"How do you eat an elephant? One bite at a time," goes the proverb that dates back to the early 1920s. Turns out the same idea may apply to advancing artificial intelligence (AI), according to a team from the Institute for Interdisciplinary Information Sciences (IIIS) at Tsinghua University.
The researchers developed a framework they call the "calculus of intelligence," or COIN, capable of breaking down the increasingly complex problems agentic AI — the systems expected to operate with minimal human oversight to do things like monitor for cybersecurity breaches, write code and much more — is tasked with solving. They published their approach on July 17 in iFuture .
"The calculus of intelligence is a mathematical framework for decomposing a complex task into smaller, well-defined subtasks that are simple enough to solve, and then combining their solutions into a coherent whole," said Yang Yuan, associate professor and corresponding author on the paper. "Just as classical calculus can calculate the area under a complicated curve by dividing it into tiny pieces and adding them up, this framework provides a way to build and understand complex systems through the step-by-step composition of numerous simple components."
The framework translates splices up the overarching AI goal into smaller tasks within a logic system called a Grothendieck topos, which Yuan described as a mathematical rulebook for facilitating the coordination of building something as complex as a large airplane.
"It must be divided into many local components — such as the wings, engines and control systems — which are designed by different teams," Yuan said, explaining that these limited perspectives are the local views, with each team only seeing the information relevant to its own task rather than every detail of the entire airplane, while still retaining the shared information needed to connect its component to the rest of the system. "This allows different teams to work independently while ensuring that the components they produce match at their shared boundaries."
The Grothendieck topos expresses the structure of local design, shared interfaces and global coordination in mathematical language. It specifies what each local component can see, what requirements it must meet and how different components must agree where they overlap. According to Yuan, COIN takes the next step by providing the rules for decomposing tasks and reliably recomposing local solutions that comply with the larger system's rulebook.
"Intelligence resides in structure, and structure can be decomposed, learned and recompose," Yuan said, noting that complexity does not mean incomprehensibility. "Many systems that appear overwhelmingly complex may simply be waiting for the right decomposition. A good structure can turn a difficult global problem into a collection of clearly bounded local problems that can be analyzed step by step. Through appropriate structural decomposition, many complex tasks can become easier to understand, execute and verify."
This structure can also lend itself to making extremely large systems with limited intelligence, Yuan pointed out.
"The truly transformative future may not be an infinitely powerful individual intelligence — how some may imagine AI — but rather humans using many limited intelligences to construct systems far beyond the scale that any single person or model could independently understand or complete," Yuan said.
According to Yuan, COIN is a step in that direction.
"The broader goal is to develop a common mathematical language for intelligence: One that can describe what models learn, how complex tasks are decomposed and how many limited intelligences can be organized into larger systems," Yuan said.
Andrew Chi-Chih Yao, professor and dean of IIIS, co-authored this paper. Yuan and Yao are also affiliated with the Shanghai QiZhi Institute.
About iFuture
iFuture is a premier open-access journal published by Tsinghua University Press on the SciOpen platform, with academic support from the Institute for Interdisciplinary Information Sciences at Tsinghua University. Led by Turing Award Laureate Prof. Andrew Chi-Chih Yao as Editor-in-Chief, the journal is the core component of the AI Open Alliance. Its core mission is to break through AI's theoretical bottlenecks and foundational infrastructure.
