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This paper presents Al Ghasali, a computational framework designed to formalize and simulate Islamic jurisprudential (Fiqh) rulings. Building upon recent set theoretic formalizations of al sabr wa al Taqsim that extract minimal operational rules as simple Horn clause logical programs, this work proves that the semantics of closed Fiqh chapters collapse completely to deterministic Finite State Automata (FSA). We establish a formal computational hierarchy, demonstrating that restricting formalization to finite propositional Horn theories guarantees decidability and constant time (O(1)) verdict extraction after offline compilation. The framework formally proves that hierarchical compositions of sub-chapter FSAs preserve this O(1) complexity, and that temporal juristic requirements, such as Tartib (order), are deterministically flattened into static ontological states, preserving the memoryless property of the automaton. To generate human-readable juristic summaries, Al Ghasali incorporates Large Language Models (LLMs) via an architecture where the deterministic FSA strictly controls the probabilistic LLM. Crucially, the paper formulates sufficient design safety conditions for this FSA LLM integration, ensuring strict adherence to the epistemological boundaries of Usul al Fiqh. To neutralize the risk of LLM hallucination, the system employs a Grammar Guided Constrained Decoding mechanism. We formally define the mapping function that dynamically constructs unambiguous right linear grammars from the FSA execution history. The paper mathematically proves that this architecture, augmented by a Bounded Neutral Lexicon and Valence Locking, guarantees the absolute absence of Decisional Hallucination, Juristic Factual Hallucination, and Semantic Inversion reducing the LLM to a pure linguistic transformer of deterministic juristic facts

For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.

This paper determines the accepting-state spectrum of the right-quotient operation for permutation automata, solving an open problem posed by Rauch and Holzer. They showed that in the unary case in which the languages whose right quotient we consider are languages over an alphabet with but one symbol, the accepting state complexities which could ensue for automata recognizing their quotient is exactly all integers between $1$ and $mn$ inclusive when $m,n \ge 1$; we show that for arbitrary alphabets the accepting-state spectrum is $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\{0\}$ if $m=0$ or $n=0$, and $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\mathbb{N}_{>0}$ if $m,n\ge 1$. Our result shows that every positive accepting-state complexity ensues for right quotient as long as both of the languages of which we consider the right quotient are nonempty. $0$ becomes the only unavoidable magic value. Moreover, we show that an alphabet with just two letters can witness any arbitrary $α\ge 1$ when $m,n \ge 1$, in that languages can be defined over a binary alphabet with accepting-state complexity $m$ and $n$ respectively for which their right quotient has accepting-state complexity $α$. What ultimately happens in our proof is that we construct $(A^{\mathrm q}_{m,α},B^{\mathrm q}_{n,α})$ such that the words accepted by $B^{\mathrm q}_{n,α}$ induce a point stabilizer in a symmetric group, and the standard quotient construction giving an automaton recognizing the quotient language then saturates the original final set of $A^{\mathrm q}_{m,α}$ to a full orbit; this gives a minimal quotient automaton with exactly $α$ final states. The complete spectrum which is our result follows from this, solving the open question of what the right quotient's accepting-state complexity for permutation automata is.

Smooth sequences over an alphabet of positive integers $\{a,b\}$ are sequences that are infinitely derivable, the emblematic example being the Oldenburger-Kolakoski sequence $κ$ over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth sequences called smooth words. In this paper we prove that the smooth words are exactly the factors of smooth sequences and we make progress towards the conjecture that the complexity of smooth words over $\{a,b\}$ grows like $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove the lower bound over any alphabet and the upper bound over even alphabets, and we improve the known upper bound over odd and mixed alphabets. We also provide a dichotomy that links the language, the morphicity and the factor complexity of $κ$.