J.Acad.(N.Y.)6,2:99-199 2016 (100 pages)

The simplified linguistic relation between syntax and semantics as intrinsic property of classic Arabic motivates a dedicated look at P vs. NP in light of efforts and solutions presented by ancient Araband Muslim scholars to facilitate logicaland mathematical deduction. In Islamic Jurisprudence (Fikh) it has recently been shown [Abdelwahab et al. 2014] that if a formal system expressing Fikh is chosen in such a way that it is both, logically complete and decidable, the question of a complete and consistent legislation is decidable. If this formal Fikh-system is additionally chosen to be at least as expressive as propositional logic, the deduction of detailed sentences is efficient while the deduction of general rules is NP-complete. Further investigation reveals that ancient scholars adopted a very efficient approach for checking the validity of assertions with regard to both, language and logical argument which was mainly characterized by the extensive use of syntactical patterns already existent in input-variables. Accordingly, the structure of input-variables in 3-SAT-problems is investigated. The discovered variable-structure divides 3-SAT-formulas into three interrelated types of which one enables efficient pattern-oriented procedures. A novel 3-SAT-solver technique is introduced which binds the resolution of 3-SAT-formulas to the construction of FBDDs using the newly proposed pattern procedures. Finally, both 3-SAT and the question of finding 2-approximation algorithms for MinFBDD (the problem of minimizing FBDDs) are addressed. Positive results are reported for both questions. A practically important consequence is the ability to construct near to minimal FBDDs with a polynomial number of nodes for all Boolean functions expressed in a compact way. Eventually, an application of this new 3-SAT-solver is shown to enable polynomial upper bounds of the number of nodes in FBDDs constructed for finite projective planes problems overhauling the currently known exponential lower bounds.