Abstract:
The concept of “state” is a foundational pillar of theoretical computer science, yet its classical formulation exhibits a representational poverty that increasingly hinders progress at the confluence of computability theory and artificial intelligence. Current models, from the finite states of automata to the high-dimensional vectors of machine learning, capture an instantaneous snapshot of a system’s manifest variables but fail to formally integrate critical information regarding its temporal, resource, semantic, and strategic context. This paper introduces the Nona-Dimensional State Formalism (NDSF) as a new paradigm to address this deficiency. We posit that a complete description of a computational state is a vector, the Nona-State, in a structured nine-dimensional vector space. These dimensions are partitioned into three logical triads: the Manifest Dimensions (Instruction, Memory, I/O), the Contextual Dimensions (Execution Trace, Resource Context, Semantic Context), and the Abstract Dimensions (Abstraction Level, Algorithmic Paradigm, Computability Horizon). The state transition is modeled as a tensor operation within this “computational phase space,” providing a rich, dynamic framework for computation. We define a corresponding abstract machine, the Nona-Dimensional Automaton (NDA), and demonstrate its Turing-completeness. The NDSF offers a more expressive language for describing computation, provides a potential solution to the state representation learning problem in artificial intelligence, and lays the groundwork for a new, geometric approach to algorithmic analysis, thereby offering a more unified and powerful foundation for the theory of computation.
Yıldırım, E. (2025). The Nona-Dimensional State Formalism: A New Foundation for Computational State Representation. Zenodo. https://doi.org/10.5281/zenodo.17046321
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