In-vitro reconfigurability of native chemical automata, the inclusiveness of their hierarchy and their thermodynamics.

Living systems process information using chemistry. Computations can be viewed as language recognition problems where both languages and automata recognizing them form an inclusive hierarchy. Chemical realizations, without using biochemistry, of the main classes of computing automata, Finite Automata (FA), 1-stack Push Down Automata (1-PDA) and Turing Machine (TM) have recently been presented. These use chemistry for the representation of input information, its processing and output information. The Turing machine uses the Belousov-Zhabotinsky (BZ) oscillatory reaction to recognize a representative Context-Sensitive Language (CSL), the 1-PDA uses a pH network to recognize a Context Free Language (CFL) and a FA for a Regular Language (RL) uses a precipitation reaction. By chemically reconfiguring them to recognize representative languages in the lower classes of the Chomsky hierarchy we illustrate the inclusiveness of the hierarchy of native chemical automata. These examples open the door for chemical programming without biochemistry. Furthermore, the thermodynamic metric originally introduced to identify the accept/reject state of the chemical output for the CSL, can equally be used for recognizing CFL and RL by the automata. Finally, we point out how the chemical and thermodynamic duality of accept/reject criteria can be used in the optimization of the energetics and efficiency of computations.

slow and the frequency with which they are visited during the consumption of reactants and production of reaction products depends on the precise reaction vessel conditions, aliquot composition, as well as the order in which the aliquots are fed. (Note that for non-linear chemical kinetics, one expects that the order in which the aliquots are fed to the reaction be important, as non-linear mathematical operations are generally non-commutative; for example, x y ≠ y x or, log sin x ≠ sin log x. (This observation is particularly interesting in the light of the properties of a computation 2

.)
In summary, a chemical reaction can be thought of as an automaton which processes the information contained in an ordered sequence of aliquots. The generation of the reaction products can therefore be viewed as the transformation of the original information in the sequence of aliquots into some output information now residing in the products and the final physical conditions of the reactor. (And these can, in turn, carry out some work or perform a function.) The above paragraph together with the mechanical (i.e. repetitive) nature of a chemical reaction allows one to think of the operation of a chemical reaction in full as a "computation carried out by chemistry": information is input by a sequence of aliquots, mechanically processed by the chemical automaton and finally delivered in the form of the information contained in the final state of the reaction and its products. We call this "native chemical computation". In other words, chemical reactions, which occur at the Angstrom scale and in a number of 6.023 ×10 23 molecules per cubic centimeter, are the ultimate molecular recognition automata 1 . But can we harness this power? A step to take would be to make them programmable using the fact that the frequency and strength with which the various internal reaction pathways are visited depend on the details of the aliquot choice and on the order with which they are fed to the reaction. Thus, we can introduce a means to program the chemical automaton in order to recognize a variety of sequences of tokens (letters of an alphabet.) As is well known abstract automata process information contained in sequences of tokens (letters in the alphabet) that belong to some language. These are classified in the inclusive four-level Chomsky language hierarchy 3 from the simplest Regular Languages (RL) to the most complex, the Recursive Enumerable Languages. Correspondingly, the automata, which identify at least one language in its class and none above this class, are also arranged in an inclusive hierarchy 4 . That is, an automaton will recognize at least one language in its class and one or more languages in each of the lower levels in the hierarchy. Hence a chemical automaton should be able to process chemical information in the same way that abstract automata.
Since any computation can be carried out as a collection of interconnected word recognition (acceptance and rejection of words) problems 5 it is then important to ask the question of whether chemical automata form an inclusive hierarchy.
In a recent paper 1,6 we have demonstrated that chemical computation at any level of the Chomsky hierarchy does not require the intervention of biochemistry. We have built individual physico-chemical realizations 1,6,7 of each of the automata in the Chomsky hierarchy without any biochemistry and without large carbon molecules. We have done this by showing (a) how the letters of an alphabet can be represented chemically, (b) how to represent words in languages at the various levels of the Chomsky hierarchy using sequences of these letters, (c) how by feeding sequences of these chemicals representing words in chemical reactions of various levels of complexity act as automata and (d) we have introduced a physico-chemical measure that characterizes the thermodynamics of chemical computation. That is, one can represent the (practical) levels of automata in the Chomsky hierarchy in terms of chemical reactions of various levels of complexity 8 . But one still needs to show that automata form an inclusive hierarchy.
For the important case of the Turing machine 9 we showed that there exists a free-energy and reaction extent related thermodynamic metric 1,6 which can be used to characterize the results from processing a sequence. The measure has dimensions of action and is related to the free-energy dissipated by the chemical reaction in the recognition of the sequence. We also saw that the chemical description of the alphabet symbols for the words in a language can be macroscopically adjusted so that all words in the language accepted by the Turing machine have the same value of this thermodynamic measure. This is fundamental, as it implies that the Turing machine can be programmed and play a fundamental role in problems that require pattern recognition.
In this paper we will prove experimentally that the chemical Chomsky hierarchy is an inclusive hierarchy, as in the case of the abstract automata. We will do this by showing that in complete parallel with their abstract counterparts, the chemical realizations of automata not only recognize languages at their level but also below in the Chomsky hierarchy. That is the hierarchy of chemical automata is also inclusive. We will also extend the use of the thermodynamic metric mentioned above and previously introduced for the Turing machine, to the automata below the Turing machine rank in the hierarchy. This then shows that the thermodynamic interpretation of a computation with chemical automata is universal, in the sense that there exists a thermodynamic interpretation applicable to all automata. Finally, we will do all the above without using any biochemistry, so that our conclusions imply that full computation with chemistry does not require biochemistry.

Results
We know 1,6 that a native chemical Finite Automaton implemented by a simple precipitation reaction recognizes a Regular Language, L 1 (see Fig. 1). Similarly, a native chemical 1-stack Pushdown automaton based on pH chemistry was shown to recognize the Dyck language, a Context-Free Language L 2 . We also designed a native Turing machine based on Belousov-Zhabotinsky chemistry capable of recognizing a well-known context sensitive language, L 3 = {a n b n c n , where n ≥ 1} (see Fig. 1). We have already demonstrated that BZ can recognize the Dyck language using the formalism of multi-tape Turing Machines 7 . In what follows we will use these three languages as reference languages to characterize recognizing automata at the appropriate levels in the Chomsky hierarchy of languages 3 . (Since these are actual material implementations of the automata, their tapes cannot be infinite (or unbounded) as in theoretical implementations. By a series of strategies (cf. caption to Fig. 1 in ref. 1) one can www.nature.com/scientificreports www.nature.com/scientificreports/ increase the tape length. As noted by Minsky in ref. 21 , an infinite tape is impossible in practice, since it would require infinite energy to be implemented.) Abstract automata recognition of the Dyck language, a context-free language. We choose the Dyck language, a context free language in the Chomsky hierarchy, to test the inclusiveness of native chemical automata. The Dyck language is a classic example in automata theory of a context free language. Indeed, abstract 1-stack PDA and Turing Machine descriptions for recognizing the Dyck language are well-known and available 10,11 .
The Dyck language or language of balanced parentheses consists of all the strings of open and closed parentheses satisfying the following two rules: Rule-1 During processing of the string the number of ")" never exceeds the number of "(". Rule-2 Once the string is fully processed there are as many open parentheses "(" as closed parentheses ")". Different classes of abstract automata check systematically these two rules in different ways thus affecting the number of steps, state transitions and computation time of the automaton.
An abstract 1-stack PDA recognizes the Dyck language by a procedure that keeps track of the difference between the number of open parentheses and closed parentheses in a limited-access (last-in first-out) type of memory called a "stack" 10,11 . Whenever a "(" is read by the PDA, an element is added ("pushed") to the top of the stack and whenever a closed parentheses ")" is read, an element is removed ("popped") from the top of the stack 10,11 . If during computation the automaton attempts to pop from an empty stack, Rule-1 above is violated, the automaton rejects the string (reject R 1 ) and halts. Once the string is processed in its entirety, i.e. the end-of-sequence symbol ('#') is read, and if the stack is nonempty, Rule-2 above is violated and the automaton rejects the string (reject R 2 ). Otherwise, the string is accepted, i.e. it is in L 2 .
An abstract Turing Machine, in contrast to the 1-stack PDA, has access to a less limited memory, its tape, with unrestricted access because the head can move forward and backwards over the tape, and can also read and write symbols on the tape. A common Turing Machine implementation 10,11 for recognizing the Dyck language is as follows: once the sequence is on the tape, the head is located at the first symbol and moves to the "right" looking for a closed parenthesis, marking it with a distinct symbol, e.g. X, and reversing the head direction (to the "left") until it finds the closest matching open parenthesis, overwriting it with an X and reversing again the head direction. These rules are then repeated iteratively. If during computation, the head reaches the beginning of sequence symbol while moving left, the automaton rejects the string (reject R 1 ) and halts. Otherwise, once all the input symbols are processed, i.e. the automaton reaches the end-of-sequence symbol, the head direction is reversed to do a final check. If an open parenthesis is encountered before reaching the beginning of the expression, the automaton rejects the string (reject R 2 ). Otherwise, the string is accepted.
We note that although the computation is executed differently by the abstract 1-stack PDA and Turing Machine, the number and types of rejects are the same: R 1 corresponds to out-of-order ")" and can occur any time during computation; and R 2 corresponds to excess "(" which in both the 1-stack PDA and the Turing machine will Figure 1. The classical automata hierarchy and the associated Chomsky hierarchy of languages. We show some representative languages and the chemical reaction systems used for their respective experimental implementation: (a) bimolecular elementary reaction for the language of all words containing at least one a and one b; a diprotic weak acid and strong base pH system for the Dyck language, and the Belousov-Zhabotinsky oscillatory chemistry for L 3 = {a n b n c n , where n ≥ 1}. At the abstract level this hierarchy is inclusive. The work in this paper shows that the chemically realized automata also constitute an inclusive hierarchy.
Chemical recognition of the Dyck language by the BZ reaction. The Belousov-Zhabotinsky reaction [12][13][14] was discovered by Belousov in the 1950s while looking for a chemical analog of the Krebs cycle 15,16 . This nonbiochemical oscillatory chemistry, consists of the oxidation of a weak organic acid in an acidic aqueous solution by bromate ions in the presence of a transition metal catalyst. (There are in the literature several variants of its kinetic mechanism 17-19 that have been extensively studied). The features of the chemical relaxation oscillations (period, amplitude, etc) depend on the reactant and intermediate concentrations, and importantly, on the order in which they are added to the reaction, hence the capability of this reaction to carry out computations 1 .
In the design and implementation of native chemical automata, one must adequately choose the chemical species and aliquots intended to represent the alphabet symbols of the language to be recognized, accepted or rejected. Our criterion is that a chemical species can represent an alphabet symbol if it affects a uniquely distinct pathway in the reaction network which, in turn, translates into a distinct non-oscillatory or oscillatory signature 1,6 . Based on our previous results 1,7 an appropriate assignment for the Dyck language alphabet is: "("-an aliquot of the oxidizer, sodium bromate; ")" -an aliquot of the reductant, malonic acid; and "#" -an aliquot of the catalyst, in this case tris(2,2′-bipyridyl) dichloro ruthenium(II), although the system will work independently of the specific chemical nature of the BZ transition metal catalyst. (More details in Methods section.) The next, and critical, step is to identify the distinct oscillatory signatures associated with the reject and accept states. To check whether Rule-2 is satisfied or violated once the complete sequence has been processed, a chemical native TM uses two descriptors (cf. Fig. 2 for a representation of these features): one related to the frequency of the oscillations (here the period T is the time interval between two consecutive peaks) and the other related to the amplitude or location in the redox range of the oscillations (here the amplitude L is the difference between a peak value and its next trough value). Hence, if the final values of the pair [T, L] fall at the nonlinear locus [T # , L # ] (cf. Fig. 3) the sequence is accepted. Otherwise, if the final oscillations have higher period and lower amplitude than the above locus, the automaton has rejected the sequence (R 2 -reject). Note also that an R 1 -reject (due to an excess of closed parentheses) is output by this chemical automaton as either a constant minimum redox potential if the system is not in an oscillatory regime yet, or as a smooth continuous period decrease (the derivative of the period with respect to time is differentiable) if the reaction is not yet in an oscillatory regime. Figure 2 illustrates the experimental results. The top panel shows two examples of accepted strings ()()() and ((())), the middle panel shows two rejected strings due to excess ")" during computation for)()((and for ()())); and the bottom panel shows two rejected strings due to an unequal number of open and closed parentheses at the end of computation, for ()((and ()()((, respectively. Figure 3 shows the [T # , L # ] plot for the set of experimental sequences that would reach the end-of-sequence symbol (acceptance and R 2 -type rejection). The locus of accepted words gives (of course to be expected) a nonlinear power law dependence between T # and L # . Rejected sequences due to excess "(" are displaced towards lower amplitudes and slightly higher periods. In comparing the BZ-based TM and the pH-based 1-stack PDA used for the recognition of the Dyck Language, we note that the pH-based 1-stack PDA has the advantage of using a one-dimensional Rule-2 criterion (pH). However, it is also interesting to note that by using a thermodynamic-based signature, the Rule-2 checking criterion in the BZ-machine may be simplified into a one-dimensional criterion 1,6 using the following area metric The A (Word) metric measures how far from the maximum attainable Gibbs free-energy (i.e. when all catalyst is in its oxidized form) ΔG, is the Gibbs free-energy associated with the chemical oscillations output by the automaton once the full input string has been processed (this assumes that both terms are integrated over an equally long time interval). Remarkably, for the chemical automata-language pairings that result in a constant A Word ( ) independently of word length, the increase in the overall extent of reaction of the oxidation pathway as word length grows is compensated by an equivalent increase in the overall extent of reaction in the reduction pathway 6 . The Rule-2 checking criterion for this implementation (and of course given its chemical recipe) is as follows: accepted words have an A (Word) value of 86.5 ± 1.5, while rejected sequences have a lower A (Word) value (see Fig. 4).

Recognition of L 1 by Belousov-Zhabotinsky and the pH network.
We denote the language of all words that contain at least one "a" and one "b" by L 1 . This is a regular language and can therefore be recognized by a finite automaton as well as by abstract automata higher in the hierarchy such as the 1-stack PDA and the Turing Machine. The experimental results for our chemical 1-stack PDA and TM are shown in Fig. 5.
A typical abstract finite automaton recognizing L 1 has three distinct states: S 1 is the active state so long as one or more "a" is read, S 2 is the active state as long as one or more "b" is read, and S 3 is the state reached when the symbol that had not yet been read is read ("b" if the previous symbols were all "a" or "a" if all the previous symbols were "b"). S 3 is the only accept state and both S 1 and S 2 are reject states.
For the recognition of L 1 by the BZ reaction we start by assigning sodium bromate to symbol "a" and malonic acid to "b". The aliquot recipes of "a" and "b" can be kept the same as the recipes for "(" and ")", respectively, in the above BZ implementation for recognition of the Dyck language. S 1 is given by a flat high redox potential (catalyst dominantly in oxidized form), S 2 by a flat low redox potential (catalyst dominantly in the reduced form) and, of www.nature.com/scientificreports www.nature.com/scientificreports/ course, S 3 by the onset of oscillations. Figure 5 shows the experimental results of accepted words "aab" and "bab", and the rejected string "aaa".
The pH network already used to recognize the Dyck language, can also recognize L 1 by assigning sodium hydroxide to symbol "a" and malonic acid to symbol "b", and keeping the same aliquot recipes that were used to recognize L 2 in ref. 1 . In this chemical automaton, S 1 is given by ascending pH step changes or maximum pH, S 2 by descending pH step changes or minimum pH, and S 3 by a change of sign in the pH step change (i.e. if the pH www.nature.com/scientificreports www.nature.com/scientificreports/ had been ascending and then decreases, or if the pH had been descending and then increases upon addition of a symbol).

Discussion
We have successfully reconfigured a BZ-based Turing Machine such that in addition to the context sensitive language, L 3 , it can recognize the context-free language of balanced parentheses and/or the regular language L 1 . Similarly, we have shown how to reconfigure the pH-based 1 stack PDA to recognize L 1 . These results completely align our chemical automata with abstract automata theory and show that the chemical automata conform to a hierarchy that is inclusive, i.e. automata with higher complexity in the hierarchy can recognize languages at its level and also languages at lower complexity levels.
We can compare the pH-based 1-stack PDA and the BZ-based TM for recognizing the Dyck Language. The pH-based machine used a strong base, sodium hydroxide, as "(", and a diprotic acid, malonic acid, as ")", together with an appropriate pH indicator as "#" 1 . The aliquots were chosen so that one aliquot of "(" and one aliquot of ")" drive the reaction to the midpoint in pH after the first equivalence point, i.e. the point at which   Table 1. The pH machine checks Rule-2 as follows: a pH at exactly the midpoint indicates accept, and a pH above the midpoint indicates excess open parentheses and vice versa (see below). The BZ-TM uses instead two descriptors (the period and amplitude of the oscillations) to carry out the final check on whether the number of open and closed parentheses is the same or not. Two descriptors are needed because an abstract TM is equivalent to a 2-stack PDA 20,21 , in which one stack simulates when the head moves to the right, and the other when the head moves to the left. Checking that [T # , L # ] lies at the correct nonlinear T # = f(L # ) locus is equivalent to checking that the two stacks are empty. In the case of an excess of closed parentheses type of reject (R 1 ), a distinct chemical signature (i.e. never occurring in accepted words) is needed: pH below midpoint provides such a distinct signature in the case of the pH-based PDA, while a smooth decrease of the period trend is the distinct signature in the case of the BZ-based automaton (as opposed to the observed discontinuous stepwise changes of period trends in accepted words).
The fact that a TM uses a more general and powerful memory than a one-stack PDA, translates in chemical terms into a more complex reaction network with a larger number of key intermediates and feasible chemical pathways. Figure 6 compares the reaction network of the pH-based 1-stack PDA with the BZ-based TM. The key intermediate in the pH network is the proton concentration H + , while BZ has several key intermediates, such as the bromide ion Br − and the bromous acid HBrO 2 , but bromine Br 2 , hypobromous acid HBrO, proton H + and bromomalonic acid are also critical intermediates for performing native computations. On the other hand, the  www.nature.com/scientificreports www.nature.com/scientificreports/ pH network has two dominant pathways: the "acid" pathway (green arrows in Fig. 6 Panel a) and the "basic" pathway (blue arrows in Fig. 6 Panel a). Of course, the BZ network has a larger number of dominant pathways and four 1 were clearly identified in the recognition of L 3 1 . It is also interesting to note that the BZ reaction network can be easily coupled to other chemistries, paving the way to increase the number of feasible pathways in the coupled network. This can be done by exploiting the high reactivity of BZ with many chemical species, including organic compounds, inorganic salts, monomers, etc. Indeed, BZ has been coupled to polymerization reaction 22 and polymerization-induced self-assembly [23][24][25] .
It is relatively straightforward in the pH network to map the accept and reject states into the dominant ionized form of malonic acid: For BZ, since it involves so many intermediate concentrations this type of mapping from states to chemical species becomes cumbersome, and therefore a mapping to dominant pathway transitions and oscillatory signatures is more convenient 1 .
In connection with the thermodynamic metric and its use, it is not surprising that the same thermodynamic accept/reject criterion that was developed for recognizing a context-sensitive language with the Turing machine can be applied to the Dyck language without any modification. This, of course, was to be expected in view of the generality of the thermodynamic variables used for this criterion. Indeed, our metric brings with it various strengths. For example, the thermodynamic-based metric for the Rule-2 criterion in the BZ-TM facilitates and translates the chemical response into a more intuitive accept/reject criterion for the user. Even without optimizing specifically the aliquot recipes, this criterion provides a nearly constant value for words in the Dyck language. Thus, words in the language accepted by the automaton, are efficient in keeping a balance between the oxidation and reduction pathways even as the word length increases. In contrast, words with excess open parentheses enhance the oxidation pathway more than the reduction pathway, and thus the change in Gibbs free energy is not kept constant and varies with word length for sequences not in the automaton's accepted language.
Following the same logic, a thermodynamic Rule-2 criterion can also be formulated for the pH-based machine: the enthalpy yield is maximum and independent of word length for Dyck words and below this maximum for rejected strings 1 . However, in this case using the thermodynamic criterion does not offer any clear advantage compared to a purely chemical interpretation.
The pH-based 1-stack PDA used a shorter time interval τ between aliquots 1 , 3 min vs. 7.5 min in the BZ-based TM. (The BZ-based TM requires a longer time interval τ between symbols because of the induction time of the reaction and τ should be longer than the induction time + 2 oscillations which was the criterion we chose in this case 1 ). Hence, language recognition was executed faster with the pH-based PDA than with the BZ-based TM. In both types of chemical automata, the aliquot recipes, initial conditions, operating conditions and reactor  Figure 6. Reaction mechanisms for the pH-based 1-stack PDA and for the BZ-based TM. Panel a) shows the pH mechanism where "(" is NaOH, activating the "basic" pathway (blue arrows), ")" is the malonic acid, activating the "acid" pathway (green arrows), and the key intermediate is H + ; Hind and Ind − represent the protonated and deprotonated configurations of the pH indicator respectively. Panel b) shows a suitably modified version of the BZ-mechanism, "(" is sodium bromate activating the catalyst oxidation pathway; ")" is malonic acid activating the catalyst reduction pathway; and the key intermediates are Br − , HBrO 2 , HBrO, Br 2 , Bromomalonic acid, and H + . Arrows in blue consume protons, in green generate protons, and in grey neither consume nor generate protons. www.nature.com/scientificreports www.nature.com/scientificreports/ configuration can be optimized to reduce the time interval and speed up computations. But ultimately the slowest reaction in the mechanism limits the maximum speed for computation. Faster kinetic rates and smaller number of reactions makes the pH-based machine faster than the BZ machine for recognizing the Dyck language. Again, this is in consonance with abstract automata theory, the 1-stack PDA is faster than the Turing Machine for recognizing balanced parentheses so long as we assume that computation time is proportional to the number of state transitions the automaton carries out. The Turing machine with all its changes of direction (right and left) involves more state transitions and reads and overwrites repeated times a symbol cell, whereas a 1-stack PDA reads only once and executes one state transition per symbol. Hence, given a language, say L 2 , the performance in computing time and efficiency is different for the two automata, with the faster automaton being the least complex.
We can also compare how the three native chemical automata recognize L 1 . Their distinct chemical signatures for accept and reject states are summarized in Table 2. The reconfiguration was straightforward since the chemical signatures are not only different quantitatively but also qualitatively (e.g. no oscillations vs. oscillation).

Conclusions
We call a computation the mechanical (i.e. systematic) processing of available input information into some output information which is then used for some appropriate purpose. All computations can be cast as combinations of language recognition problems by suitable computing automata 5 . These automata parallel the Chomsky hierarchy of languages which form an inclusive hierarchy 4 .
Using chemistry, the most complex systems that we know, living systems, have the processing of information at their very core. They can efficiently process information for a variety of tasks which include molecular replication, transcription, translation, regulation, metabolism or epigenetics 26 . Information expressed with chemistry is one of the hallmarks of life. This "information is not a disembodied abstract entity; it is tied to a physical representation" 27 . It is chemical information being processed directly by chemical automata without having to resort to any kind of auxiliary "simulation" 28 . Indeed, living systems outperform supercomputers in their thermodynamic efficiency, as shown in ref. 29 for the efficiency of translation in the central dogma of biology.
Purely chemical computation would exclusively use chemical "hardware" and "software": with input information expressed chemically, its processing taking place via the pathways of suitable chemical reactions and producing an output that is also chemical (in terms of molecular states) or chemically usable, such as the values of suitable chemical state functions for the system of computing chemicals.
In this paper we have shown that native chemical automata can be configured to recognize languages at their level of the Chomsky hierarchy as well as in lower levels, suggesting that chemical reaction networks can also be categorized in an inclusive hierarchy in terms of their computational capabilities. Here we have seen experimentally that the example of the BZ network is an easily reconfigurable and capable network. Known 1 to recognize L 3 we have shown how it can be reconfigured to recognize the Dyck language (L 2 ) or the regular language L 1 , both at lower levels in the Chomsky hierarchy than the level of L 3 . The pH-automaton, known to recognize L 2 , can also be reconfigured to recognize L 1 . However, as expected, the chemical FA based on a simple bimolecular reaction cannot be reconfigured to recognize L 2 or the context-sensitive language L 3 . Similarly, the pH-based automaton cannot be reconfigured to recognize L 3 . This is because these simpler reaction networks do not have the necessary chemical intermediates or sufficiently distinct dominant pathways. Oscillatory pH-based networks, or other chemical oscillators, whose reaction networks enable at least 4 distinct reactant-dependent dominant pathways can be configured to recognize the L 3 language since this is the chemical requirement to achieve conformity with the abstract automata description (e.g. congruence between the observed and abstract state transitions).
We have also seen here that the accept/reject criteria for each of the above automata, from the FA to the 1-stack PDA to the Turing machine can be formulated in terms of a thermodynamic metric introduced in ref. 1 , which we call "Area". This thermodynamic metric not only simplifies the accept/reject interpretation from evaluating two oscillatory features into a single valued metric, but also enables optimization of the chemical automaton such that the change in Gibbs free energy, for example, is maintained constant for words in the language regardless of their length, and with subsequent implications for the energetics of computation 6 and the simplicity of design of effective native chemical computing machinery.
New hardware and computer frameworks are needed for carrying out more complex computations, for accelerating computation speed, for parallelizing computations in novel ways, for computing more efficiently in energetic terms or to control and execute the design of new materials using chemistry [22][23][24] in ways in which silicon computing cannot compete. All the above features can be improved using native chemical computation, although the specific details and associated strategies for practical applications are still undeveloped.
Indeed, bioinspired and chemical unconventional computing frameworks are thus promising, but to really compete with silicon-based systems they must be reconfigurable, "scalable and capable" 30 . Here we have shown how the reconfigurability of native chemical automata is indeed possible. This opens up the many opportunities provided by chemical computing 31 .   1 eco and 2Mag Mixcontrol eco). The change in the oxidation-reduction (redox) potential of the reaction mixture was monitored with an electrode system composed of a Pt-working electrode and a mercury sulfate reference electrode (Koslow 5100 A) connected to a benchtop meter (SperScientific). The temperature in the solution was monitored with an RTD sensor probe (Omega PR-13-2-100-1 and signal conditioner RTD SPRTX-S1) and was maintained by the circulating bath at 22.0 ± 0.3 °C during the experiment. Redox and temperature data were recorded with Labview Signal Express at a frequency of 5 data points per second. The refrigerated circulating bath opening was covered with aluminum foil to avoid light interferences since the used catalyst is photosensitive. Experimental procedure. The initial solution is kept in the bath until temperature is stabilized. Then, the initial catalyst aliquot (0.800 ± 0.006 mL of 0.0125 M stock ruthenium complex solution) is pipetted. (Note that this is equivalent to introducing the beginning and end of sequence symbol, #, to denote in this case the beginning of sequence.). The sequence checking procedure starts 450 s later by pipetting the first parenthesis in the expression. All subsequent symbols were pipetted after the previous symbol had been processing in the reactor during 450 s. The precision of the additions of the aliquots was within ±2 s. The 450 seconds long interval was selected to give enough time for the chemical system to react and compute the symbol but at the same time minimizing the gas production that could interfere with the redox measurement. The recipes for the initial catalyst concentration, open parenthesis and closed parenthesis were chosen to ensure that the reaction would start to oscillate as soon as both types of parentheses were present in the solution and within the 450 s time interval, and to give measurable changes in the amplitude and frequency of oscillations with the available setup and monitoring system.
Data analysis. For each expression, the experiment was run three times. The recorded data were analyzed, visualized and plotted in Matlab ® (R2015b v. 8.6.0.267246). Matlab ® Signal Processing Toolbox was used to read the amplitude and period from the recorded data for each of the three repetitions of the same expression. The procedure is as follows. First, the peaks of the oscillations are detected, followed by the detection of the troughs. Then, the period is obtained as the time elapsed between two consecutive peaks. The amplitude is defined as the distance between the redox value of the peak minus the redox value of the previous most near trough. Error bars were then estimated as symmetric error bars in the form of: mean ± standard deviation, where the standard deviation was normalized by the number of observations (i.e. 3). Hence, the mean final period T # and the standard deviation of the final period std T ( ) # , the mean final amplitude L # , and the standard deviation of the final amplitude std(L # ,) were estimated respectively as: