SAT Requires Exhaustive Search

In this paper, by constructing extremely hard examples of CSP (with large domains) and SAT (with long clauses), we prove that such examples cannot be solved without exhaustive search, which implies a weaker conclusion P $\neq$ NP. This constructive approach for proving impossibility results is very different (and missing) from those currently used in computational complexity theory, but is similar to that used by Kurt G\"{o}del in proving his famous logical impossibility results. Just as shown by G\"{o}del's results that proving formal unprovability is feasible in mathematics, the results of this paper show that proving computational hardness is not hard in mathematics. Specifically, proving lower bounds for many problems, such as 3-SAT, can be challenging because these problems have various effective strategies available for avoiding exhaustive search. However, in cases of extremely hard examples, exhaustive search may be the only viable option, and proving its necessity becomes more straightforward. Consequently, it makes the separation between SAT (with long clauses) and 3-SAT much easier than that between 3-SAT and 2-SAT. Finally, the main results of this paper demonstrate that the fundamental difference between the syntax and the semantics revealed by G\"{o}del's results also exists in CSP and SAT.Comment: 12 pages, revised Definition 2.2 and the example in Fig.

Prediction of “Aggregation-Prone” Peptides with Hybrid Classification Approach

Protein aggregation is a biological phenomenon caused by misfolding proteins aggregation and is associated with a wide variety of diseases, such as Alzheimer’s, Parkinson’s, and prion diseases. Many studies indicate that protein aggregation is mediated by short “aggregation-prone” peptide segments. Thus, the prediction of aggregation-prone sites plays a crucial role in the research of drug targets. Compared with the labor-intensive and time-consuming experiment approaches, the computational prediction of aggregation-prone sites is much desirable due to their convenience and high efficiency. In this study, we introduce two computational approaches Aggre_Easy and Aggre_Balance for predicting aggregation residues from the sequence information; here, the protein samples are represented by the composition of k-spaced amino acid pairs (CKSAAP). And we use the hybrid classification approach to predict aggregation-prone residues, which integrates the naïve Bayes classification to reduce the number of features, and two undersampling approaches EasyEnsemble and BalanceCascade to deal with samples imbalance problem. The Aggre_Easy achieves a promising performance with a sensitivity of 79.47%, a specificity of 80.70% and a MCC of 0.42; the sensitivity, specificity, and MCC of Aggre_Balance reach 70.32%, 80.70% and 0.42. Experimental results show that the performance of Aggre_Easy and Aggre_Balance predictor is better than several other state-of-the-art predictors. A user-friendly web server is built for prediction of aggregation-prone which is freely accessible to public at the website