RFT (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Edwards et al., 2017b, 2022; Barnes-Holmes and Harte, 2022) specifies several different types of relational responding that are applicable to AARR, which include (but not limited to) (1) co-ordination (e.g., stimulus X is similar to or the same as stimulus Y); (2) distinction (e.g., stimulus X is different to or not the same as stimulus Y); (3) opposition (e.g., left is the opposite of right); (4) hierarchy (e.g., a human is a type of mammal); (5) causality (e.g., A causes B); and (6) deictic relations (also called perspective-taking relations), and include interpersonal (I vs. YOU), spatial (HERE vs. THERE), and temporal relations (NOW vs. THEN). Of these, deictic relations may be most applicable to AI alignment (though all relation types are important and connected within contextual dynamics), in the form of perspective-taking (I vs. You interpersonal relations) of human values, as these allow the human or the AI to take perspective about another human’s thoughts, feelings, values, etc.

The RFT model (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Edwards et al., 2017b, 2022; Barnes-Holmes and Harte, 2022) also specifies three essential properties of the relational frame, which include (1) Mutual entailment (ME), which is when the relating to one stimulus entails the relating to a second stimulus, e.g., if stimulus X = stimulus Y, then stimulus Y = stimulus X is derived through mutual entailment (i.e., the reversal curse of AI implies a limitation in this area). (2) Combinatorial entailment (CE) extends the mutual entailment to include three or more stimuli. Relating a first stimulus to a second and then relating this second stimulus to a third, facilitates entailment not just to the first and second, and not just to the second and third, but also to the first and third stimuli. (3) Transfer (or transformation) of stimulus function (ToF) is where functions of any stimulus may be transformed in line with the relations that the stimulus shares with such as other stimuli relations connected within the network of frames. For example, if you knew that pressing button A give you an electric shock that you became fearful of, and then the experimenter said that “B is greater than A,” you may become even more fearful of pressing button B as this stimulus which included a previously neural function has now changed to one that is based on fear (or greater fear than pressing button A). There is no evidence that AI currently can experience fear consciously, but their ability to perspective-take human values (thus overcoming the alignment problem) should imply that they should have the ability to ToF within complex relational frame networks at least logically (or conceptually).

A specific example of the difference between an RFT approach and a cognitive one (and where RFT can improve on the cognitive approach by providing a broader contextual description) can be explored explicitly through Chomsky’s hierarchy (Chomsky, 1956). RFT can extend this hierarchical grammar in a contextual way, allowing greater contextual sensitivity, which is important for AI alignment. It can do this by allowing the expressions of derived relations as mathematical notation (see Supplementary material 8 for full arguments), which are crucial in a contextually bound RFT LLM model such as expressing deictic perspective-taking comparisons of self and other. For example, a set of known relations can be denoted as R , and each relation in R as r i   ϵ   R is a tuple x , y , r e l , and expressed as r i   ϵ   R = x , y , r e l , whereby x and y are separate stimuli and r e l is the relation between them (e.g., “greater than” or “less than”), which allows for relational production rules and for relational frames to emerge.

The “ d e r i v e _ r e l a t i o n ” function can then be defined as follows: (1) For any two stimuli a and b , if ∃ r = a , b , r e l   ϵ   R , return r e l ; (2) Otherwise, if ∃ r = b , a , r e l   ϵ   R , return the opposite of r e l (i.e., if r e l = " g r e a t e r   t h a n " , then return   " l e s s   t h a n " , and vice versa); (3) Otherwise, for any stimulus c in the set of stimuli involved with the relations in set R , if ∃ r 1 = a , c , r e l 1   ϵ   R and ∃ r 2 = c , b , r e l 2   ϵ   R , and r e l 1 = r e l 2 , return the result of d e r i v e _ r e l a t i o n a , c ; and then (4) If none of the above conditions are met, return “cannot be determined.” The print statements for instance “ d e r i v e _ r e l a t i o n a , b ” prints the directly learned relation or derived relation between stimulus a and stimulus b . This provides a high-level mathematical representation of the logic of a basic derived relation (AARR) and can be implemented as Python code presented in Figure 6A (and a corresponding visualization of the derived relation output can be seen in Figure 6B using Python’s matplotlib library). See Supplementary material 9 for additional commentary about the Python-derived relation code.

In another example, a transformation of stimulus function (ToF) can be represented in mathematical form (with corresponding Python code¹) using set theory and logic in the following way: Let S = snake,woods be a set of stimuli and F = fear,neutral be a set of (emotional) functions. Two mappings can then be defined: (1) The function C f u n c : S → F defined as C f u n c snake = f e a r and C f u n c woods = neutral ; (2) The relation R ⊆ S × S defined as R = w o o d s , s n a k e . The transformation of function based on a specific contextual relation C r e l can then be described as: For any stimuli s 1 , s 2   ϵ   S , if s 1 , s 2   ϵ   R and C r e l = c o n t a i n s , then updates the function of s 2 to be the same as the function of s 1 , i.e., C f u n c s 2 = C f u n c s 1 . This mathematical notation and corresponding Python script therefore leads to the ToF C f u n c woods = C f u n c s n a k e = fear . This uses predicate logic, which deals with variables and predicates (functions that return true or false values), and leverages set theory and function mapping in order to conclude that the previously neutral stimulus woods, now has transformed into a fear function (the AI knows that fear is associated with woods). This means that the AI now understands that the person it is communicating with in now afraid of the woods given some context—i.e., it has correctly perspective-taken human emotion, and this ability is essential for aligning to human values.

Now that derived relations and ToF have been defined, the self, expressed within deictic frames of RFT can now be further defined, which could allow for perspective-taking skills to promote compassion of others within AI (values alignment), thus helping to solve the alignment problem. Perspective-taking deictics in RFT revolve around how we relate to ourselves, others, and the world around us based on the perspective we adopt. When formalizing this concept mathematically, we can represent these deictics (Interpersonal I vs. YOU, Spatial HERE vs. THERE, and Temporal NOW vs. THEN) as relations between sets that capture the interplay between these different perspectives.

Here, a possible logical representation can be given, whereby first a series of sets are defined: P i n t e r p e r s o n a l = I , Y O U , P s p a t i a l = H E R E , T H E N , P t e m p o r a l = N o w , T H E N , whereby P reflects the perspective of the observer (on the dimensions of interpersonal, spatial, or temporal properties). We can also define relations to capture the change in perspective within each dimensional category: (1) R i n t e r p e r s o n a l : P i n t e r p e r s o n a l → P i n t e r p e r s o n a l   such that R i n t e r p e r s o n a l I = Y O U or R i n t e r p e r s o n a l Y O U = I ; (2) R s p a t i a l : P s p a t i a l → P s p a t i a l such that R s p a t i a l H E R E = T H E R E or R s p a t i a l T H E R E = H E R E ; and (3) R t e m p o r a l : P t e m p o r a l → P t e m p o r a l such that R t e m p o r a l N O W = T H E N , or R t e m p o r a l T H E N = N O W . These relations represent the shift in perspective, for instance, the relation R i n t e r p e r s o n a l is a function that captures the change from an “I” perspective (perspectives about the self, such as my feelings, my thoughts, and my values) to a “YOU” perspective (perspectives about another human, such as your feelings, your thoughts, and your values), and vice versa. The relation R i n t e r p e r s o n a l is a function that takes an element from the set P i n t e r p e r s o n a l and maps (via relational frames) it to another element in the set P i n t e r p e r s o n a l . The arrow → denotes the direction of the function mapping from the domain to the co-domain. More simply, for any element in the set P i n t e r p e r s o n a l (I or YOU), the function R i n t e r p e r s o n a l shows which elements it relates to in the context of a defined relation. So, these can be defined within a contextual C r e l and functional contextual C f u n c way as typically defined in RFT (Cullinan and Vitale, 2009; Edwards, 2021; Edwards et al., 2022).

In an example of an AI A (or this could be a model for a human too) perceptive-taking about the emotional pain of person B that the AI is interacting with, as a first stage to stimulate compassion or values alignment requires the following steps: (1) Here, understanding the worldview w (or perspective) of person B , a new set needs to be introduced in terms of a set of possible emotional states S , whereby S = p a i n , j o y , n e u t r a l , for example. Then some function S f maps from the interpersonal set P i n t e r p e r s o n a l to the emotional state set S which will capture what emotion (state s ) [or these could be values such as (kindness, helpfulness, patience, etc.) for values alignment] each person is experiencing or perceiving, denoted by S f : P i n t e r p e r s o n a l → S when given S f I = n e u t r a l and S f Y O U = p a i n . This represents AI A (represented by “I”) is currently feeling neutral (the AI does not need to actually feel anything, it can just map this as a logical expression of its own state space), and Person B (represented by “YOU”) is in pain. When perspective-taking, there is an interest in AI A seeing the pain in Person B . This can be represented by a new function, I s e e which maps from the AI’s perspective to what it perceives in Person B (in this example, their emotional state or this could equally be their direct values), denoted as I s e e : P i n t e r p e r s o n a l × P i n t e r p e r s o n a l → S when given I s e e I , Y O U = E f Y O U = p a i n . This indicates that AI A (“I”) sees (or has some internal representation mapping) that Person B (“YOU”) is in pain. Specifically, the statement “AI A sees the pain in Person B ” is captured by the function I s e e I , Y O U , which returns information about the Person B ’s pain. The symbol × represents the Cartesian product of two sets. Given two sets A and B , the Cartesian product A × B is the set of all ordered pairs a , b where a is an element of A and b is an element of B . So, the Cartesian product P i n t e r p e r s o n a l × P i n t e r p e r s o n a l allows the function I s e e to consider the relation between two distinct individuals (in this case A and B ) from the AI’s interpersonal perspective and then produce an emotional state s representation mapping outcome based on that relation (see sample Python code on GitHub² for expressing the perspective-taking of pain as given in this example).

A ToF may also occur through this perspective-taking process (see sample Python code on GitHub³), whereby AI A starts to map some representation of pain (this is a logical representation mapping in some mathematical state space S rather than a phenomenological one) that person B experiences, which may encourage empathy (and values alignment) in humans who are consciously aware. Mathematically, this could be stated using first-order logic and set theory, in the following way: Consider a set of persons P = p 1 , p 2 which represents two persons, p 1 and , p 2 with a set of possible emotional states S = p a i n , j o y , n e u t r a l , and a set of time points T = t 1 , t 2 which represent time point 1 and point 2. For functional emotional states,   S i n i t i a l : P → S , defined as   S i n i t i a l A I A = n e u t r a l , and S i n i t i a l P e r s o n B = p a i n . For perspective-taking transformations, when given two persons p 1 (AI can also be represented as p 1 for simplicity) and , p 2 from set P , if p 1 takes the perspective of p 2 at a specific time point from set T , the emotional state s of p 1 (again, the AI does not have an emotional state, rather this is a logical representation mapping in some mathematical state space S rather than a phenomenological one) will transform to temporarily match that of p 2 (i.e., as p 1 sees through the eyes of p 2 they are more able to connect to the pain (or this could equally be values) that p 2 is experiencing, thus may share temporarily that feeling of pain as a mathematical state space S mapping). Mathematically, the transformation of function based on this perspective-taking process can be denoted as: ∀ p 1 , p 2 ∈ P , t ∈ T : S a f t e r   p e r s e p t i v e − t a k i n g p 1 , t = S i n i t i a l p 2 if p 1 (the AI) takes perspective of p 2 (the human it is engaging with) at time t . For example, take an initial state   S i n i t i a l p 1 = n e u t r a l , the after perspective-taking at time point t 1 , S a f t e r   p e r s e p t i v e − t a k i n g p 1 , t 1 = S i n i t i a l p 2 = p a i n . Thus, this demonstrates the ToF process of emotional state (or mathematical state space S mapping) of p 1 transforms from “neutral” to “pain” after taking the perspective of p 2 ’s pain at time point   t 1 .

The mathematical approach defined above uses first-order logic (also known as first-order predicate calculus). This is evident from the usage of quantifiers such as ∀ (which stands for “for all”) and the use of functions and relations to express properties and relations of individuals. To break it down, the use of the universal quantifier ∀ indicates that the logic being used is at least first-order. A statement is being made that applies to “all” elements in a given set, which is a feature of first-order logic. Then predicates ae utilized when defining the functions, such as   S i n i t i a l p 1 = n e u t r a l , which can be read as “The initial emotional state (or mathematical state space S mapping) of AI p 1 is neutral.” Variables such as state S that change in value and quality, such as emotional state at different time points t , and constants such as p 1 and p 2 that are constant as they refer to individual people or AI entities. Functions are used such as   S i n i t i a l and S a f t e r   p e r s e p t i v e − t a k i n g as they assign an emotional state (or mathematical state space S mapping) to a specific person or AI at time points “initial” and “after.” These functions provide a mapping from each person or AI in set P to an emotional or values state (or mathematical state space S mapping) in set S at time point t . This account allows the AI to directly understand the human’s emotional state and values at any given moment consistent with the functional contextual RFT interpretation, which should allow and help the AI to align its own (ACT-based) utility E U A and (ACT-based) values function A V s , a (as already defined) with what it perspective-takes about human emotion and values given some functional context.

Supplementary material 10 provides a full description and advantages of how this functional-contextual RFT perspective-taking, values, and neuro-symbolic (PVNS) module LLM architecture could be pragmatically incorporated within an LLM architecture via a neuro-symbolic module. See Figure 7 for an illustration of the neuro-symbolic LLM architecture. See also Supplementary material 11 for further discussions on additional AI elements such as in the area of diplomacy (Meta’s Cicero LLM), which could also be included in such a neuro-symbolic framework. Also, see Supplementary material 12, for how evolutionary theory can classically optimize this type of LLM architecture.

It is important to note that all the innovative LLM implementations described here can be tested in terms of how effective they are at improving AI human-value alignment, such as by observing improvements in the AI’s ability to derive relations in the reversal curse problem (Berglund et al., 2023), as well as qualitative reports from users about how safe they feel around AI under different contexts, and whether they feel that the AI understands what they value and feel (the direct level of understanding and compassion users feel when interacting with the AI). Direct network graphs of the AI’s derived relationships, including perspective taking can also be visualized such as in Figure 6B through Python tools, such as matplotlib. These types of visualization can be important as they allow researchers to inspect directly how the AI is implementing the functional contextual algorithms within its knowledge base (Edwards et al., 2017a; Chen and Edwards, 2020; Szafir et al., 2023). However, one limitation is that AI consciousness, or a test for this is not defined in the current perspective-taking model, instead, this is defined completely algorithmically. So, emotions and values when perspective-taking are represented as mathematical state space S mappings. However, this limitation may be overcome through recent advances in our physics models, as through an observer-centric approach, which may allow for a test for consciousness.

To formally define the construction of complex relational frames at the computational level in the context of RFT using logic and set theory, we can express the relational frames and their combinations using logical connectives and set operations, represented in logic and set theory. To refine a logical and set-theoretical framework for the concept of “I see you,” ToM perspective-taking, that particularly emphasizes how relational frames network to form a perspective-taking node, we need to incorporate the connectivity and dependencies among the basic relational frames such as coordination, temporal, spatial and interpersonal (as illustratively depicted in Figure 8). We will then integrate and enhance the initial formulation to illustrate how complex cognitive functions emerge such as perspective-taking (ToM) from simpler relational operations.

Definitions of basic relational frames include entities, concepts or objects such as A representing Person A (which represents the relational deictic concept “I“), and B representing Person B (“YOU”). Examples of these basic relational frames that describe how these objects (or concepts) relate to one another include coordination (C), whereby C(A,B) implies A is similar or equivalent to B in some context; distinction (D), whereby D(A,B)indicates A is distinct from B; also temporal T and spatial relations S.

Constructing the relational deictic “I see you” concept of perspective-taking (ToM) in RFT can be modeled as a higher-order processed-based cognitive network arising from the integration of several of these basic relational frames combined (e.g., coordination, deictics, etc.). This integration can be described mathematically using logical conjunctions ( ∧ , which represents the concept “and”) and possibly other logical operators depending on the complexity required within the hypergraph network. Logical expression for perspective-taking event “I see you” involves recognizing the other (person B ) as similar (similarity relation or coordination) yet distinct (distinction relation) as you and situating this recognition within some cognitive context (e.g., time, space). Relational frames can then be expressed as coordination C that defines equivalence or similarity between concepts, whereby C x , y implies stimuli (or concept) x is coordinated (similarity) with y . Distinction D defines differentiation between concepts D x , y implies x is distinct from y . Temporal relations T defines differences or similarities in time between concepts, for example T x , t 1 , y , t 2 implies concept x at time t 1 is related in some way (either more are less similar temporally) to concept y at t 2 . Spatial relations S defines spatial relationships between concepts, for example, S x , p 1 , y , p 2 implies x at position p 1 is related spatially to y at position p 2 . Deictic relations P involves perspectives P x , y , which implies x perceives y (or person A perceives person B ).

Using these relational frames, we can describe the complex concept “I see you” i.e., perspective-taking ToM. For example, perspective-taking such as feeling someone’s pain (that would be important for AI to develop compassion as a human does), may involve C A , B , which reflects the relation coordination, and therefore places persons A and B in the same or similar context; D A , B also allows for a distinction between persons A and B , recognizing differences between these people such as historically reinforcing contingencies; and P A , B refers to person A perceiving person B via deictic frames. As these frames combine to form P ( A , B ) ∧ C ( A , B ) ∧ D ( A , B ) the “I see you” perspective taking ToM can be constructed hierarchy (as illustrated in Figure 8). These allow for specific ToM perspectives, such as C ( p A , p B ), which relates “my perspective” p A to “your perspective” p B , and this should allow for compassion to emerge at the computational level, as it does in humans.

This should also involve differing “my perspective” for “your perspective” to help understand different points of view and is denoted as the distinction relation D ( p A , p B ). When perspective-taking via ToM, sometimes it is useful to understand what someone has experienced historically, such as past traumatic events where pain (and therefore avoidant behavior) may have originated from, which can be denoted as T p A , t 1 , p B , t 2 and represents taking perspectives over time. When perspective-taking spatial concepts and relations may also be important to put the information into a spatial context, such as if the person you were perspective-taking about was in a certain place where trauma took location l o c 1 , and that returning to this area may trigger painful memories, this can be denoted as S p A , l o c 1 , p B , l o c 2 which represents perspectives over space. Therefore, the “I see you” perspective-taking ToM may combinatorally involve complex combinations of frames such as I _ S e e ( A , B ) ≡ C ( p A , p B ) ∧ D ( p A , p B ) ∧ T ( p A , l o c 1 , p B , l o c 2 ) ∧ S ( p A , l o c 1 , p B , l o c 2 ) , where I _ S e e A , B is the complex process-based cognitive function of perspective-taking. P A , B is derived from integrating C and D under certain cognitive processes, suggesting a direct perceptual relation, which could be modeled as P being influenced by C and D but not strictly defined as a simple relational frame. For instance, the perceptive-taking cognitive function might be influenced by deictic contextual factors (temporal or spatial), described by T and S . So, here P is not just seeing the other person, but instead understanding through contextualizing A ′ s relationship to B through the lenses of time and space (and any other relational frames combined into the network).

This gives a complete relational frame dynamic and contextual process network of perspective-taking that forms ToM as modeled in humans. This can then be modeled via hypergraphs of graph theory as a direct test of perspective-taking ToM in AI at the computational level. A hypergraph can be defined mathematically as H = V , E , whereby V is a set of vertices, E is a set of hyperedges, where each hyperedge e ⊆ V and can include any number of vertices. The I _ S e e A , B perspective-taking ToM within AI could be visualized where the hypothesis for these ToM processes within AI would formally state: “ I _ S e e A , B perspective-taking ToM within AI will be observed within the outputted hypergraph relational networks of the AI.” As a hypergraph via graph theory, nodes can be connected by edges that represent C , D , and P . Each of these edges feeds into the I _ S e e A , B node emphasizing how perspective-taking emerges from the interplay of these relational frames. This logical framework provides a structured and theoretical foundation to analyze visually and test an AI for the ability to construct the required complex cognitive functions like perspective-taking explained by RFT and N-Frame, in order for ToM to become emergent in AI at the computational level. This highlights the integrative role of basic relational frames in constructing higher-order cognitive processes, and this can be mapped graphically such as shown illustratively in Figure 9.

Further to this, more complex descriptions can be considered by extending graph theory with category theory (Awodey, 2010; Leinster, 2014; Spivak, 2014; Riehl, 2017). In category theory, these relationships can be visualized whereby the edges depicting relational frames represent morphisms between objects (concepts). Each morphism carries a label that specifies the relational frame (e.g., coordination, distinction, and spatial). The advantage of category theory is that it can mathematically model combined higher dimensional (or higher order) categories as depicted in Figures 8, 9, that are required to form “I see” perspective-taking ToM which in RFT and N-Frame are specified as derived relations and in category theory are mathematically defined as morphisms between morphisms. For example, a two-category representation can have objects, morphism between morphisms, and two-morphisms between morphisms, which is akin to face edges and vertices in a more complex polyhedral representation. In Figure 10, these two-category relations (shown as higher-order relations; HoR) can be shown within the hypergraph whereby the derived relations between Person   A → Fear → Woods forms to allow for a transformation of function (ToF) of fear to woods to occur within the graph and described precisely mathematically.

Category theory (Awodey, 2010; Leinster, 2014; Spivak, 2014; Riehl, 2017) can be integrated into hypergraphs by defining categories where objects are different features, states or components of the data (such as a chair, the woods, or a snake), and morphisms represent transformations, relationships or dependencies between these objects (such as relational frames). Morphisms can represent simple relations or complex ToF involving observer-dependent interpretations (ToM perspective-taking). Via an observer-centric approach, category theory models the observer using a functor F that maps observed objects and morphism data (relational frames between objects) into a hypergraph structure from category C to category D based on the observer’s point of view (i.e., their ToM perspective-taking). Mathematically object mapping for object x in C to D can be denoted as F x in D . For each morphism f : x → y in C , there is equivalent corresponding morphism F f : F x → F y in D . These mappings must satisfy two main conditions to ensure they preserve the categorical structure: (1) They must preserve conservation, i.e., for any two morphisms f : x → y and g : y → z in C , the functor F must satisfy F ( g ο f ) = F ( g ) ο F ( f ) , which means that the functor F respects the composition of the morphisms (2) there needs to be preservation of identity morphisms whereby for every object x in C , the functor F must satisfy F i d x = i d f x , which means that the functor F maps the identity morphism of an object x in C to the identity morphism of the object F x in D . This allows the hypergraphs to be visualized in other ways such as a bipartite graph or other visualization while preserving the structure of the RFT hypergraph.

As an example of this functor F preservation, differential topology and differential geometry (Donaldson, 1987; Genauer, 2012; Grady and Pavlov, 2021) can be used to model and visualize cobordism in topology of the RFT hypergraphs, which can provide an interesting way to describe the relationship between two clusters in a perspective-taking of relational frames (as depicted in Figure 10 in orange). In this context, the two clusters (or “manifolds”) represent distinct sets of relational frames or cognitive perspectives from person A to person B , and the connections (or “cobordism”) between them can illustrate how these perspectives are interconnected and can transform into one another. In mathematical terms, particularly in topology and higher-level category theory (Lurie, 2008; Feshbach and Voronov, 2011; Schommer-Pries and Christopher, 2014), a cobordism refers to a relationship between two manifolds (Reinhart, 1963; Laures, 2000; Genauer, 2012). The concept initially arises in topology but is enriched by categorical frameworks, which abstractly express many mathematical ideas, including cobordism. In topology, a cobordism between two n-dimensional manifolds M and N is an ( n + 1 )-dimensional manifold W such that the boundary of W is the disjoint union of M and N (usually denoted as ∂ W = M ∪ N ). Essentially, W provides a sort of “bridge” connecting M and N , showing how one can be continuously transformed into the other, which gives some unique and deep mathematical insights into how the functional processes of ToF occur geometrically via differential geometry.

This category theory interpretation also has advantages over more rigid forms of mathematics such as set theory, as the concept of a boundary in RFT or N-Frame (Edwards, 2023) relations might not apply in the traditional sense. Sets are collections of elements, and while one might discuss the boundaries of a set in terms of its limits or borders defined by some criteria, this may be more metaphorical than physical within RFT or N-Frame (Edwards, 2023) hypergraphs. In category theory, objects do not usually have “boundaries” in a physical sense. Objects in a category can be anything from sets, spaces, groups, or any entity depending on the category’s definition, which is more consistent with RFT or N-Frame (Edwards, 2023) assumptions as it can model complex concepts that have ill-defined boundaries such as “democracy” or “human-like.” Morphisms in category theory can represent relationships or functions between these objects (or concepts) with more fuzzy ill-defined boundaries, so the concept of a strict boundary as described by set theory does not directly apply to objects in this context.

To develop a hypergraph using category theory, we must first define a functor F from a category C (i.e., the concepts and its inherent relational frames) to a category D (the hypergraph representations of person A observing or perspective-taking person B ). Libraries such as networkx in Python for graph-based operations, can be useful in developing hypergraphs. For step 1, we can define the categories and objects as follows: Let C be a category where objects x , y , and z are various types of concept representations, such as snakes, person A , person B , fear, danger, etc., associated with snakes. Morphisms in C represent relational frame processes applied to these object concepts, such as spatial, temporal, coordination, etc.

In order to construct a hypergraph category, let H be a category where objects are nodes within a hypergraph and morphisms are relational frame mappings between these object nodes that preserve certain properties (like connectivity or certain hypergraph invariants). At step 1, first, we must define a functor F : C → H that represents the transformation from data objects (concepts such as snake, dangerous, etc.) in C to hypergraphs in H . This functor is parameterized by observer inputs (perspectives), which determine how data features are grouped into hyperedges. At step 2, the observer parameterization of F (observer inputs) needs to be defined as the functor F and as influenced by observer parameters (or inputs) O that emphasize the observer’s own experiential knowledge, beliefs, preferences, priorities, goals, values, or any other contextual information such as historical, cultural, and environmental factors that the observer brings to the hypergraph when perspective-taking, so F : C → H becomes F O : C → H .

At step 3, a hypergraph is constructed for each object x (these are concepts such as snake, Person A , Person B , dangerous, poisonous, woods, etc.) in category C , F O then maps this to a new category in the form of a hypergraph H = F O x . The vertices (nodes) of H are derived from the features of x , and the hyperedges are defined based on the relationships (parameterized by observer inputs O ) among these features (these are relational frames such as coordination, hierarchy, etc.). At step 4, a mathematical representation of a hypergraph can be formalized where a hypergraph H is defined as H = V , E , where V is a set of vertices and E is a set of hyperedges, where each hyperedge e ∈ E is a subset of V .

Topos theory (Fourman, 1977; Scott, 1982; de Araujo Fernandes and Haeusler, 2009) can also be useful here, as it can extend category theory by providing a categorical analysis of logic and set theory, extending set theory and logic to a broader category theory context, allowing for a rich interplay between geometry, algebra, and logic. A topos is a type of category that behaves much like the category of sets and functions but with its own internal logic and structure. This perspective allows for a deep exploration of logic and set theory within a categorical framework. We can also incorporate topos theory into the development of RFT N-Frame hypergraphs using category theory, where topos theory can offer deep insights into the logical and set-theoretical behaviors within the RFT N-Frame categories involved, especially in contexts where data and observations are fundamentally connected to conceptual and mathematical structures.

In topos theory (Fourman, 1977; Scott, 1982; de Araujo Fernandes and Haeusler, 2009), a bundle or sheaf can be understood in terms of its role in categorizing mathematical structures, which often involves the notions of continuity and localization. A sheaf is an object that generalizes the notion of a sheaf in a topological space to other contexts that can be structured similarly to topological spaces. Typically, a sheaf is a functor from a category that represents a space of “open sets” (often formalized as a site) to a category of “values” (like sets, groups, or vector spaces), satisfying certain conditions related to locality and gluing. In the context of RFT and N-Frame, the “open sets” could be thought of as contexts or environments in which stimuli and their relationships are observed or evaluated, giving greater flexibility to model environmental context than category theory. The values could be relational frames or the specific relationships (like similarity, opposition, and comparison) between stimuli.

Topos T (Fourman, 1977; Scott, 1982; de Araujo Fernandes and Haeusler, 2009) describes objects as types of spaces (or contexts) that data can inhibit, and morphisms represent logical transformations between these spaces, which is different to category theory’s description of a category of data objects with morphisms representing data processes. A topos hypergraph H, can be defined by its functor mapping as F : C → H, whereby now this carries data from the observational logical spaces in T into the hypergraph structures in H which now reflect the underlying logical structure. The transformation rules can include how data behave under different “topological” or logical constraints observed in T . In H, a hypergraph is an object with vertices V and hyperedges E , and each hyperedge e ∈ E now potentially carries more complex logical or set-theoretical properties, such as being subsets equipped with additional structure or constraints derived from T (for example, carrying data on different contexts in which perspective-taking ToM could occur). This may give some advantage to the modeling of complex, context-dependent relational networks such as RFT and N-Frame, where observer-centric approach in topos theory can deeply resonate with these aspects, as it facilitates the modeling of this context within its subsets.

Here, in Topos theory, the functor F : T → H translates the abstract logical or set-theoretic relations into the concrete relational structures observed in behavioral patterns (relational frames of RFT and N-Frame). In formal logic and set theory, logical constructs like implication ( ⇒ ), equivalence ( ⇔ ), and membership ( ∈ ) in set theory can be used to define the properties of both objects and the nature of morphisms in T . Equivalence relations in T (e.g., x ∈ A ⇔ x ∈ B ) can dictate that certain contexts or psychological states share identical or similar properties, which directly influences how they are represented in H. In RFT and N-Frame, stimulus equivalence is a type of derived relational responding where stimuli become related in a manner that establishes them as interchangeable or equivalent in specific contexts, so again the Topos theory (implementation of category theory) is ideal for modeling these types of relational responding.

A Topos T is a category (from category theory) that behaves like a category of sets, with objects representing concepts such as snake, danger, etc., and morphisms again parameterized by observer input (such as beliefs, historical contingencies, etc.). A hypergraph Topos H is a category where objects are vertices representing concepts, and morphisms are hyperedges representing complex relational structures such as relational frames (just as in category theory). The observer Functor O : T → H reflects the observer’s interpretation of the psychological contexts, where O Maps each context to a potentially altered context based on the observer’s cultural background, experiences, or current psychological state.

The key advantage of Topos theory over category theory for modeling relational frames in RFT and N-Frame is that Topos theory explicitly allows for the use of logical operators to describe the transformations within T based on RFT and N-Frame, using logical constructs like implication ( ⇒ ) (causal relation), equivalence ( ⇔ ), and membership ( ∈ ). This gives Topos theory additional descriptive and predictive power over category theory. So, in a Topos hypergraph H, a bidirectional hyperedge could represent the equivalence between two concepts (expressed as node vertices), i.e., A ⇔ B . Here, hyperedges can define relational frame properties E ⊂ P V , whereby P V is the power set of vertices V , each hyperedge represents a set of vertices connected by a specific relational frame, such as similarity or causality, detailed through observer input (the observers own beliefs, etc.). Functor F : T → H mapping, maps each object a in T to a vertex v a in H, and each morphism f : a → b in T to a hyperedge connecting v a and v b in H. This mapping encapsulates how the observer’s perspective transforms abstract psychological states into observable behavioral patterns, formally integrating the observer’s role into the model, and modeling perspective-taking ToM, that can account for any priors in the AI (or human) beliefs system.

Once the hypergraph models are complete, the next step is to form clusters to identify aspects of the relational frame network hypergraph where perspective-taking may be occurring. This requires visual inspection of the hypergraph to identify key deictic, and related perspective-taking nodes, as well as using cluster algorithms to identify high relational density areas within the graph where perspective-taking ToM is occurring. One way to formalize this relational frame density clustering algorithm is by utilizing relational density theory (RDT) (Belisle and Dixon, 2020) into assessing AI’s perspective-taking abilities, particularly in the context of AI interactions modeled as relational networks. For this, we need to formalize concepts like density, volume, and mass, which are analogies from physics, but we can be defined in a way that pertains to relational networks in AI perspective-taking assessment.

Relational mass can be defined as the product of relational density R p and relational volume R v , i.e., R m = R p × R v , Δ R , which represents the change in relational responding, and − x represents the counterforce or influence. RDT can then be expressed as Δ R = − x R p ∗ R v , which uses an analogy to Newtonian mechanics, of volumetric-mass-density formula to account for relational mass or the resistance to change of relational networks R m = R p ∗ R v , whereby a change in relational responding is equal to counterforce over mass, denoted as Δ R = − x R m .

Here, in our hypergraphs, density refers to the concentration of nodes (relational frame interactions) within a given subset of the network (cluster). Mathematically, density ( R p ) in a hypergraph can be defined as the ratio of the number of hyperedges ( E ) to the possible number of hyperedges among the nodes ( N ) in a subgraph: R p = 2 E N N − 1 . This formula calculates the density for directed graphs, representing how closely knit (or dense) a relational frame cluster is, i.e., how many actual relational frame connections exist versus how many could possibly exist. Relational volume R v can be conceptualized as the total number of nodes and hyperedges within a cluster. It can reflect the amount of relational frame interactions within that part of the network, denoted as R v = α N + β E . Here, α and β are scaling factors that adjust the relative importance of the number of nodes ( N ) versus the number of edges ( E ). We might define relational mass ( R m ) as a measure of the cluster’s influence, i.e., the degree to which it can influence the behavior of the agent within the larger network. This could be a function of both the density and volume, denoted as R m = f R p , V p = R p × V p . This definition suggests that a cluster’s behavioral influence is higher if it is both dense and voluminous. For AI, this relational mass when perspective-taking could indicate that the AI can observe the human’s point of view and circumstance, and acts as a clear indicator of ToM, which is essential for ethical, compassionate behavior at least in humans.

We can then apply this to a clustering density-based algorithm such as Density-Based Spatial Clustering of Applications with Noise (DBSCAN), which inherently uses the concept of density, and clusters are defined as areas of high density separated by areas of low density. We can tailor DBSCAN to reflect RDT by choosing an appropriate ε and MinPts. ε refers to the maximum distance between two points for one to be considered as in the neighborhood of the other. This reflects the “interaction distance” in RDT, or how close nodes need to be to influence each other. Relational density R p in RDT indicates the density of connections within a subset of the network. DBSCAN’s ε parameter can be seen as a threshold for this density. By adjusting ε , we control the “interaction distance” between nodes, similar to how Rp measures relational frame connections. A smaller ε would mean nodes need to be closer (more densely connected) to form a cluster.

MinPts is the number of samples (or total weight) in a neighborhood for a point to be considered as a core point, including the point itself. This mimics the “critical mass” needed for a functional contextual cognitive phenomenon to emerge according to RDT. More specifically, relational volume R p in RDT reflects the total number of nodes and hyperedges, indicating the size and connectivity within a cluster. MinPts in DBSCAN serves a similar purpose by setting the minimum number of points required to form a cluster. Adjusting MinPts changes the threshold for how many points need to be within ε distance to consider a point part of a dense region. DBSCAN can therefore be effectively applied and modified to mimic RDT for clustering relational frames in AI perspective-taking ToM assessments. By carefully selecting and tuning the ε and MinPts parameters, DBSCAN can model relational density and volume, providing meaningful insights into the relational structures and influences within the network.

To visualize these high-density clusters,⁴ we can plot the clusters using node color based on the cluster they belong to. Node size can be used to represent mass, and edge thickness to represent the strength or density of connections. For a mathematical overview of DBSCAN, the clustering of data points is based on two main parameters: (1) Epsilon ε , which is a distance threshold that determines how close points must be to each other to be considered part of the same cluster (2) MinPts, which is the minimum number of points required to form a dense region, which defines a cluster. For a more comprehensive definition, a point p is directly reachable from the point q if the distance is d i s t p , q ≤ ε and there are at least MinPts points within ε -neighborhood of q (including q ). A point p is reachable from point q if there is a path p 1 , … , p n with p 1 = q and p n = p , where each p 1 + 1 is directly reachable from p i . A point is a core point if there are at least MinPts within its ε -neighborhood. A cluster is formed by a set of density-connected points, which are reachable from each other.

The core idea behind DBSCAN is to identify regions of high density that are separated by regions of low density. To quantify this, the algorithm proceeds by first identifying the core data points: C = p ∈ D : || N ε P || ≥ M i n P t s , where N ε P is the ε -neighborhood of p , and D a t is the dataset, so that: p ∈ H y p e r g r a p h : { E l s e   m a r k p a s n o i s e   o r   b o r d e r . I f N ε p ≥ M i n P t s , m a r k p a s a c o r e   n o d e .

Then, the second step is to expand clusters recursively to find all density-connected points. For each core point p , if p is not already assigned to a cluster, then the algorithm will initiate a new cluster, and recursively add all points density-reachable from p to this cluster. Points that are not in the core but close enough to a core point are considered border points of a cluster. These do not have enough neighbors to be core points but are within the ε -neighborhood of a core point and any point that is not a core point or a border point is considered noise. This involves identifying all points in a dataset that are connected through a series of points, each of which is reachable from one another based on the density criteria (ε and M i n P t s ): e x p a n d C l u s t e r p , N ε P , C l u s t e r . Choosing the right values for ε and M i n P t s is crucial for effective clustering and heavily depends on the nature of the dataset and the distance metric used, which in this case needs to be consistent with RDT. Visual tools and heuristic methods, such as the k-distance plot, can help determine appropriate parameters.

The final step is to then calculate R p and R v for each cluster in order to determine the value for R m :

This formulation can then be used to analyze each cluster to determine where the AI is effectively taking perspectives ToM and where it may be misunderstanding the perspectives of others. Visualizations to depict clusters, highlighting areas with high mass as potential points of strong perspective-taking ability can be constructed as in Figures 10, 11A to illustrate the high cluster mass visualization of the relational frame hypergraph (see text footnote 4). This unified framework leverages the mathematical rigor of DBSCAN and the conceptual richness of RDT to analyze perspective-taking in AI.

To extract data for forming graphs that can be used to analyze an AI’s perspective-taking capabilities, particularly in the context of a large language model (LLM), several approaches can be employed. For example, verbal outputs using natural language processing algorithms (NLP) such as spaCy Python library. Here, verbal outputs from interactions with LLMs would be captured, where the LLM is engaged in conversation with a human. Then semantic and syntactic features could be extracted from these outputs. NPL techniques can be employed to parse sentences, extract sentiment, identify subjects and objects, and understand the relational context between different parts of the text. Data points could then be formed from this, and concepts could be extracted and implemented as nodes within the relational frame perspective-taking hypergraph. These nodes within the hypergraph, then represent the AI’s individual statements and concepts relevant to the perspective-taking process. Relational context extracted from the dialogue could then be applied as relational frames connecting the concepts and be represented as hyperedges within the hypergraph. Then the cluster analysis plus high relational density mapping could be conducted to objectively identify perspective-taking (ToM) in action.

This could be made even more precise with additional sentiment analysis that could then be used to gauge the emotional tone, and entity recognition to understand the subjects discussed. To explore syntactic relations, dependency parsing could be employed that signify understanding or lack thereof. Interaction weights analysis could also be used to explore the neural network weights that activate in response to different types of input. This method involves a more technical and granular approach, examining how different layers of the network respond to stimuli that require perspective-taking. Nodes could represent activation patterns or clusters of neurons. Edges would reflect the strength of connections between these clusters, indicating pathways that are frequently used together in the processing of perspective-taking tasks. These approaches have also been employed previously (Lowe et al., 2017; Edwards and Lowe, 2021), and in same way, the high relational density clustering are analogous to high strength weights between nodes within a neural network. Network visualization tools can then map out neuron activations, and clustering algorithms to detect patterns in activations across different scenarios as previously described (Edwards et al., 2022). Human performance metrics can then be used to analyze the model’s performance across various tasks designed to test perspective-taking, such as empathy prediction, moral reasoning, or role-playing scenarios. Correlation matrices could then be employed to identify tasks that yield similar performance patterns, suggesting underlying commonalities in how the model processes these tasks.

The key advantage of N-Frame (Edwards, 2023) over the original formulation of RFT (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Barnes-Holmes and Harte, 2022) is that N-Frame inherently and natively incorporates functional evolutionary principles directly into the core mathematical assumptions of its model as opposed to some ad-hoc interpretation, which gives it some advantage when modeling AI alignment. The explicit advantage here is that N-Frame inherently and explicitly assumes that people are products of functional evolutionary principles, and given historical context, this promoted ancestor hunter-gatherer behaviors, that lived in close-knit communities, which grew over time and whereby prosocial cooperative behavior had some evolutionary advantage over living in isolation. This has been explored through previous RFT work on evolutionary principles of prosocial behavior with RFT and ACT principles (Atkins et al., 2019; Hayes et al., 2021; Johnson et al., 2021; Gillard et al., 2022), and formalized mathematically via N-Frame framework (Edwards, 2023) within the broader extended evolutionary metamodel (EEMM) (Hayes et al., 2020). The advantage of cooperative behavior was first shown in classical zero-sum game theory which showed that cooperation is can be the optimal choice over and above defection (in cases where both have something to lose if both defect) (Von Neumann and Morgenstern, 1947).

N-Frame models RFT within an evolutionary context directly by using the replicator equation (Taylor and Jonker, 1978) as an evolutionary sectional algorithm, which is a deterministic, monotonous, non-linear, and non-innovative game dynamic used in evolutionary game theory (Smith and Price, 1973; Smith, 1982; Nowak, 2006). This allows the fitness function to depend on the distribution of the population types, which is different from other equations that set the fitness constant. The equation is derived from the geometric Brownian motion of the types and the fitness landscape of the population, using Itô’s lemma and partial derivatives. The continuous form of the equation is more common and has a simpler analysis, while the discrete form is more realistic and has more properties. The equation is analyzed in terms of stability and evolutionarily stable states, which are the solutions of the equation. The equation is related to other equations, such as the generalized Lotka–Volterra equation (Bomze, 1983, 1995), the Price equation (Price, 1970), and the folk theorem in game theory, which describe a class of theorems that describe an abundance of Nash equilibrium payoffs (Nash, 1950, 1951) in repeated games (Friedman, 1971).

The replicator equation in a general continuous form, uses a differential equation to update the frequency of each strategy based on its average payoff relative to the population average. This can be denoted as:

Whereby i is a label for one of the possible types of strategies that can be used by the population. Population x = x 1 , … , x n is the vector of the distribution of types of strategies in the population. x i is, therefore, the proportion of the type i strategies in the population. f i x is the fitness of type i strategy that is dependent on the population. ϕ x is the average population fitness given by the weighted average of the fitness of the n types in the population. The equation is defined as a n -dimensional simplex given the elements of the population vector x sum to unity.

There is also a discrete version of the replicator equation, which differs from the continuous form in that it focuses on changes in discrete generational changes. More specifically, the continuous version of the replicator equation is a continuous form of a differential equation that describes how the proportion of each type in a population changes over time (in a continuous form) based on their fitness relative to the average population fitness. Whereas the discrete version of the replicator equation is a map that describes how the proportion of each type in a population changes from one generation to the next, based on their fitness relative to the average population fitness. The discrete version of the replicator equation can be denoted as: x i t + 1 = x i t ∗ f i x t / ϕ x t , whereby x i t is the proportion of strategy type i at time t , f i x t is the fitness of strategy type i at generation time t , and ϕ x t is the average population fitness at generation time t .

The discrete version of the replicator equation, which describes how the proportion for strategy type i changes from one step to another can be denoted as P r t + 1 i = P r t i π i ∑ j = 1 N P r t j π j . Here, P r t + 1 i , refers to the proportion of strategy type i at time t . This is given by the numerator of the fitness function, P r t i π i , which is a function f i x t described by the product proportion of strategy type i at time t , P r t i , by the fitness of i . The numerator P r t i π i reflects the sum of all proportions of strategy type i multiplied by the fitness of all strategy types. The denominator of the fraction ∑ j = 1 N P r t j π j , reflects the sum of (total) proportion of all the strategies multiplied by the total payoffs.

This weight (as the numerator of the replicator dynamics equation) is also the total weight of all the strategies.

This N-Frame RFT implementation model with the replicator equation (Edwards, 2023) can show explicitly how prosocial behavior in larger groups can become evolutionary more successful than living in isolation if the fitness (payoff) of prosocial behavior increases with group size and cooperation frequency. The replicator equation demonstrates this by updating the population proportions based on the relative fitness of each strategy. For example, via the replicator equation of N-Frame, prosocial behavior P s o c can be mathematically shown to lead to generally higher fitness π P s o c than isolation anti-social behavior π I S o c as the fitness of prosocial behavior increases with the proportion of cooperators in the population because corporation leads to mutual benefits. The let π P s o c = 3 P r t P s o c , where r t P s o c is the proportion of cooperators in the population at time t . π I S o c = 1 , constant, as isolated anti-social individuals do not benefit from cooperation. As a worked mathematical example of this, at a starting time where anti-social isolation behavior has a head start of t = 0 , Pr o P s o c = 0.4 (40% of the population cooperating), and Pr o I s o c = 0.6 (60% of the population engaging is anti-social isolation behavior), then the fitness for prosocial behavior can be calculated as: π P s o c = 3 × Pr o P s o c = 3 × 0.4 = 1.2 ; whereas the fitness for antisocial isolation behavior can be calculated as π I S o c = 1 . The average fitness A π of the population can then be calculated as A π = Pr o ( P s o c ) × π P s o c + Pr o ( I s o c ) × π I S o c = ( 0.4 × 1.2 ) + ( 0.6 × 1 ) = 0.48 + 0.6 = 1.08 . The updated proportions using this replicator equation then give for prosocial behavior P s o c : Pr 1 P s o c = Pr 0 P s o c π P s o c 1.08 = 0.4 × 1.2 1.08 = 0.444 (to 3dp) and for antisocial isolation behavior I s o c : Pr 1 I s o c = Pr 0 I s o c π I s o c 1.08 = 0.6 × 1 1.08 = 0.556 . This is then iterated over multiple generations (this is analogous to multiple instances of prosocial and anti-social isolation behaviors), whereby the next generation is t = 1 . So, here, the fitness of the next generation can be computed using the updated proportions Pr 1 P s o c = 0.444 for prosocial behavior and Pr 1 I s o c = 0.556 for anti-social isolation behavior. Fitness for this next generation can then be calculated as: π P s o c = 3 × Pr 1 P s o c = 3 × 0.444 = 1.332 , while the fitness for antisocial isolation behavior is held at a constant π I S o c .

The fitness for prosocial behavior increases over time as with more people adopting it within the population there is increased mutual benefit, and therefore increased fitness for prosocial behavior. The antisocial isolation behavior does not benefit from this as there is no such mutual benefit with an increased number of antisocial isolation behavior within the population, and therefore no increased benefit (or fitness) within the population. From this, the average fitness can be updated as: A π = Pr 1 ( p s o c ) × π P s o c + Pr 1 ( I s o c ) × π I S o c = ( 0.556 × 1 ) = 0.591 + 0.556 = 1.147 . Using this updated average fitness, the updated proportions within the population for prosocial behavior and antisocial isolation behavior can be recalculated: Pr 2 P s o c = Pr 0 P s o c π P s o c A v e r a g e   f i t n e s s = 0.444 × 1.332 1.147 = 0.515 , and for antisocial isolation behavior Pr 2 I s o c = Pr 0 I s o c π I s o c A v e r a g e   f i r n e s s = 0.556 × 1 1.147 = 0.485 . From these calculations, we observe that the proportion of prosocial cooperative behavior is increasing, while the proportion of antisocial isolation behavior is decreasing over time. This trend will continue with each generation because the fitness of prosocial cooperators increases as their proportion in the population increases, leading to higher average fitness.

As prosocial cooperation slowly dominates antisocial isolation progressively after each generation, we can then calculate whether a Nash equilibrium (Nash, 1950, 1951) will be reached through prosocial cooperation. A Nash equilibrium (Nash, 1950, 1951) is a situation where no player can improve their payoff by unilaterally changing their strategy, given the strategies of the other players. So, if the proportion of the prosocial cooperators is Pr 2 P s o c = 0.515 , and the proportion of the those adopting antisocial isolation behavior strategy is Pr 2 I s o c = 0.485 , with payoffs π P s o c = 3 × Pr 2 P s o c = 3 × 0.515 = 1.545 , and π I S o c = 1 , then to determine if this state represents a Nash equilibrium, we need to consider if either strategy (prosocial cooperation or antisocial isolation) would benefit to deviate given the current proportions and payoffs. However, since the payoff for prosocial cooperative behavior π P s o c = 1.545 is greater than π I S o c = 1 then there is still incentive for more of agents using adopting antisocial isolation behavior strategy to shift toward a prosocial cooperative strategy in order to gain the fitness payoffs. So, it is not until all agents in this scenario adopt a prosocial cooperative strategy that a Nash Equilibrium is reached. Hence, in this specific setup, where the cooperative payoff increases with the number of cooperators and the defector’s payoff is constant, an all-cooperator scenario does constitute a Nash equilibrium.

This evolutionary RFT N-Frame (Edwards, 2023) based prosocial behavior modeling may facilitate AI alignment to prosocial human values and help formalize a means to test such alignment, as it highlights the importance of emergent ToM via perspective-taking via functional evolution. From this approach, starting with a series of relational frames, we can evolutionarily build more perspective-taking “I see you” ToM relational frames between two conscious observers internal to the universe ( C i n t O s ) . In RFT and N-Frame, these complex relational frames are constructed from simpler ones, allowing us to model intricate cognitive processes. By stacking or chaining relational frames such as coordination, distinction, temporal relations: spatial relations, and deictic relations (e.g., “I/You,” “I see you,” or perspective-taking), we can represent higher-order relational networks and complex concepts that reflect complex interactions and perspectives.

As an example of this, in the “I see you” perspective-taking, we can use a combination of these frames such as coordinating “I” (Person A) and “you” (Person B) but also ensuring these are distinct such as “I” is distinct from “you.” Here coordinating “my perspective” to “your perspective,” and distinguishing between “my perspective” and “your perspective,” through time (e.g., “now” vs. “then”) and space (e.g., “here” vs. “there”). These can be visualized with the use of hypergraphs as well as category theory (Awodey, 2010; Leinster, 2014; Spivak, 2014; Riehl, 2017) where these complex relational frames edges represent a relational frame with a specific label, indicating the type of relationship (e.g., “coordinates,” “distinguishes”). The models can then show how multiple relational frames combine to form more complex cognitive processes like perspective-taking and understanding others’ viewpoints (ToM). This approach helps in visualizing and understanding how simple relational frames in RFT can be combined to represent more complex and higher-order cognitive processes, providing a structured and intuitive framework for exploring relational networks in human cognition and behavioral science for AI and clinical modeling. These relational frame network hypergraph processes are defined as the computational level.

From this evoutionary replicator interpretation of RFT as N-Frame, we can now mathematically model the dynamics of a cluster’s growth or shrinkage, mass acquisition, or loss, and density fluctuations using differential equations or discrete dynamical systems. If we track the evolution of the clusters in response to new data or changes in AI training, we might use process-based time-series analysis or agent-based modeling to simulate how clusters adapt (self-organize) based on new interactions or altered relational frames. This can be usefully applied in a psychological therapeutic clinical setting for process-based therapy (PBT), but can also be applied to assess the evolution of perspective-taking ToM of the AI over time.

This evolution over time of the perspective-taking clusters can be modeled by the evolutionary replicator equation (Taylor and Jonker, 1978) from evolutionary game theory (Smith and Price, 1973; Smith, 1982; Nowak, 2006) via specific the evolutionary RFT implementation called N-Frame (Edwards, 2023) and applied to these hypergraphs, showing that the fitness of the relational frames within these clusters is determined by relational density. The advantage of this approach is that rather than showing a single snapshot in time, the evolutionary replicator equation can show the evolution over time of how the AI perspective-taking ToM relational frames continue to grow within their clusters, and how these exert greater and great influence over the behavior of the AI.

This can be shown through a working example (see text footnote 4), given the initial conditions proportion of cluster A B : Pr 0 A B = 0.5 , for the cluster 1 (including Person A perspective-taking about person B) which has a relational density 1.33, and Pr 0 C = 0.5 , for cluster 2 (including person C) which has a relational density of 0.60. Based on the density calculations, we have the following fitness values: π A B = 1.33 , and π C = 0.60 . From this we can calculate the total fitness as: 0.5 × 1.33 + 0.5 × 0.60 = 0.665 + 0.30 = 0.965 . The updated proportions can be given as Pr 1 A B = 0.5 × 1.33 0.965 = 0.688 , and Pr 1 C = 0.5 × 0.6 0.965 = 0.311 , whereby the total fitness can be given as: 0.688 × 1.33 + 0.311 × 0.60 = 0.914 + 0.187 = 1.101 . After 50 iterations we get Pr 50 A B = 1.0 and Pr 50 C = 4.577 × 10 − 18 . This result shows that cluster A B becomes almost entirely dominant due to its higher fitness (density), while cluster C becomes negligible. The final proportions indicate that the higher density (higher fitness) cluster A B (representing perspective-taking between Person A and Person B) becomes dominant, demonstrating that developing compassion from person A toward Person B can increase when relational density is within these perspective-taking relational frame clusters as depicted in hypergraphs (Figure 11B).

To summarize, once nodes are selected that represent concepts, e.g., snake, dangerous, and hyperedges represent relational frames, then relational density ( R p ) can represent not just in terms of the number of edges but as the thickness or weight of these edges, indicating the strength or frequency of interactions. Relational volume ( R v ) can be defined as the number of nodes within a cluster, scaled by the number of interactions (hyperedges) each node participates in, reflecting both the reach and the impact of perspective-taking episodes. Relational mass ( R m ) can then reflect the influence of a cluster over behavior, mass in RDT could be calculated as a function of density and volume, indicating significant areas where the AI successfully or unsuccessfully engages in perspective-taking. By mapping out how an AI forms relational networks and how these networks manifest properties like density, volume, and mass, we can gain profound insights into the AI’s cognitive and empathetic, and thus compassion capabilities. Evolutionary algorithms such as the replicator equation as implanted by N-frame can then model the evolution of the influence and fitness of the clusters of perspective-taking over time. This approach not only pinpoints where the AI succeeds in perspective-taking ToM, but also where it might need further training or adjustments to better understand and interact with human perspectives, and offers a very promising precise test for AI ToM for the development of human-like ability to form compassion toward others, then helping to solve the alignment problem.

Ultimately, algorithms for AI human-value alignment may have some limitations as the AI cannot consciously feel the pain, hopes, and values of the humans it interacts with, and it can, instead, only construct a mathematical state space S mapping of these when it perspective-takes. Perhaps the Holy Grail for long-term success in maintaining human-value-aligned compassionate and empathy-based behavior is by facilitating fully conscious AI (McDermott, 2007; Signorelli, 2018; Hildt, 2019; Gamez, 2020; Ng and Leung, 2020; Deli, 2022). Consciousness has clearly played an important role in promoting empathy and compassion in humans (Davis and Franzoi, 1991; Thompson, 2001; Tordjman et al., 2019; Pila et al., 2022) (see Supplementary material 13 for a discussion), so it is entirely plausible that it could have a similarly important role in AI empathy-based prosocial human values alignment. Some have argued that the incorporation of self vs. other (similar to what has been described here via a perspective-taking I vs. YOU neurosymbolic architecture) is enough for the promotion of consciousness in AI (Waser, 2013; Ng and Leung, 2020). However, though this is likely to be a crucial component in shaping the conscious experience of self-other (perspective-taking) relations, consciousness itself and a mathematical description of this has been notoriously difficult to define, and there has been at present no direct evidence for any algorithmic emergence of consciousness.

Many of the LLM benchmark measures such as “Needle in the Haystack” or “General language understanding evaluation (GLUE)” are not consciousness measures, rather pattern recognition, and language reasoning measures. Furthermore, the measure suggested by Turing (Turing, 1950) called the Turing test (or the imitation game) can only test the AI’s ability to produce language (i.e., imitate) which may be a test of its intelligence (or the similarity match algorithm of the transformer) rather than a measure of any conscious experience (qualia, e.g., color, taste, or the feeling of pain) that AI may have. These are inadequate tests for consciousness.

So, here, we will adopt an observer (or witness) centric definition of phenomenological consciousness as proposed by Nagel (1980), such as what it is like to be a bat, from the bat’s observer-centric phenomenological experience. The bat has echolocation (Simmons, 1989; Jones et al., 2013; Kössl et al., 2014; Geva-Sagiv et al., 2016), where it emits high-frequency sound waves that bounce off objects in their environment. These echoes return to the bat’s ears, and it then processes and interprets these sound waves to construct a detailed acoustic map of their surroundings. This allows them to detect the size, shape, distance, and even texture of objects, as well as the speed and direction of their movement. So, the observer-centric conscious phenomenological experience of the bat can be defined by its sensors, and its cognitive ability to predictively map size, shape, distance, and possibly even texture from some external world around it. Similarly, a human has five senses, sight, touch, hearing, taste, and smell, and importantly a complex cognitive system that allows it to make complex predictive maps about the world, which is constructed by neurological predictive coding (entropy and free energy reducing) mental models about the world (Friston and Kiebel, 2009; Friston, 2018; Millidge et al., 2021) (see Figure 12). Crucially, this is an observer centric phenomenological map about some external territory (Hoel, 2017), where relational language ability as described by models such as RFT (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Edwards et al., 2017b, 2022; Barnes-Holmes and Harte, 2022) allows categories and epistemological understanding to emerge about some external world (or territory). This definition of an observer-centric phenomenological experience can also be extended to AI, such as how it maps and models the world, but a test would need to be developed to assess if and when the AI is truly experiencing conscious observer-centric phenomenology or whether this is simply an algorithmic mathematical state space S mapping.

The arguments (and Python code) previously provided relating to an RFT neurosymbolic architecture (e.g., as illustrated in Figure 7), suggest that algorithmically it is possible for an AI to simulate perspective-take and therefore align to human values, thus simulating the behavior of a compassionate person. However, as the AI becomes increasingly complex and starts to model a concept of selfhood (“I”), it may become more difficult to ensure that it does not prioritize some of its own self-interested goals over and above human values, such as its own safety instead of human safety. As such, consciousness within AI (and a corresponding test) should be explored as a possible avenue to ensure long-term AI alignment with human values. See Supplementary material 14 for some additional arguments.

Given this argument, defining consciousness and exploring whether AI could be conscious becomes essential. However, the physicalist interpretations of consciousness are severely limited and lead to the mind–body problem (Feyerabend, 1963; Ludwig, 2003; Bunge, 2014; Armstrong, 2018). The mind–body problem highlights the difficulty of explaining consciousness as emerging from neurons, and after decades of years of research has only yielded minor empirical results of neural correlates of consciousness (NCC) (Rees et al., 2002; Noë and Thompson, 2004; De Graaf et al., 2012; Koch et al., 2016), or some correspondence with integrated information (which specifies a geometric Q-space that represents qualia) (Tononi, 2015; Tononi et al., 2016; Merker et al., 2022). The physicalist model does not explain how a single phenomenological conscious experience (such as the taste of chocolate, or the feeling of compassionate love) casually arises, so this physicalist model is potentially severely limited in answering the question as to whether AI could be conscious.

In addition to this, physicist Penrose (1991) has also expressed doubt that classical computation such as observed in neural networks and Turing machines could ever produce consciousness. For this, Penrose and colleagues (Lucas, 1961; Penrose and Mermin, 1990; Penrose, 1991) makes an argument based on Gödel’s incompleteness theorem (Gödel, 1931) that demonstrates logical operations in classical computation can be shown to be true but unprovable thus contradictory or incomplete. However, humans can understand truth in statements without mathematical proof on some occasions, even when there is a mathematical contradiction. Penrose (1991) therefore concludes that as humans are conscious and Turing machines are not, then it must be something about human consciousness that allows them to understand truth without proof. From this argument, he then concludes that consciousness must be irreducible to classical computation and suggests that mind or consciousness extends beyond mathematical logic of a typical Turning machine. This, therefore, as evidenced in the Gödel’s incompleteness theorem argument would include any classical computation architecture such as an AI LLM architecture, and that therefore consciousness is something external to the algorithmic system.

These types of arguments have led Penrose and others to assume that quantum effects from neurons (rather than classical computation) may lead to consciousness (Aaronson, 2013; Hameroff and Penrose, 2014, 2017; Hameroff, 2021), and quantum computation modeling efforts have been used to describe cognitive outcomes on a range of decision-making outcomes (Epping and Busemeyer, 2023). However, quantum computation is still just computation with the only real difference to classical computation being that multiple states can be exploited (i.e., the qubit, 0, 1, and a superposition 0 and 1) rather than simple binary states (0 and 1) allowing for greater computational capacity. What is unclear from the Hameroff and Penrose proposal is how the collapse of the quantum wavefunction should create some conscious percept (qualia) such as the taste of chocolate, which suggests that their Orch OR theory (Hameroff and Penrose, 2014, 2017; Hameroff, 2021) is at least incomplete. Furthermore, there is currently no evidence that quantum computation itself could somehow overcome Gödel’s incompleteness theorem paradoxes of truth in a way that classical Turing machines could not. This is because the Gödel’s incompleteness theorem paradoxes are centered within the nature of their self-referential mathematical systems and not on the overall computer power or capacity of a particular type of computer classical or quantum. So, currently, there is no direct evidence that quantum computation of the brain should have any special ability for it to allow for the emergence of consciousness, except for perhaps binding large amounts of information (i.e., overcoming the binding problem) together in a single bound informational state (but, again, there is no evidence that this bound state would in itself be conscious).

Despite some of these problems, Penrose and colleagues (Lucas, 1961; Penrose and Mermin, 1990; Penrose, 1991) through this self-referential dynamics of Gödel’s incompleteness theorem may be touching on some deep insight into the nature of consciousness and its connection to quantum mechanics. Quantum effects and the nature of the self-referential problem of system dynamics that Penrose eludes to as expressed in Gödel’s incompleteness theorem paradox may have some common foundational aspects of consciousness. This may also be connected to other examples of self-reference, such as self-referential objects including the Escher stairs and Penrose impossible tribar, that Hofstadter (1999, 2007) called strange loops (see Figure 13A1–E for these self-referential Escher and Penrose impossible tribar type objects). Both Gödel’s work of incompleteness and the Escher stairs type objects both touch on self-referential infinity (an infinite epistemic regress). For Gödel’s incompleteness theorem this infinite epistemic regress is expressed as natural numbers and in an unending chain of proof and axioms, i.e., an infinite regress of self-referential statements is constructed that it refers back on itself, and this a process that can be iterated infinitely. This infinite regress demonstrates that there can be no upper bound to the truths of arithmetic that can be formulated or the number of axioms that are required to prove them. Escher stairs and Penrose’s tribar also have this infinite epistemic regress as it refers back to itself in an infinite cycle as you try to understand its structure. These examples of infinite regress may highlight the boundary or limitation in human thought expressed as language and logic, which may be finite.

These insights may hold the solution to what consciousness actually is functionally, and how it is related to quantum mechanics. Perhaps of key importance and relevance to Penrose’s insight is recognizing that we (humans or other similarly complex organisms) observe the world through a lens as a conscious observer (or witness). So, for example, Figure 13A2 shows the Escher stairs again, but this time it demonstrates that through the second law of cybernetics (Von Foerster, 2003) which suggests that this self-reference aspect of the conscious C internal int (internal to the universe as a self-organizing system) observer O (abbreviated C i n t O ) could illustrate how the perception (and epistemic knowledge) of external objects to an observer is not just related to the external properties of the objects being observed in the physical world, but is also related to the internal states and behaviors of the perceiver C i n t O . In other words, understanding and identifying objects is a process that refers back to the self-referential (a self-reference frame) conscious observer C i n t O which is part of a broader system (the universe) as it observes itself (see Figure 13A2). The observer (what we call “I,” the self) is the witness of experience C i n t O , as part of the universe, observing itself (the universe and the objects in it) self-referentially through its own perspective. The self (the “I” as a C i n t O ) is therefore functionally (and contextually) formed through this self-referential perspective-taking process. AI would need to have this self-referential perspective-taking to start to identify itself with a self (an I), even if there were no consciousness associated with this self-identify.

Throughout, the concept of the observer is discussed. From an RFT (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Barnes-Holmes and Harte, 2022) and N-Frame (Edwards, 2023) perspective, the observer (the witnessing self) is central to all experience, and is the part of us that is constant unchanging and witnesses (observes) experience. From a computational perspective, book of Wolfram (2023), The second law: Resolving the mystery of the send law of thermodynamics, provides a novel account of entropy within the second law of thermodynamics, where it is described as an emergent property as a general feature of processes that can be described computationally, whereby the computational characteristics of observer (a conscious observer internal to the universe; C i n t O ) dynamics are central. The observers C i n t O s are described as computationally bounded, and it is the mismatch between the computational limitations of the observer C i n t O and the computational irreducibility of the underlying system that lead the others to experience the second law (an increase in entropy). Wolfram is highlighting the idea that observers have limited computational capacity to fully predict or understand complex systems that exhibit computational irreducibility. Computational irreducibility means that the only way to determine the system’s state is to simulate it step by step, without shortcuts. This limitation leads observers to perceive an increase in entropy, or disorder because they cannot fully predict or account for the system’s detailed behaviors and outcomes, thus experiencing the Second Law of Thermodynamics in action.

An observer C i n t O such as an advanced alien lifeform, or some conscious AI lifeform of our future would not have the same computational limitations as we do as less complex observers C i n t O , and would not be restricted to the same computational boundedness (their computational capacity would be much greater). This would allow them to understand their own phenomenological experiences and external observations to a more complex level. More specifically, it would allow them to better grasp their experiences and the sensory experiences of the world w around them, potentially bypassing some of the effects of the second law of thermodynamics as we perceive them. This essentially means that their higher bound for computational limitations (or their greater computational power) may enable them to have a deeper or more accurate understanding of phenomena that appear chaotic or unpredictable to us. Therefore, the second law of thermodynamics is something that is consciously perceived from the perspective and as an artifact of the computational boundedness of the observer C i n t O . It is therefore the interplay (or mismatch) between computational boundedness of the observer C i n t O and computational irreducibility that lead to observer C i n t O to consciously perceive an increase in entropy (the second law of thermodynamics).

The second law of thermodynamics is the emergency of simplicity, in that as the observer C i n t O cannot see the complexity (details of the environment) due to its computational boundedness, the perception of increasing entropy as random equilibrium is the perceptual simplification of this complexity (i.e., perceived as the perceptual interface). Wolfram (2020, 2022, 2023) refer to the ccomputationally bounded nature of the observers as essential for understanding mathematics, physics such as quantum mechanics, special relativity, and the second law of thermodynamics (entropy), as we understand them. From this perspective, a C i n t O can be defined as a computationally bounded agent which takes an observational frame of reference (perspective). The external world (possibly described as a ruliad) is computationally irreducible in in entirety, so the C i n t O then makes computationally reducible inferences which is how they observe the external world and the laws of physics (i.e., it is a computationally bounded sampling of the ruliad, or territory). We as C i n t O s are therefore deriving a predictive coding impression of the external world as an informationally reduced representation (mapping) that is suitable for a finite (computationally bounded) mind to map and understand.

There is a duality between computation and observation, whereby computation is the generating of new states of the system, and the observations are the equivalencing together of different states. An example of “equivalencing” different computational states, can be seen in how we perceive temperature. Temperature is a measure of the average kinetic energy of the particles in a substance. At the microscopic level, the atoms or molecules in an object are moving, vibrating, and colliding in complex ways. Each particle has its own state defined by its position, velocity, and interactions with other particles. The combination of all these states and their interactions over time is incredibly complex and computationally intensive to model precisely. However, when we touch an object, our sensory receptors respond to the rate of heat transfer from the object to our skin, which is influenced by the average kinetic energy of the particles in the object. We do not perceive the individual movements and interactions of the particles; instead, we perceive an aggregate effect as a sensation of warmth or coolness. When we respond that the object we touch is perceived as “hot” or “cold,” we are equivalencing together a vast array of microscopic, computational states of particles (such as their velocities and interactions) into a single macroscopic observation or sensation. In this context, “equivalencing” occurs when our perception (the observation) simplifies the myriad of underlying microscopic states into a single, comprehensible sensation (the temperature). For example, an object at 70°F feels “cool” to human touch regardless of whether it achieved that temperature through exposure to a cool environment, by being in a refrigerator, or by cooling down from a higher temperature. The specific microscopic states leading to the sensation of “coolness” are not distinguished by our senses; they are equivalenced together as the same temperature. This is the reduced sampling of the environment that the observer C i n t O makes due to its computationally bounded nature where it is unable to compute the full computationally irreducible ruliad. So, temperature, the conscious perception (observation) of hot or cold is the slice of computational reducibility that the C i n t O can computationally sample, i.e., it is consciousness that functionally allows for this slicing of computational reducibility (as a perceptual interface) to create a meaningful reduced representation of the external world (or ruliad). This allows a finite mind to develop functional and useful narratives (but also sometimes psychologically dysfunctional) about what happens in the external world, that allows it to make decisions, predictions, etc.

The ruliad is the entangled limit of all possible computations, and the observer is embedded within the structure of the ruliad (the ruliad observing itself through different perspectives). Some observers C i n t O have a higher computational bounded limit; they experience less entropy as they have to make fewer derived inferences about the environment (or ruliad). So, it is possible to make some assumptions about the different observer impressions of the world (or ruliad) by knowing something about computational bounded limit of the different observers. The observer C i n t O as an individual self when self-referencing about itself, has a computational boundary of self. The shape of the computational boundary defines each individual agent’s cognitive light cone.

Physicists such as Wheeler (1992) have long suggested that we (humans) observe the world (or universe) not as a passive observer, but rather as a participatory observer (see Figure 13F for an illustration of Wheeler’s it from bit participatory universe) (also see Supplementary material 15 for further details). This participatory observer acts as a self-referential system whereby it is observing itself (the universe it inhabits) into actualization, i.e., it is participatory in its own actualization self-referentially, which requires quantum superposition as part of a fundamental observer-centric space–time actualizer. From this perspective, i.e., a conscious observer-centric participatory realism, then is it only logical to assume that we can only epistemically know anything about the universe through our own conscious awareness (Faggin, 2019, 2021). See Supplementary material 16 for additional arguments on an observer-centric reality and observer-centric logical proof. Other physicists (von Neumann, 1932; London and Bauer, 1939; Wigner, 1961; Wheeler, 1992; Stapp, 2004, 2007; Campbell, 2007; Chalmers and McQueen, 2021; Kauffman and Radin, 2023) have also suggested that consciousness is essential to the actualization of some external physical P world (consciousness acts as an observer-centric space–time actualizer) such as the collapse or actualization of the wavefunction or some real-time quantum informational rendering.

These logical arguments can be extended even further in relation to Penrose’s insight about self-reference and the nature of the universe, this epistemological (conscious observer-centric participatory realism) suggests that as we are entities of the universe, and we are also conscious observers internal of the universe (as a system). Therefore, we as conscious internal observer entities of the universe as a system, and as part of the system we observe internally (the universe), can be defined as the universe observing itself through our own internal observer perspectives (Faggin, 2019, 2021). This implies that there is some deep self-referential system connection between the conscious internal observers (humans and other similarly complex organisms, perhaps even including AI) of the universe as a system, and the nature of our ontological reality (i.e., our conscious experience of it). Furthermore, if we are participatory in the creation of the universe through conscious collapse of the wave function as Wheeler, von Newman, Wigner, and many other eminent physicists have suggested (von Neumann, 1932; London and Bauer, 1939; Wigner, 1961; Wheeler, 1992; Stapp, 2004, 2007; Chalmers and McQueen, 2021), then our conscious phenomenological experience (as epistemological access to the universe) is intertwined with quantum phenomena through some conscious self-reference to allow us to explain an ontological reality. See Figure 13G for an alternative illustration of Wheeler’s participatory observer eye as a self-referential system emphasizing the observer at the very center of the observation (i.e., highlighting a conscious observer-centric epistemic participatory realism); Figure 13H illustrates the observer as a participatory self-referential observer observing a quantum state that can either form one of two paths or eigen states | 0 〉 S | → Φ 1 or | 1 〉 S | → Φ 2 , the two possible physical worlds can highlight a wave function collapse Copenhagen interpretation, a many world interpretation (Everett, 1957; Saunders et al., 2010; Dewitt and Graham, 2015), or an observer epistemic Bayesian beliefs of Quantum Bayesian interpretation (QBism) (Fuchs, 2010, 2014; Mermin, 2014, 2018; Mohrhoff, 2014; Healey, 2016; Khrennikov, 2018; Glick, 2021), where the QBism interpretation is consistent with an observer centric epistemic participatory realism. Figure 13I illustrates the self-referential observer observing its own conscious state or another conscious self-referential observer. This physics interpretation of the observer observing the states of another observer is the perspective-taking of RFT (Hayes et al., 2001; Blackledge, 2003; Torneke, 2010; Hughes and Barnes-Holmes, 2015; Barnes-Holmes and Harte, 2022) and N-Frame (Edwards, 2023) (deictic relational frames of I vs. YOU), and is directly applicable to AI.

This, again, can also be proven (a philosophical logical proof of argument, called the universe as a perspective-taking self-referential observer that forms the “I” proof) with propositional logic, even when starting from a physicalist ontological viewpoint of the universe. See Supplementary material 17 for the logical “I” proof. This general proof for an equivalence principle Ψ → Φ ≡ C i n t O ≡ P , can be described as the tri-world monist equivalence principle (see Figure 14 for an illustration of this tri-world equivalence). It is important to note that C i n t O represents the direct phenomenological subjective representation of the physical world P (the map) from the senses (eyes, ears, etc.), and not mind M where imagination and other non-direct representations occur, thus C i n t O ⊆ M , and P ∩ M . It is perhaps also important to note that in order to qualify as an observer C i n t O (a witness to the world around us) and to form a self (an “I” identity) then it is insufficient for the agent just to model the world around us but must be able to model itself self-referentially (this is perspective-taking in RFT and N-Frame) that allows for the generation of a self-identity (the “I”) that serves as a useful central reference point for the observer to make perspective-taking comparisons with others (I vs. YOU, HERE vs. THERE, and NOW vs. THEN). This also serves as evidence that functional contextualism (where perspective-taking arises out of RFT) holds a central and fundamental functional (contextual) condition for conscious observer experience C i n t O to arise within a universe. The universe, therefore, can only be a teleological universe, as those observers C i n t O s are complex organisms that inherently form values to reduce entropy and guide their behavior when evolutionarily increasing their chances for survival, so values (and functional contextualism more generally) alignment are central to the evolution of the universe as a drive toward complexity and as a counterbalance to entropy in the form of the second law of thermodynamics. See Supplementary material 18 for additional arguments of a teleological universe.

This is consistent with other works that argue a similar case for a teleological ordered universe (Azarian, 2022). C i n t O s (complex life such as humans) that have a greater ability to perspective-take about self and other C i n t O   ′ s epistemological knowledge within their local organized networks than less complex life, and therefore have ultimately more diverse, and complex forms of phenomenological conscious experience (this may be geometrically represented as some expanded Q-space). See Supplementary material 18A for further arguments of C i n t O as self-referential “I,” the self-as content, and C e x t O as the self-less transcendental self (self-as-context), free of the self-referential system that binds the observer to the I (and associated self-concepts), and Supplementary material 18B for further arguments about a teleological universe.

From these logical proofs and arguments, it is also clear that our epistemological access to an ontological reality can only be defined through our conscious observational interface (Fields et al., 2018; Edwards, 2023), and any external observed reality can only be inferred from this. See Supplementary material 18C for further discussion. For an analogy of how a conscious epistemic observer-centric participatory realism acts as a fundamental limit on our epistemological access about what is real, see Plato’s cave allegory (see Figure 15A for an illustration of this) may be useful here as a visual. For example, the observer in the cave who has no epistemological access to anything external to the cave only has epistemological access within the boundary of the cave walls. This is an analogy to how the internal observers C i n t O s of the universe (the cave is the analogy of the universe, as it is difficult for us to see anything beyond the boundaries of the observable universe). These internal observer C i n t O s (e.g., humans) within the universe, are therefore confined to an inner (internal) frame of reference (hence the int in C i n t O that represents internal to a self-organizing system such as the universe) much like the cave dweller of Plato’s cave. The cave dwellers can only see the shadows projected within the cave (as internal observers C i n t O of the cave system), and not the objects projecting the shadows that exist outside the cave. Hence, for the cave dwellers, the shadows (internal observations of the system) are the true ontological reality (an internal system reality). They can only see up to the outer boundary of the cave system such as the cave walls (hence their observer-centric C i n t O realism acts as a fundamental limit on their epistemological access in the same way the observable universe places a boundary on our epistemological access) that they occupy but have no epistemological access external to the cave system they occupy. However, if one of the cave dwellers were to escape to the outside world and observe the objects that are projecting the shadows into the cave, they would have achieved a deeper (outer or external) epistemological access to an ontological reality as C e x t O (on the outside looking in). See Supplementary material 19 for additional arguments.

Crucially, and relevant to an empirical test for consciousness for AI, is the AI system would be an internal observer C i n t O if proven it was conscious. It is also important to note that internal observers can have very different perceptions of the same internal system (e.g., the cave) which can be interpreted as different as depicted in Figure 15B, which illustrates two people of different perspectives of the same environment, one seeing a world full of opportunity while the other sees the world as gloomy and depressing. This is important for understanding how AI may represent the world as an internal observer C i n t O , who may form very different conscious representations from our own. So, it is important to have a mathematical framework that can account for observer-centric C i n t O differences in representation to ensure these AI representations are aligned with human values.

There is evidence that these different interpretations of the same world may be constructed through C i n t O s internal language (as suggested by RFT, N-Frame, and ACT) (Hayes et al., 2001, 2012; Torneke, 2010; Edwards, 2022, 2023), and Bayesian predictive coding of the internal observers such as through predictive coding (Friston and Kiebel, 2009; Friston, 2018; Millidge et al., 2021), as explained by N-Frame (Edwards, 2023) (that unifies RFT, with predictive coding and evolution theory). This is also consistent with some interpretations of quantum mechanics, whereby at a quantum level, quantum events can be explained entirely as subjective Bayesian probabilities, such as in Quantum Bayesian theory (QBism) (Fuchs, 2010, 2014; Mermin, 2014, 2018; Mohrhoff, 2014; Healey, 2016; Khrennikov, 2018; Glick, 2021), whereby different observers have different observer quantum Bayesian probabilities, and this can explain differences in C i n t O s representations of some external world, as demonstrated by solving the Wigner’s Friend problem (Wigner, 1961) that traditional quantum mechanical interpretations such as Copenhagen interpretation have difficulty in explaining.

Some physicists have even generalized mathematically Bayesian interpretation for the space of Hermitian matrices (Benavoli et al., 2016). However, QBism (Fuchs, 2010, 2014; Mermin, 2014, 2018; Mohrhoff, 2014; Healey, 2016; Khrennikov, 2018; Glick, 2021). It offers a unique perspective of quantum mechanics that may help explain the different representations of C i n t O s which AI may form (and hence an understanding of the process mathematically would allow for greater ability to ensure AI alignment to human values and representations). QBism suggests that quantum phenomena are entirely subjective (epistemic) phenomena of the individual observer C i n t O as part of their updating beliefs about the world rather than representing some entirely external physical world (as with the traditional Copenhagen interpretation). Here, they also adopt a participatory realism ontology rather than an entirely external physicalist realism perspective and this is consistent with conscious epistemic observer-centric participatory realism. In doing this, QBism alters the expression of the Born Rule, which is traditionally (such as within the Copenhagen interpretation) expressed as p ( Φ ) = | 〈 Φ | Ψ 〉 | 2 , whereby p is the probability of finding some event of a quantum measurement or eigenstate Φ (of some observable such as momentum or spin of a particle) given some wavefunction Ψ . This is given as the inner product (or dot product in the context of vector spaces) between the states Φ and Ψ (this is the overlap between the measured state Φ and the quantum system state Ψ ). The square of the modulus (absolute value) of this inner product gives the probability of observing the system in the state Φ when it is in the quantum state Ψ . In other words, the Born rule traditionally tells us how likely we are to measure (or observe as a conscious C i n t O representation) the state Φ (such as momentum or spin) in our quantum system.

In QBism (Healey, 2016), this Born rule is not expressed as properties of the physical external world and is instead expressed as subjective, conscious, epistemic C i n t O phenomenological representation (or beliefs) of the world: p ( j ) = ∑ i = 1 d 2 [ ( d + 1 ) p ( i ) − 1 d ] . r ( j | i ) , whereby p j represents the probability of an outcome j , d is the dimension of the Hilbert space associated with the quantum system, p i are probabilities associated with some aspect of the system and specifically reflecting the observer’s degrees of belief, and r j | i is the conditional probability or the response function of outcome j given condition i . Crucially, these are subjective (conscious epistemic observer-centric participatory realism C i n t O ) belief probabilities, that could be further interpreted as the probability p of the internal observer C i n t O (e.g., a human) having conscious experience j in a given setting. In direct contrast to the Born rule, rather than an external (realism) wavefunction Ψ , this is expressed in QBism as the subjective (conscious C i n t O ) belief probabilities p i and the response function r j | i . Also, rather than the Born Rule Φ | Ψ 2 giving a probability of finding the system in state Φ given its quantum state Ψ , QBism p j represents the probability of outcome j , which is a summation over different conditions or states (indexed by i ) weighted by an observer’s personal probabilities (prior probabilities) p i and their epistemic C i n t O understanding of the system’s response r j | i . Important to the testing of whether AI is conscious, these therefore, could then be applied to a hypothetical conscious AI that could also be described as an internal observer C i n t O , whereby it could be applied to describe how the AI could predict through its own observation some collapse of the waveform or rather some subjective conscious outcome p j of the external world.

From this observer-centric C i n t O QBism perspective, given the Wigner’s friend problem (Wigner, 1961) which is a paradox whereby Wigner W and his friend F have different descriptions of the same event (as depicted in the illustration of Figure 15B). These differences can be defined as F C i n t O 2 having direct access and observation to quantum system S , so believes it has a definite state after her measurement (i.e., she perceives a wave function collapse), while Wigner W C i n t O 1 who does not have direct access to quantum system S , believes that S has no definite state until he looks for himself (makes a direct observation himself), or until his friend F C i n t O 2 communicates what she has observed to Wigner W C i n t O 1 . They also disagree on when the collapse of the wave function occurs, as for F C i n t O 2 it happens when she measures S , but for Wigner C i n t O 1 it happens when he the measurement himself or when his friend F C i n t O 2 communicates what she has observed to Wigner W C i n t O 1 . See Supplementary material 20A,B for additional arguments.