Theoretical Foundations of Computational Islamic Jurisprudence (Fiqh)

This paper presents the theoretical foundations and a comprehensive formal system for Computational Islamic Jurisprudence (Fiqh). We begin by establishing six foundational principles that distinguish computational Fiqh from conventional natural language processing and general theological formalization. These principles define Islamic jurisprudence as a normative deductive science operating on empirically grounded facts, necessitating a dual inductive-deductive methodology, and requiring a logical foundation specifically engineered for Fiqh rather than adopted from existing formalisms. Based on these principles, we construct a three-layer stratified computational language, F = D ◦ A ◦ G, consisting of: (1) an Arabic Morphological/Grammatical Grounding Engine that transforms fully vocalized Classical Arabic into language-independent Grounded Action Structures (GAS);(2) the Arabic Reified Action Calculus (ARAC) that normalizes these structures into canonical logical representations; and (3) a function-free Horn-clause Data log engine for executable legal deduction. Furthermore, we introduce a bridging methodology (Algorithm Ω) that dynamically generates a Fiqh-Categorized Attribute Grammar from inductively extracted Horn clauses via an Ontological Meta Map (Monto), seamlessly connecting the inductive extraction of legal rules to their deductive application. The paper rigorously proves the system’s semantic correctness, representational completeness, termination, and decidability. Crucially, by leveraging morphosyntactic inflection markers (δ modes) as structural constraints, we prove the strict unambiguity of the proposed grammar, achieving deterministic O(n) linear parsing complexity. This work provides a mathematically verifiable, epistemologically grounded, and computationally tractable foundation for automated Islamic jurisprudence.