Formal Models in AGI Research Pei Wang Temple University Philadelphia, USA

AGI Needs Formal Model A complete AGI work should consist of 1) a theory of intelligence, in a natural language 2) a formal model of the theory, in a symbolic language 3) a computer implementation of the model, in a programming language Formal models provide clarity and accuracy

Formal Model is not Everything An AGI work whose model has desired formal or mathematical properties is not necessarily superior to other AGI works, because The model may formalize an improper theory of intelligence It may fail to guide a computer implementation AGI is not mathematics, but has clear empirical content and engineering demand

Formal Model for AGI For AGI, A model must start from some idealized and simplified assumptions, but it does not mean all assumptions are equally valid A model with improper fundamental assumptions is unlikely to be useful A model's success in other domains does not guarantee its success here

Traditional Models In the current AI/AGI research, the major traditions of formal model are: Mathematical logic Theory of computation Probability theory But each of them has serious limitations when applied to AGI. This talk will address the first two

The Logic of Mathematics Mathematical logic was designed to formalize theorem-proving process, not thinking process in general Theorem proving is the process of deriving new truth from given truth (axioms), while reasoning outside mathematics usually can neither depend on true premises, nor deliver true conclusions Therefore, the logic of mathematics is not the logic of cognition, nor an idealization of the latter

Issues for Mathematical Logic Uncertainty: randomness, fuzziness, ignorance, inconsistency, etc. must be handled Ampliativity: induction, abduction, and analogy seems to say more in the conclusions Openness: new evidence may challenge the previous beliefs of the system Relevance: premises and conclusions cannot merely be related in truth-values, not in contents

Non-Classical Logics are not Enough Each of the issues has been addressed by some non- classical logic, it is not enough: The issues are addressed in isolation from each other The modifications and extensions typically happen in grammar rules, inference rules, or axioms, without touching the semantical foundation of the logic

Non-Axiomatic Logic: Assumption NAL assume the system has insufficient knowledge and resources, so Empirical theory of truth: The truth-value of a statement indicates the extent to which the statement agrees with the system's experience. Validity as evidence-preserving: An inference rule is valid if and only if its conclusion is supported by the evidence provided by its premises. NAL provides a unified solution to the issues

Theory of Computation: Origin Theory of computation (automata, algorithm, computability, computational complexity) was established to specify “computational” procedures in mathematics, which repeatably maps a problem instance to a predetermined solution. Outside mathematics, a problem-solving process often cannot be specified as such a mapping, since it depends on the past history and current context, which are usually not described as part of the problem instance.

Time in Computation Time is not a necessary part of a problem: a problem can appear in any moment, and usually it does not include a demand for response time Time is not a necessary part of a solution: whether a response is a “solution” has nothing to do with when it is produced, as long as it takes finite time In summary, “computation” is fundamentally time- independent, which is desired in mathematics, where procedures should be universally repeatable

Time in Adaptation An adaptive system evolves over time, and may never repeats its internal state. Time is a necessary part of a problem: the value of a solution decreases over time Time is a necessary part of a solution: whether a response is a “solution” has a lot to do with when it is produced In summary, “adaptation” is fundamentally time- dependent, where the problem-solving processes are usually not universally repeatable

Assumption of NARS NARS (Non-Axiomatic Reasoning System) is designed by treating “intelligence” as adaptation with insufficient knowledge and resources, i.e., the system is finite, real-time, and open. For each problem (instance), the system solves it using the available knowledge and resources at that moment. Since neither the internal state nor the external state repeat, the problem-solving process does repeat.

Conclusions Though AGI needs formal models, the traditional models do not meet its requirements, since they are mainly built for mathematics AGI needs formal models that are based on realistic assumptions.The system has to act according to available knowledge and resources Models assuming sufficient knowledge and resources do not even provide proper idealizations or simplification for AGI