When I first saw OpenAI’s plan for superalignment, I consider it as an impossible goal based on theory and practices.

After several days investigation, I found several possible ways to approximate the superalignment (but not a complete solution).

I asked five questions for that.

1. What do the maths look like under superalignment?
2. What do the logics look like under superalignment?
3. What do the game look like under superalignment?
4. What do the society look like under superalignment?
5. What do the law look like under superalignment?
6. What do the economics look like under superalignment?

PART I Maths

1. Here is an outline of a more rigorous research agenda for developing superalignment theory using algebraic topology:
   • Formalize notions of alignment between AI objective functions and human values using topological similarity metrics. Can we define quantitative alignment measures grounded in topological invariants?
   • Characterize the space of possible human value functions and proxy functions using algebraic topology. Identify key features like dimensionality, curvature, homology groups, etc.
   • Develop techniques to iteratively update proxy functions to better approximate the human value function space based on experience. Use topological metrics to assess convergence and alignment.
   • Study the optimization landscapes induced by different classes of objective functions and learning algorithms using Morse theory. Identify topological obstacles to alignment. Design new algorithms with better topological properties.
   • Use persistent homology and multilayer networks to understand how proxy functions need to adapt as AI systems gain capabilities. Ensure robust trajectories that maintain alignment.
   • Leverage homotopy theory to prove fundamental constraints on aligning advanced AI systems with complex human values. Identify upper bounds on capabilities for provable alignment.
   • Analyze game theoretic dynamics between AI agents using algebraic topology on strategy spaces. Design incentives and mechanisms for cooperation and avoiding adversaries.
   • Synthesize insights from theoretical alignment research into practical training techniques and algorithms that can be implemented in real-world systems. Focus on scalable methods with topological guarantees.
   • This agenda covers both theoretical core research on using topology to formally characterize alignment as well as guidance for engineering practical AI systems that can learn values robustly. The goal is to make superalignment mathematically rigorous while also applicable to real-world problems.

References:

1. Formally define a metric for alignment based on topological similarity of proxy and human value functions. Refer to metrics on function spaces from topological data analysis (e.g. Carlsson 2009).
2. Characterize the topology of human value function space. Refer to results on topological properties of neural networks and other function classes (Bianchini et al 2014).
3. Prove guarantees on convergence of proxy functions using results on topological persistence and stability (Cohen-Steiner et al 2007).
4. Use Morse theory to analyze optimization landscapes of objective functions (Milnor 1963). Identify connections to gradient descent convergence.
5. Apply constructions from multi-scale topological persistence across AI capability gains (Carlsson & Zomorodian 2009). Track alignment over time.
6. Leverage fundamental groups and covering space theory to constrain possible alignments (Hatcher 2002). Identify capability limits for provable alignment.
7. Model multi-agent game dynamics using agent strategy spaces with topological constructions (Krantz et al 2019).
8. Translate theoretical insights into practical algorithms using topological data analysis libraries like GUDHI (Carriere et al 2015).

PART II

Here is a high-level overview of how logical formalisms could be used to represent and reason about superalignment:

Propositional Logic:

• Represent key properties of AI systems, human values, capabilities etc. as propositions with Boolean truth values
• Encode alignment between AI objectives and human values as logical relations between propositions
• Use logical connectives and truth tables to infer alignment from other properties
• Prove alignment theorems through propositional logic proofs

First-Order Logic:

• Represent objectives, values, capabilities as predicates over variables with quantified domains
• State alignment as a relation between predicates binding variables appropriately
• Leverage first-order logic rules of inference to derive alignment properties
• Prove more complex alignment theorems using first-order proof systems

Modal Logic:

• Model different possible capabilities and outcomes as alternative worlds/models
• Frame alignment statements using modal operators like necessity, possibility, obligation
• Develop possible worlds models that guarantee alignment by construction
• Use modal logic axioms and theorems to reason about alignment across capabilities

Higher-Order Logic:

• Represent objectives, values and capabilities as higher-order functions or relations
• Express alignment as relational correspondence between composed functions
• Manipulate alignment relations using lambda calculus and function abstraction
• Prove powerful theorems about alignment in expressive higher-order systems

This provides a starting point for leveraging logical languages and proof systems to make alignment arguments more precise and rigorous. There are many open research questions in mapping superalignment concepts into formal logics.