Computational Depth and Reducibility

This paper investigates Bennett\u27s notions of strong and weak computational depth (also called logical depth) for infinite binary sequences. Roughly, an infinite binary sequence x is defined to be weakly useful if every element of a non-negligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett\u27s observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost every infinite binary sequence is weakly deep, but not strongly deep

Reactamole: Functional Reactive Molecular Programming

Chemical reaction networks (CRNs) are an important tool for molecular programming, a field that is rapidly expanding our ability to deploy computer programs into biological systems for a variety of applications. However, CRNs are also difficult to work with due to their massively parallel nature, leading to the need for higher-level languages that allow for easier computation with CRNs. Recently, research has been conducted into a variety of higher-level languages for deterministic CRNs but modeling CRN parallelism, managing error accumulation, and finding natural CRN representations are ongoing challenges. We introduce Reactamole, a higher-level language for deterministic CRNs that utilizes the functional reactive programming (FRP) paradigm to represent CRNs as a reactive dataflow network. Reactamole equates a CRN with a functional reactive program, implementing the key primitives of the FRP paradigm directly as CRNs. The functional nature of Reactamole makes reasoning about molecular programs easier, and its strong static typing allows us to ensure that a CRN is well-formed by virtue of being well-typed. In this paper, we describe the design of Reactamole and how we use CRNs to represent the common datatypes and operations found in FRP. We also demonstrate the potential of this functional reactive approach to molecular programming by giving an extended example where a CRN is constructed using FRP to modulate and demodulate an amplitude modulated signal

Population-Induced Phase Transitions and the Verification of Chemical Reaction Networks

We show that very simple molecular systems, modeled as chemical reaction networks, can have behaviors that exhibit dramatic phase transitions at certain population thresholds. Moreover, the magnitudes of these thresholds can thwart attempts to use simulation, model checking, or approximation by differential equations to formally verify the behaviors of such systems at realistic populations. We show how formal theorem provers can successfully verify some such systems at populations where other verification methods fail

ALCH: An Imperative Language for Chemical Reaction Network-Controlled Tile Assembly

In 2015 Schiefer and Winfree introduced the chemical reaction network-controlled tile assembly model (CRN-TAM), a variant of the abstract tile assembly model (aTAM), where tile reactions are mediated via non-local chemical signals. In this paper, we introduce ALCH, an imperative programming language for specifying CRN-TAM programs. ALCH contains common features like Boolean variables, conditionals, and loops. It also supports CRN-TAM-specific features such as adding and removing tiles. A unique feature of the language is the branch statement, a nondeterministic control structure that allows us to query the current state of tile assemblies. We also developed a compiler that translates ALCH to the CRN-TAM, and a simulator that simulates and visualizes the self-assembly of a CRN-TAM program. Using this language, we show that the discrete Sierpinski triangle can be strictly self-assembled in the CRN-TAM. This solves an open problem that the CRN-TAM is capable of self-assembling infinite shapes at scale one that the aTAM cannot. ALCH allows us to present this construction at a high level, abstracting species and reactions into C-like code that is simpler to understand. Our construction utilizes two new CRN-TAM techniques that allow us to tackle this open problem. First, it employs the branching feature of ALCH to probe the previously placed tiles of the assembly and detect the presence and absence of tiles. Second, it uses scaffolding tiles to precisely control tile placement by occluding any undesired binding sites