Abstract: Biomolecular computation is the scientific field focusing on the theory and practice of encoding combinatorial problems in ordinary DNA strands and applying standard biology lab operations such as cleansing and complementary sequence generation to them in order to compute an exact solution. The primary advantage offered by this computational paradigm is massive parallelism as the solution space is simultaneously searched. On the other hand, factors that need to addressed under this model are the DNA volume growth and computational errors attributed to inexact DNA matching. Biomolecular computation additionally paves the way for two- and three-dimensional self assemblying biological tiles which are closely linked at a theoretical level to a Turing machine, establishing thus its computational power. Applications include medium sized instances of TSP and the evaluation of the output of bounded fan-out Boolean circuits.

Abstract: This paper solves a long standing open problem of whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine by showing that the indistinguishable binomial decision tree can be formed in a 3-SAT instance. This paper describes how to construct the decision tree and explains why 3-SAT has no polynomial-time algorithm when the decision tree is formed in the 3-SAT instance. The indistinguishable binomial decision tree consists of polynomial numbers of nodes containing an indistinguishable variable pair but generates exponentially many paths connecting the clauses to be used for sequences of resolution steps. The number of paths starting from the root node and arriving at a child node forms a binomial coefficient. In addition, each path has an indistinguishable property from one another. Due to the exponential number of paths and their indistinguishability, if an indistinguishable binomial decision tree is constructed in which there exist one or more paths generating an empty clause, the number of calculation steps needed to extract the empty clause is not polynomially bounded. This result leads to the conclusion that class P is a proper subset of class NP.

Abstract: Two theorems about the P versus NP problem be proved in this article (1) There exists a language $L$, that the statement $L \in \textbf{P}$ is independent of ZFC. (2) There exists a language $L \in \textbf{NP}$, for any polynomial time deterministic Turing machine $M$, we cannot prove $L$ is decidable on $M$.

Abstract: The P versus NP problem is studied under the relational model of E F Codd I found that the term "complete configuration" is unnecessary and harmful in computational complexity theory because of excessive symbol redundancy For an input, its valid sequences of complete configurations are normalized into a relational model of shared trichoices with no redundancy To simplify the problem, a polynomial time nondeterministic Turing machine is polynomially reduced to a periodic machine, which only reverses its tape head displacement at the tape ends By enumerating all the O(p(n)) shared trichoices, a polynomial time p(n) periodic machine is simulated in time O((p(n))^4) under logarithmic cost A simple elementary proof of P=NP is obtained

Abstract: We look at the problem of counting the exact number of accepting computation paths of a given nondeterministic Turing machine (NTM). We give a deterministic algorithm that runs in time \(\widetilde{O}(\sqrt{S})\), where S is the size (number of vertices) of the configuration graph of the NTM, and prove its correctness. Our result implies a deterministic simulation of probabilistic time classes like \(\mathsf {PP}\), \(\mathsf {BPP}\), and \(\mathsf {BQP}\) in the same running time. This is an improvement over the currently best known simulation by van Melkebeek and Santhanam [SIAM J. Comput., 35(1), 2006], which uses time \(\widetilde{O}(S^{1 - \delta })\). It also implies a faster deterministic simulation of the complexity classes \(\mathsf {\oplus P}\) and \(\mathsf {Mod_k P}\).

Abstract: This paper is devoted to the complexity of the Boolean satisfiability problem. We consider a version of this problem, where the Boolean formula is specified in the conjunctive normal form. We prove an unexpected result that the CNF-satisfiability problem can be solved by a deterministic Turing machine in polynomial time.

Abstract: In this paper, we consider the computational power of a new variant of networks of splicing processors in which each processor as well as the data navigating throughout the network are now considered to be polarized. While the polarization of every processor is predefined (negative, neutral, positive), the polarization of data is dynamically computed by means of a valuation mapping. Consequently, the protocol of communication is naturally defined by means of this polarization. We show that networks of polarized splicing processors (NPSP) of size 2 are computationally complete, which immediately settles the question of designing computationally complete NPSPs of minimal size. With two more nodes we can simulate every nondeterministic Turing machine without increasing the time complexity. Particularly, we prove that NPSP of size 4 can accept all languages in NP in polynomial time. Furthermore, another computational model that is universal, namely the 2-tag system, can be simulated by NPSP of size 3 preserving the time complexity. All these results can be obtained with NPSPs with valuations in the set \(\{-1,0,1\}\) as well. We finally show that Turing machines can simulate a variant of NPSPs and discuss the time complexity of this simulation.