SAND.2023.5}, annote = {Keywords: Gadgets, robots, undecidability, Petri nets} }

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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

There have been many advances in molecular computation that offer benefits such as targeted drug delivery, nanoscale mapping, and improved classification of nanoscale organisms. This power led to recent work exploring privacy in the computation, specifically, covert computation in self-assembling circuits. Here, we prove several important results related to the concept of a hidden computation in the most well-known model of self-assembly, the Abstract Tile-Assembly Model (aTAM). We show that in 2D, surprisingly, the model is capable of covert computation, but only with an exponential-sized assembly. We also show that the model is capable of covert computation with polynomial-sized assemblies with only one step in the third dimension (just-barely 3D). Finally, we investigate types of functions that can be covertly computed as members of P/Poly.

Robert M. Alaniz, David Caballero, Timothy Gomez, Elise Grizzell, Andrew Rodriguez, Robert Schweller, and Tim Wylie. Covert Computation in the Abstract Tile-Assembly Model. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alaniz_et_al:LIPIcs.SAND.2023.12, author = {Alaniz, Robert M. and Caballero, David and Gomez, Timothy and Grizzell, Elise and Rodriguez, Andrew and Schweller, Robert and Wylie, Tim}, title = {{Covert Computation in the Abstract Tile-Assembly Model}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.12}, URN = {urn:nbn:de:0030-drops-179482}, doi = {10.4230/LIPIcs.SAND.2023.12}, annote = {Keywords: self-assembly, covert computation, atam} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

One of the most fundamental and well-studied problems in tile self-assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity of this problem in the 2-Handed Assembly Model (2HAM) at a constant temperature is a long-standing open problem since the model was introduced. Previously, only membership in the class coNP was known and that the problem is in P if the temperature is one (τ = 1). The problem is known to be hard for many generalizations of the model, such as allowing one step into the third dimension or allowing the temperature of the system to be a variable, but the most fundamental version has remained open.
In this paper, we prove the UAV problem in the 2HAM is hard even with a small constant temperature (τ = 2), and finally answer the complexity of this problem (open since 2013). Further, this result proves that UAV in the staged self-assembly model is coNP-complete with a single bin and stage (open since 2007), and that UAV in the q-tile model is also coNP-complete (open since 2004). We reduce from Monotone Planar 3-SAT with Neighboring Variable Pairs, a special case of 3SAT recently proven to be NP-hard. We accompany this reduction with a positive result showing that UAV is solvable in polynomial time with the promise that the given target assembly will have a tree-shaped bond graph, i.e., contains no cycles. We provide a 𝒪(n⁵) algorithm for UAV on tree-bonded assemblies when the temperature is fixed to 2, and a 𝒪(n⁵log τ) time algorithm when the temperature is part of the input.

David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Unique Assembly Verification in Two-Handed Self-Assembly. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caballero_et_al:LIPIcs.ICALP.2022.34, author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim}, title = {{Unique Assembly Verification in Two-Handed Self-Assembly}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {34:1--34:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.34}, URN = {urn:nbn:de:0030-drops-163751}, doi = {10.4230/LIPIcs.ICALP.2022.34}, annote = {Keywords: self-assembly, unique assembly verification, 2-handed assembly model} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Tile Automata is a recently defined model of self-assembly that borrows many concepts from cellular automata to create active self-assembling systems where changes may be occurring within an assembly without requiring attachment. This model has been shown to be powerful, but many fundamental questions have yet to be explored. Here, we study the state complexity of assembling n × n squares in seeded Tile Automata systems where growth starts from a seed and tiles may attach one at a time, similar to the abstract Tile Assembly Model. We provide optimal bounds for three classes of seeded Tile Automata systems (all without detachment), which vary in the amount of complexity allowed in the transition rules. We show that, in general, seeded Tile Automata systems require Θ(log^{1/4} n) states. For Single-Transition systems, where only one state may change in a transition rule, we show a bound of Θ(log^{1/3} n), and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of Θ(({log n}/{log log n})^{1/2}).

Robert M. Alaniz, David Caballero, Sonya C. Cirlos, Timothy Gomez, Elise Grizzell, Andrew Rodriguez, Robert Schweller, Armando Tenorio, and Tim Wylie. Building Squares with Optimal State Complexity in Restricted Active Self-Assembly. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alaniz_et_al:LIPIcs.SAND.2022.6, author = {Alaniz, Robert M. and Caballero, David and Cirlos, Sonya C. and Gomez, Timothy and Grizzell, Elise and Rodriguez, Andrew and Schweller, Robert and Tenorio, Armando and Wylie, Tim}, title = {{Building Squares with Optimal State Complexity in Restricted Active Self-Assembly}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.6}, URN = {urn:nbn:de:0030-drops-159482}, doi = {10.4230/LIPIcs.SAND.2022.6}, annote = {Keywords: Active Self-Assembly, State Complexity, Tile Automata} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger pre-built assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP^{NP}-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is in coNP for 𝒪(1) bounded temperature and coNP-complete when temperature is part of the input. We further provide preliminary results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, and provide polynomial time solutions.

David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Complexity of Verification in Self-Assembly with Prebuilt Assemblies. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caballero_et_al:LIPIcs.SAND.2022.8, author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim}, title = {{Complexity of Verification in Self-Assembly with Prebuilt Assemblies}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.8}, URN = {urn:nbn:de:0030-drops-159503}, doi = {10.4230/LIPIcs.SAND.2022.8}, annote = {Keywords: 2-handed assembly, verification, prebuilt} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Staged self-assembly has proven to be a powerful abstract model of self-assembly by modeling laboratory techniques where several nanoscale systems are allowed to assemble separately and then be mixed at a later stage. A fundamental problem in self-assembly is Unique Assembly Verification (UAV), which asks whether a single final assembly is uniquely constructed. This has previously been shown to be Π^{p}₂-hard in staged self-assembly with a constant number of stages, but a more precise complexity classification was left open related to the polynomial hierarchy.
Covert Computation was recently introduced as a way to compute a function while hiding the input to that function for self-assembly systems. These Tile Assembly Computers (TACs), in a growth only negative aTAM system, can compute arbitrary circuits, which proves UAV is coNP-hard in that model. Here, we show that the staged assembly model is capable of covert computation using only 3 stages. We then utilize this construction to show UAV with only 3 stages is Π^{p}₂-hard. We then extend this technique to open problems and prove that general staged UAV is PSPACE-complete. Measuring the complexity of n stage UAV, we show Π^{p}_{n - 1}-hardness. We finish by showing a Π^{p}_{n + 1} algorithm to solve n stage UAV leaving only a constant gap between membership and hardness.

David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{caballero_et_al:LIPIcs.ESA.2021.23, author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim}, title = {{Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.23}, URN = {urn:nbn:de:0030-drops-146047}, doi = {10.4230/LIPIcs.ESA.2021.23}, annote = {Keywords: self-assembly, covert computation, staged self-assembly, assembly verification} }

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**Published in:** LIPIcs, Volume 174, 26th International Conference on DNA Computing and Molecular Programming (DNA 26) (2020)

Many models of self-assembly have been shown to be capable of performing computation. Tile Automata was recently introduced combining features of both Celluar Automata and the 2-Handed Model of self-assembly both capable of universal computation. In this work we study the complexity of Tile Automata utilizing features inherited from the two models mentioned above. We first present a construction for simulating Turing Machines that performs both covert and fuel efficient computation. We then explore the capabilities of limited Tile Automata systems such as 1-Dimensional systems (all assemblies are of height 1) and freezing Systems (tiles may not repeat states). Using these results we provide a connection between the problem of finding the largest uniquely producible assembly using n states and the busy beaver problem for non-freezing systems and provide a freezing system capable of uniquely assembling an assembly whose length is exponential in the number of states of the system. We finish by exploring the complexity of the Unique Assembly Verification problem in Tile Automata with different limitations such as freezing and systems without the power of detachment.

David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Verification and Computation in Restricted Tile Automata. In 26th International Conference on DNA Computing and Molecular Programming (DNA 26). Leibniz International Proceedings in Informatics (LIPIcs), Volume 174, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{caballero_et_al:LIPIcs.DNA.2020.10, author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim}, title = {{Verification and Computation in Restricted Tile Automata}}, booktitle = {26th International Conference on DNA Computing and Molecular Programming (DNA 26)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-163-4}, ISSN = {1868-8969}, year = {2020}, volume = {174}, editor = {Geary, Cody and Patitz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.2020.10}, URN = {urn:nbn:de:0030-drops-129635}, doi = {10.4230/LIPIcs.DNA.2020.10}, annote = {Keywords: Tile Automata, Turing Machines, Unique Assembly Verification} }