9 papers
XML Prompting as Grammar-Constrained Interaction: Fixed-Point Semantics, Convergence Guarantees, and Human-AI Protocols
Faruk Alpay, Taylan Alpay
Structured prompting with XML tags has emerged as an effective way to steer large language models (LLMs) toward parseable, schema-adherent outputs in real-world systems. We develop…
Manipulating Transformer-Based Models: Controllability, Steerability, and Robust Interventions
Faruk Alpay, Taylan Alpay
Transformer-based language models excel in NLP tasks, but fine-grained control remains challenging. This paper explores methods for manipulating transformer models through principl…
Nonlinear Continuum of States and Intuitionistic Flows in a Cognitive Space
Faruk Alpay, Taylan Alpay
We construct a rigorous mathematical framework for an abstract continuous evolution of an internal state, inspired by the intuitive notion of a flowing thought sequence. Using tool…
Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates
Faruk Alpay, Bugra Kilictas, Taylan Alpay
We develop a synthesis of orthomodular logic (projections as propositions) with operator fixed-point theory in Hilbert spaces. First, we introduce an anchored implication connectiv…
Logically Contractive Mappings: Fixed Points and Event-Indexed Rates
Faruk Alpay, Taylan Alpay
We introduce "logically contractive mappings" nonexpansive self-maps that contract along a subsequence of iterates and prove a fixed-point theorem that extends Banach's principle.…
Transfinite Iteration of Operator Transforms and Spectral Projections in Hilbert and Banach Spaces
Faruk Alpay, Taylan Alpay, Hamdi Alakkad
We study ordinal-indexed, multi-layer iterations of bounded operator transforms and prove convergence to spectral/ergodic projections under functional-calculus hypotheses. For norm…