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A morphogenetic (M) system is an abstract computational model combining properties of membrane (P) systems, such as computing via abstract particles in separate compartments regulating their workflow, with algorithmic self-assembly generalizing original Wang tiles to arbitrary polytopes forming complex shapes in 2D/3D (or generally, dD) space. Even very simple morphogenetic systems can exhibit complex self-organizing behaviour and, at the abstract level, one can observe characteristic properties of morphogenetic phenomena such as controlled growth, self-reproduction, homeostasis and self-healing. Here we focus on computational aspects of the morphogenetic systems. After summarizing a series of results related to their computational universality (in the Turing sense) and computational complexity, we present two small universal M systems (one of them self-healing) and we also demonstrate how morphogenetic systems relate to the classes P and NP. [ABSTRACT FROM AUTHOR]

Currently, a complexity-class problem is proving the existence of one-way permutations: one-to-one and onto maps that are computationally ‘easy’, while their inverses are computationally ‘hard’. In what follows, we make use of Bennett’s algorithm of the reversible Turing machine (quantum information heat engine) to perform a cascade of two controlled-NOT gates to physically create a permutation operation. We show that by running this input-saving (Turing) machine backwards the critical inequality of Landauer’s thermodynamic limit is reversed, which provokes the symmetry-breaking of the quantum circuit based on two successive controlled-NOT quantum gates. This finding reveals that a permutation of controlled-NOT gates becomes one-way, provided that adiabatically immersed in a heat bath, which determines the condition of existence of a thermodynamically non-invertible bijection in polynomial-time, that would otherwise be mathematically invertible. This one-way bijection can also be particularly important because it shows nonlinearities in quantum mechanics, which are detectable by watching that the mathematical reversibility of controlled-NOT gates does not work physically. [ABSTRACT FROM AUTHOR]

The more unambiguous statement of the P versus NP problem and the judgement of its hardness, are the key ways to find the full proof of the P versus NP problem. There are two sub-problems in the P versus NP problem. The first is the classifications of different mathematical problems (languages), and the second is the distinction between a non-deterministic Turing machine (NTM) and a deterministic Turing machine (DTM). The process of an NTM can be a power set of the corresponding DTM, which proves that the states of an NTM can be a power set of the corresponding DTM. If combining this viewpoint with Cantor's theorem, it is shown that an NTM is not equipotent to a DTM. This means that 'generating the power set P(A) of a set A' is a non-canonical example to support that P is not equal to NP. [ABSTRACT FROM AUTHOR]