Turing models, classic

Classic Turing Models. The most well studied models of spontaneous pattern formation are based on Turing s original idea that symmetry breaking in biomorphogenesis occurs via an instability in a reaction-diffusion system. In order for this to be operative the time scales for reaction and diffusion must be comparable. Since cellular systems are in general quite small... 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

Subsequently Sorensen critized Olah s method of comparing the observed chemical shifts with thc expected on the basis of comparisons with models. To choose between one nonclassical ion 135 and rapid equilibration of classical ions 134 the equilibration of modified ions 136 and 137 was studied. Addition of another two CH3 groups to should not, the author believes, greatly affect the cation stmc-ture. In the PMR spectra e n at — IM °C no parate signals of isomers 136 and 137... 

The first corresponds to the classical activator-inhibitor system, where the elements fy<0 and g > 0 represent, respectively, Y (the inhibitor) inhibiting the formation of X (the activator), and X promoting the formation of Y. The second, with the opposite sign pattern for these off-diagonal elements, corresponds to a positive-feedback system such as the Gray-Scott model, where X is the autocatalyst and Y is typically a consumable reactant. In this case, both the autocatalyst and the reactant promote the formation of the autocatalyst, and, in turn, both species participate in the consumption of the reactant. In either case, a Turing instability can exist. 

The previous theorem can be generalized to several other paraconsistent logics, that is, to those where a conservatively translation function from CPL can be defined taking into account that such translation function must be effectively calculated. In the other direction, the existence of uniform families of boolean circuits to every uniform family of L-circuits (for any logic L provided with PRC) is guaranteed by the classical computability of roots for polynomials over finite fields. Then, the L-circuits model does not invalidate Church-Turing s thesis. 

Cook s theorem (cf. [10]) states that any NP-problem can be converted to the satisfiability problem in CPL in poljmomial time. The proof shows, in a constructive way, how to translate a Turing machine into a set of CPL formulas in such a way that the machine outputs T if, and only if, the formulas are consistent. As mentioned in the introduction, a model of paraconsistent Turing machines presented in [1] was proved to solve Deutsch-Jozsa problem in an efficient way. We conjecture that a similar result as Cook s theorem can be proven to paraconsistent Turing machines. In this way, paraconsistent circuits could be shown to efficiently solve Deutsch-Jozsa problem. Consequences of this approach would be the definition of non-standard complexity classes relative to such unconventional models of computation founded over non-classical logics. 

In [8] it is shown that this problem can be solved exponentially faster in the quantum model than with a classical probabilistic Turing machine. 

---