준스켈레톤 배선도 그래프와 해시 다이어그램의 동형성 및 전략 추출 알고리즘
📝 Abstract
A wiring diagram is a labeled directed graph that represents an abstract concept such as a temporal process. In this article, we introduce the notion of a quasi-skeleton wiring diagram graph, and prove that quasi-skeleton wiring diagram graphs correspond to Hasse diagrams. Using this result, we designed algorithms that extract wiring diagrams from sequential data. We used our algorithms in analyzing the behavior of an autonomous agent playing a computer game, and the algorithms correctly identified the winning strategies. We compared the performance of our main algorithm with two other algorithms based on standard clustering techniques (DBSCAN and agglomerative hierarchical), including when some of the data was perturbed. Overall, this article brings together techniques in category theory, graph theory, clustering, reinforcement learning, and data engineering.
💡 Analysis
A wiring diagram is a labeled directed graph that represents an abstract concept such as a temporal process. In this article, we introduce the notion of a quasi-skeleton wiring diagram graph, and prove that quasi-skeleton wiring diagram graphs correspond to Hasse diagrams. Using this result, we designed algorithms that extract wiring diagrams from sequential data. We used our algorithms in analyzing the behavior of an autonomous agent playing a computer game, and the algorithms correctly identified the winning strategies. We compared the performance of our main algorithm with two other algorithms based on standard clustering techniques (DBSCAN and agglomerative hierarchical), including when some of the data was perturbed. Overall, this article brings together techniques in category theory, graph theory, clustering, reinforcement learning, and data engineering.
📄 Content
1.1. Ologs and wiring diagrams. The notion of an ontology log, or olog for short, was defined by Spivak in 2012 as a method for knowledge representation [13]. Since ologs are based on category theory in mathematics, it means that all the tools and techniques from category theory are at one’s disposal when using ologs. In addition, since ologs are authored using words (in any written language of the author’s choice), they are easily understandable to humans. By design, an olog also represents a database schema in a natural way, meaning they provide a framework for organizing data in an autonomous system. Even though ologs are very similar to knowledge graphs in appearance, they are different in a fundamental way: arrows in an olog are always composable. This means, in practice, that the arrows in ologs often correspond to functions.
In 2024, the first author built on the notion of an olog and considered the notion of a wiring diagram, defined to be a directed graph whose labels come from an olog [9]. Wiring diagrams, in the sense as defined in [9], make it easier than ologs to represent complex concepts that may involve ‘before-and-after’ (e.g. temporal) relations among their components. For example, if p denotes a person, s a coffee shop, and c a cup of coffee, then the concept of ‘a person buying a cup of coffee’ (or simply ‘buying coffee’) can be represented by the wiring diagram Date: November 26, 2025.
This wiring diagram conveys the idea, that in order for the concept of ‘buying coffee’ to occur, all these four events must occur:
• A: p enters a coffee shop; • B: p pays for the coffee; • C: p receives the coffee; • D: p leaves the coffee shop.
The arrows in the wiring diagram represent before-andafter relations, so the events A, B, C, D are required to satisfy these order relations:
• A must occur before B as well as C;
• both B and C must occur before D.
Overall, the concept ‘buying coffee’ is considered to have occurred if all the individual events A, B, C, D have occurred, and all the order relations above are satisfied.
In practice, one can think of a wiring diagram as a directed acyclic graph where the vertex labels correspond to sensor readings [9]. The sensor could be a simple, physical sensor such as a thermometer, in which case an associated label in a wiring diagram could be ’the ambient temperature reaches 30 degrees Celsius’. The sensor could also be a complex, non-physical sensor such as a software detecting anomalies in online user behaviors, in which case an associated label in a wiring diagram could be ‘customer X makes a highly atypical financial transaction’.
Ologs themselves can already be used to perform basic deductive reasoning [13]. Since wiring diagrams are extensions of ologs (in the sense that the label of each node in a wiring diagram must come from an olog), wiring diagrams can be used to represent more complicated types of reasoning, such as analogical reasoning and problem solving [9].
When it comes to using wiring diagrams to perform reasoning within an autonomous system, two fundamental problems arise:
• Problem I: How does the autonomous system translate sensor data to wiring diagrams? • Problem II: How does the autonomous system perform general reasoning by manipulating wiring diagrams?
Problem I is part of the broader problem of, how does an autonomous system understand its environment by defining concepts on its own, based on data collected through its sensors? In our context, this means the following: suppose an autonomous system makes multiple observations of a human buying coffee from a coffee shop; how then does the autonomous system form the concept of ‘buying coffee’ using the language of wiring diagrams? In cognitive psychology, for example, Schank and Abelson suggested that children learn to grasp abstract concepts such as ’eating at a restaurant’ by experiencing it multiple times and then forming script-like structures associated to the concept [12]. In this paper, we attempt to realize this process by designing algorithms that can be implemented in autonomous systems.
In designing such algorithms, however, one faces the following mathematical problems: What are the wiring diagrams we are dealing with, and how can we find all of them? That is, how do we characterize the wiring diagrams that are relevant to constructing our algorithms, and how do we enumerate all such wiring diagrams? We answer these questions in Theorem 3.7 by showing that the underlying graphs of the wiring diagrams that we work with -which we call quasi-skeleton wiring diagram graphs -correspond to Hasse diagrams in graph theory, the enumeration problem of which is already well-known [4].
Once we understood the space of wiring diagrams we are dealing with, we were able to design algorithms that extract wiring diagrams from data. Our main algorithm, Algorithm 6.7, takes as input a collection of sequences representing multiple observations, and produces as output collections of
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