A Mathematical Formulation of AGI in the (C, U, V) Framework

Di He; Cong Fang; Yisen Wang; Yujia Peng; Yizhou Wang; Song-Chun Zhu

doi:10.1016/j.eng.2025.08.034

Engineering ›› DOI: 10.1016/j.eng.2025.08.034

review-article

A Mathematical Formulation of AGI in the (C, U, V) Framework

Di He ^a^,^c
, Cong Fang ^a^,^c
, Yisen Wang ^a^,^c
, Yujia Peng ^b^,^c^,^d
, Yizhou Wang ^c^,^e
, Song-Chun Zhu ^a^,^c^,^d^,^*

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Abstract

This article seeks to explore a general formulation of artificial general intelligence (AGI) under a unified framework that defines AGI agents as points in a joint ( $C, U, V$ ) space. An agent is characterized by three components: ① a cognitive architecture $C,$ which represents the modules (mathematical functions) inside the agent’s mind, as well as the connections and communication protocols between these modules, including the theory of mind (ToM); ② a set of potential functions $U$ , which represents the skills of perception, cognition, and planning (e.g., a potential function can be a neural network trained for visual object recognition, or embodied motion planning); and ③ a set of value functions $V$ , which includes the agent’s urges, preferences, and social affections, as well as benefits for individual agents or a group of agents. In this setting, “intelligence” is defined as a wide range of phenomena exhibited by agents when they interact with complex environments (i.e., physical intelligence) and other agents (i.e., social intelligence). Given an initial point in the ( $C, U, V$ ) space, an agent can explore new $V$ -dimensions, which in turn drives the acquisition and learning of skills by enabling the learning of new potential functions $U$ in the environment and by updating the cognitive model. We have developed a Tong test as a benchmark and evaluation criteria: An agent that has reached the human level ( $C, U, V$ ) is called a “Tong Agent.” The convergence of this process defines the limits of the agent’s evolution; we name this the “stopping problem” of Tong Agents, based on the analogy of the halting problem in a Turing machine.

Keywords

Artificial general intelligence / Value alignment / Machine learning / Evolutionary learning

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Di He, Cong Fang, Yisen Wang, Yujia Peng, Yizhou Wang, Song-Chun Zhu. A Mathematical Formulation of AGI in the (C, U, V) Framework. Engineering DOI:10.1016/j.eng.2025.08.034

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References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	Silver D, Schrittwieser J, Simonyan K, Antonoglou I, Huang A, Guez A, et al.Mastering the game of go without human knowledge.Nature 2017; 550(7676):354-359.

[2]	Devlin J.Bert: pre-training of deep bidirectional transformers for language understanding.2018. arXiv: 1810.04805.

[3]	He K, Chen X, Xie S, Li Y, Dollár P, Girshick R. Masked autoencoders are scalable vision learners, IEEE, New Orleans, LA, USA. New York City (2022), pp. 16000-16009

[4]	Open AI, Achiam J, Adler S, Agarwal S, Ahmad L, Akkaya I, et al. GPT-4 technical report., arXiv:2303.08774 (2023)

[5]	Anil R, Borgeaud S, Alayrac JB, Yu J, Soricut R, Schalkwyk Jet al.Gemini: a family of highly capable multimodal models.2023. arXiv: 2312.11805.

[6]	Zhu Y, Gao T, Fan L, Huang S, Edmonds M, Liu H, et al.Dark, beyond deep: a paradigm shift to cognitive AI with humanlike common sense.Engineering 2020; 6(3):310-345.

[7]	Yuan L, Zhu SC.Communicative learning: a unified learning formalism.Engineering 2023; 25:77-100.

[8]	Lieto A, Bhatt M, Oltramari A, Vernon D.The role of cognitive architectures in general artificial intelligence.Cogn Syst Res 2018; 48:1-3.

[9]	Fan L, Xu M, Cao Z, Zhu Y, Zhu SC.Artificial social intelligence: a comparative and holistic view.CAAI Artif Intel Res 2022; 1:144-160.

[10]	Laird JE.The SOAR cognitive architecture. MIT Press, Cambridge (2019)

[11]	Anderson JR, Bothell D, Byrne MD, Douglass S, Lebiere C, Qin Y.An integrated theory of the mind.Psychol Rev 2004; 111(4):1036.

[12]	Sun R.The CLARION cognitive architecture: extending cognitive modeling to social simulation.R. Sun (Ed.), Cognition and multi-agent interaction: from cognitive modeling to social simulation, Cambridge University Press, Cambridge 2006; 79-99.

[13]	Terman LM, Merrill MA.Stanford–Binet Intelligence Scale: manual for the third revision, form lM. Houghton Mifflin Harcourt, Boston (1960)

[14]	Bayley N.Bayley-III: Bayley Scales of infant and toddler development. Giunti OS, Florence (2009)

[15]	Tomasello M.Origins of human communication. MIT Press, Cambridge (2010)

[16]	Holyoak KJ, Thagard P.Mental leaps: analogy in creative thought–. MIT Press, Cambridge (1996)

[17]	Lake BM, Salakhutdinov R, Tenenbaum JB.Human-level concept learning through probabilistic program induction.Science 2015; 350(6266):1332-1338.

[18]	Kaufman AS, Lichtenberger EO.Assessing adolescent and adult intelligence. (3rd ed.), Wiley, Hoboken (2006)

[19]	Boake C.From the Binet–Simon to the Wechsler–Bellevue: tracing the history of intelligence testing.J Clin Exp Neurpsychol 2002; 24(3):383-405.

[20]	Gardner H, Hatch T.Educational implications of the theory of multiple intelligences.Educ Res 1989; 18(8):4-10.

[21]	Cherniss C, Extein M, Goleman D, Weissberg RP.Emotional intelligence: what does the research really indicate?.Educ Psychol 2006; 41(4):239-245.

[22]	Raven JC, Court JH.Raven’s progressive matrices. Western Psychological Services, Los Angeles (1938)

[23]	Peng Y, Han J, Zhang Z, Fan L, Liu T, Qi S, et al.The Tong test: evaluating artificial general intelligence through dynamic embodied physical and social interactions.Engineering 2024; 34:12-22.