arXiv subjects

Elnaserledinellah Mahmood Abdelwahab

Publications and source records attributed to Elnaserledinellah Mahmood Abdelwahab.

3 recordsLinked to original sources

arXiv2023

Three Dogmas, a Puzzle and its Solution

Modern Logics, as formulated notably by Frege, Russell and Tarski involved basic assumptions about Natural Languages in general and Indo-European Languages in particular, which are contested by Linguists. Based upon those assumptions, formal Languages were designed to overcome what Logicians claimed to be 'defects' of Natural Language. In this paper we show that those assumptions contradict basic principles of Arabic. More specifically: The Logicians ideas, that within Natural Language words refer to objects, 'ToBe'-constructions represent identity statements, Indefinite Descriptions must be replaced by existential quantifiers to form meaningful Sentences and Symbols can have no interpretation-independent meanings, are all falsified using undisputed principles of Arabic. The here presented falsification serves two purposes. First, it is used as a factual basis for the rejection of approaches adopting Semantic axioms of Mathematical Logics as Models for meaning of Arabic Syntax. Second, it shows a way to approach the important computational problem: Satisfiability (SAT). The described way is based upon the realization that parsing Arabic utilizes the existence of 'meaning-particles' within Syntax to efficiently recognize words, phrases and Sentences. Similar meaning-particles are shown to exist in 3CNF formulas, which, when properly handled within the machinery of 3SAT-Solvers, enable structural conditions to be imposed on formulas, sufficient alone to guarantee the efficient production of non-exponentially sized Free Binary Decision Diagrams (FBDDs). We show, why known exponential Lower Bounds on sizes of FBDDs do not contradict our results and reveal practical evidence, obtained for multiplication circuits, supporting our claims.

cs.CL↗

arXiv2016

Constructive Patterns of Logical Truth

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.

cs.CC↗

arXiv2016

The Algorithm of Islamic Jurisprudence (Fiqh) with Validation of an Entscheidungsproblem

The historic background of algorithmic processing with regard to etymology and methodology is translated into terms of mathematical logic and Computer Science. A formal logic structure is introduced by exemplaryquestions posed to Fiqh-chapters to define alogic query language. As a foundation, ageneric algorithm for deciding Fiqh-rulings is designed to enable and further leverage rule of law (vs. rule by law) with full transparency and complete algorithmic coverage of Islamic law eventually providing legal security, legal equality, and full legal accountability.This is implemented by disentangling and reinstating classic Fiqh-methodology (usul al-Fiqh) with the expressive power of subsets of First Order Logic (FOL)sustainably substituting ad hoc reasoning with falsifiable rational argumentation. The results are discussed in formal terms of completeness, decidability and complexity of formal Fiqh-systems. AnEntscheidungsproblem for formal Fiqh-Systems is formulated and validated.

cs.AI↗