Constructive Patterns of Logical Truth

TL;DR

This paper explores the logical and computational aspects of Arabic linguistic structures and Islamic jurisprudence, introducing efficient pattern-based algorithms for SAT problems and FBDD constructions, with implications for complexity bounds.

Contribution

It presents novel pattern-oriented procedures for 3-SAT and approximation algorithms for FBDD, linking ancient logical methods to modern computational complexity.

Findings

Efficient pattern-based algorithms for 3-SAT problems.

Polynomial-sized FBDD constructions for Boolean functions.

Polynomial bounds for FBDD nodes in finite projective plane problems.

Abstract

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 Arab- and Muslim scholars to facilitate logical- and 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, this paper introduces efficient pattern-oriented procedures for 3-SAT as well as 2-approximation algorithms for MinFBDD. It opens up the possibility of constructing polynomial sized FBDDs 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.