Description
Sure, let's reformulate your overall idea by incorporating the definitions we've discussed so far, as well as introducing some additional concepts that could be helpful:
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We start with a base space B, which represents the program code, input data, and any other relevant information that defines the initial state of a system.
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The execution of the program can be viewed as a continuous map or function p: E → B, where E is the execution space representing all possible states the system can transition through during its execution.
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This function p is a quasifibration, which is a generalization of a fibration in algebraic topology. It means that while p may not be a true fibration, it still shares some properties regarding how the (relative) homotopy groups of E, B, and the fibers p^-1(x) are related.
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The fibers p^-1(x) represent the transitions or connections between states that occur during execution for a given point x in the base space B (e.g., a specific point in the program code or input data).
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These fibers can be seen as embeddings of a higher-dimensional structure (the quasifibration p) expressed in a lower dimension, similar to how instances of ASTs, MetaCoq terms, or JSON data graphs can be viewed as embeddings of higher-dimensional concepts.
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The system is comprehensive, meaning that fibers in one domain can be interpreted as code or quasifibrations that generate fibers in another domain. This creates a recursive hierarchy where fibers at a higher level act as quasifibrations that produce fibers (state transitions) at a lower level.
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To capture this comprehensiveness, we can introduce the concept of an ontology or a strongly typed data structure that defines and relates the various elements (scalars, arrays, dictionaries, etc.) within the system. This ontology can be seen as a meta-narrative or a higher-dimensional structure that all the lower-dimensional representations (fibers, quasifibrations, etc.) are embedded in.
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Additionally, we can incorporate techniques like deep graph embeddings, where nodes (states) are assigned numerical IDs based on their importance or frequency, with lower IDs representing more common or core states. Hyperedges can be used to connect more than two nodes, representing artificial nodes with a chain of connections.
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Different types of transitions between states can be represented as different edge types, allowing for a more detailed analysis of the system's behavior.
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Textual analysis of code via binary compiler execution introspection can be employed to automatically mark up and analyze the program's textual representation based on its execution path and the meaning extracted from the code.
Please let me know if this reformulation accurately captures your overall idea and if you would like to add, modify, or clarify any part of it.