Ouroboros Spaces: An Intuitive Approach to Self-Referential Functional Analysis with Applications to Probability Theory
aa r X i v : . [ m a t h . F A ] F e b Ouroboros Spaces:An Intuitive Approach to Self-ReferentialFunctional Analysis with Applications toProbability Theory
Nathan Thomas Provost ∗ Abstract
In this paper, we aim to introduce the concept of the Ouroboros space and thecomplimentary concept of the Ouroboros function by using the Ouroboros equation[1] as our starting point. We start with a few univariate definitions, and then move onto multivariate definitions. We contextualize and motivate these concepts with a fewexamples. Then, we discuss a few aspects from probability theory that are relevant tothe idea of an Ouroboros space, eventually proving two critical theorems. We brieflyintroduce the case of mixed domains, and summarize our findings, while emphasizingthe importance of self-referential functions.
Introduction
Ancient history and historical artifacts have had an indescribable impact on thedevelopment of modern society. Recently, the symbol of a self-consuming snake(historically known as the
Ouroboros ) has been the inspiration for various ideasin mathematics, computer science, and biology. Chiefly, it has been used toname an equation that embodies the idea of a self-reference, the likes of whichwas the central topic of a fascinating paper, written within the last decade,focusing on the “the various manifestations or ways in which [the] Ouroborosequation has emerged.” [1] We instead aim to give a new, intuitive definitionto the function spaces that consist of solutions to this enigmatic equation. Nat-urally, we begin with the univariate case for which we prove some interestingproperties, but we quickly advance to the multivariate case, which allows us totie probability theory to Ouroboros spaces in an interesting way. ∗ Student of Applied Mathematics and Statistics at Brown University. University Email:nathan [email protected] uroboros Spaces We are chiefly concerned with the idea of the
Ouroboros Equation [1]: f ( f ) = f We will first build our definitions using the univariate case, given similarly by f ( f ( x )) = f ( x ) , ∀ x . Suppose that A is a set and B is another set. Then wewill call O ( A ) the Ouroboros Function Space , or more laconically, the
OuroborosSpace for the domain given by A . To be more explicit, we can deduce that: f ∈ O ( A ) −→ f : A → B ∋ f ( f ( x )) = f ( x ) , ∀ x ∈ A We call such a function an
Ouroboros function . It obviously follows that O ( A )is the set of all functions f that satisfy this condition. From this realization, wecan rigorously define the Ouroboros Space like so, though we will later reviseand perfect this definition: O ( A ) = { f : A → B | f ( f ( x )) = f ( x ) , ∀ x ∈ A } To motivate this concept, we provide a brief example.
Example 1 : Suppose that f : R → R ∋ f ( x ) = x . Let A = R = B . Thenwe can now obviously write f : A → B ∋ f ( x ) = x . Let x ∈ R = A be anarbitrary value in the domain for x . By definition, f ( x ) = x , which meansthat f ( f ( x )) = f ( x ) = x , ∀ x ∈ A = R . Therefore, it holds that f ∈ O ( R )from our definition. A nearly identical proof holds for a complex domain.The previous case gives an immediate and quite obvious example of anOuroboros function, but it does so under a particular condition. Namely, thelast example unfolds under the condition that A = B , but this seems a bit boldof an assumption to make. In the following example, we prove that f : A → B can be an Ouroboros function without the condition that A = B . Example 2 : Suppose we have some real number c ∈ R which determines thefunction g : R → { c } ∋ g ( x ) = c . Obviously, we can let A = R and B = { c } ,observing that B = { c } ⊂ R = A . From this, we have a function of the form g : A → B again. If we let x ∈ R = A once more, we see that g ( x ) = c , whichmeans that g ( g ( x )) = g ( c ), but from our definition of this constant function, g ( c ) = c , so g ( g ( x )) = g ( c ) = c = g ( x ) , ∀ x ∈ R . From this deduction, weultimately conclude that g ∈ O ( R ).We have now shown that it is possible for an Ouroboros function to exist inthe form f : A → B where B ⊆ A . A natural question after all of this is: canwe have an Ouroboros function where A ⊂ B ? In short, the answer is no, whichleads to the following lemma: 2 emma 1 : Suppose we have a univariate function f . If f ∈ O ( A ), wherewe have O ( A ) = { f : A → B : f ( f ( x )) = f ( x ) , ∀ x ∈ A } and f : A → B , then B ⊆ A . Proof : We will prove this lemma by employing the method of proof by con-tradiction. We assume we have an Ouroboros function f ∈ O ( A ), where f : A → B and A ⊂ B , noting that f ( x ) is defined for inputs only in thedomain A . Let us define C = B/A such that C ⊂ B , but C * A and A * C .This further implies that A ∪ C = B . From these definitions, we deduce that ∃ x ∈ A ∋ f ( x ) ∈ C , which since C ⊂ B , means that f ( x ) ∈ B also. How-ever, we stated that C * A and A * C , which means that since f ( x ) ∈ C ,it also holds that f ( x ) / ∈ A . This means that f ( f ( x )) is undefined because f ( x ) / ∈ A , implying that f ( f ( x )) = f ( x ). This would mean f is not anOuroboros function, which is a clear contradiction of our assumptions. There-fore, if f ∈ O ( A ) where f : A → B , then B ⊆ A .This lemma gives us valuable information about Ouroboros functions andOuroboros spaces. Naturally, we will now revise our initial definition to providethe true definition of an Ouroboros space for the domain given by A . Definition 1 : The
Ouroboros Space for the Domain Given by A is afunction space given by: O ( A ) = { f : A → B | f ( f ( x )) = f ( x ) , ∀ x ∈ A, ∀ B ⊆ A } If f ∈ O ( A ), then f ( x ) is said to be an Ouroboros Function for the Do-main Given by A .Now that we have obtained the proper definition of an Ouroboros space, wecan apply example 1 to several iterations of the parent linear function. If f : Z → Z ∋ f ( x ) = x, ∀ x ∈ Z , then f ∈ O ( Z ). We could keep changing the domain andcodomain of our function to find examples of functions from O ( N ), O ( Q ), and O ( C ), but these examples would just be boring iterations of the same idea. Thenext progression regarding Ouroboros spaces and Ouroboros functions comesfrom the concept of a multivariate extension. From this extension, as we willsee, comes a relatively shocking result about a commonly used numerical valuethat is absolutely essential to the formal and natural sciences. To start thisprocess, we begin with a new, rough definition of a multivariate Ouroborosspace that we will later perfect as we did with our univariate definition: O ( A n ) = { f : A n → B | f ( f ( x ) , ..., f ( x )) = f ( x ) , ∀ x ∈ A n } wherein we assume x = [ x ... x n ] T ∈ A n and f ( x ) = f ( x , ..., x n ), so that wealso mean f ( f ( x ) , ..., f ( x )) = f ( f ( x , ..., x n ) , ..., f ( x , ..., x n )). We also assumethat for i = 1 , ..., n ∈ N , x i ∈ A . We will now state the multivariate analog oflemma 1 and give a proof of its implications.3 emma 2 : Suppose we have a multivariate function f . If f ∈ O ( A n ),where we have O ( A n ) = { f : A n → B | f ( f ( x ) , ..., f ( x )) = f ( x ) , ∀ x ∈ A n } and f : A n → B , then B ⊆ A . Proof : We again prove this lemma through proof by contradiction. Assume f ∈ O ( A n ), f : A n → B , and A ⊂ B , again noting that f is defined only forinputs from its domain A . Then we again define C = B/A such that C ⊂ B ,but C * A and A * C . This further implies that A ∪ C = B . From these defi-nitions, we deduce that ∃ c , ..., c n ∈ A ∋ f ( c , ..., c n ) ∈ C , which since C ⊂ B ,means that f ( c , ..., c n ) ∈ B also. Since C * A and A * C , f ( c , ..., c n ) ∈ C means that f ( c , ..., c n ) / ∈ A . This means that f ( f ( c , ..., c n ) , ..., f ( c , ..., c n ))is undefined, since f ( c , ..., c n ) / ∈ A , implying that this value is not in the do-main, which in turn means that f ( f ( c , ..., c n ) , ..., f ( c , ..., c n )) = f ( c , ..., c n ).This means that f is not an Ouroboros function, which contradicts our initialassumption. Therefore, if f ∈ O ( A n ) and f : A n → B , then B ⊆ A .From this lemma, we can make our second definition. This general state-ment covers Ouroboros spaces and functions of an arbitrary number of variables. Definition 2 : The
General
Ouroboros Space for the Domain Givenby A n is a function space given by: O ( A n ) = { f : A n → B | f ( f ( x ) , ..., f ( x )) = f ( x ) , ∀ x ∈ A n , ∀ B ⊆ A } If f ∈ O ( A n ), then f ( x ) is said to be an Ouroboros Function for the Do-main Given by A n , where x = [ x ... x n ] T ∈ A n Naturally, we might want to motivate this abstract higher level definitionwith a few examples. First, we consider two examples with particular dimen-sion sizes (namely n = 2 and n = 3). Then, we will prove a general result thatties the ideas of an Ouroboros space and an Ouroboros function to a fundamen-tal concept in probability theory. Example 3 : Consider the function f : R → R where f ( x, y ) = ( x + y ).We can choose any x ∈ R and y ∈ R , noting that A = R = B whichmeans f : A → B . By definition, f ( x , y ) = ( x + y ), so it follows that f ( f ( x , y ) , f ( x , y )) = ( ( x + y ) + ( x + y )) = ( x + y ) = f ( x , y ).Thus, by definition, f ∈ O ( R ). Example 4 : Now, consider the function g : R → R where g ( x, y, z ) = ( x + y + z ). Again, we choose any x ∈ R , y ∈ R , and z ∈ R , noting that A = R = B such that g : A → B . By definition, g ( x , y , z ) = ( x + y + z ), so it natu-rally follows that we can go on to write g ( g ( x , y , z ) , g ( x , y , z ) , g ( x , y , z )) = ( ( x + y + z )+ ( x + y + z )+ ( x + y + z )) = ( x + y + z ) = g ( x , y , z ).Therefore, following again from our definition, we can say that g ∈ O ( R ).4 robability Theory with Ouroboros Functions We are starting to notice a pattern in our examples from the previous sectionregarding higher dimensional Ouroboros functions. In example 3, our functionis essentially just the average of two numbers ( x and y ), and in example 4, ourfunction is just the average of three numbers ( x , y , and z ). This leads us to ageneral theorem regarding arithmetic averages and Ouroboros spaces. Theorem 1 : Suppose we have a general arithmetic average function of n arbitrary variables x i ∈ R , ∀ i = 1 , ..., n ∈ N , denoted by:A : R n → R ∋ A( x , ..., x n ) = 1 n n X i =1 x i . For any arbitrary dimension n ∈ N , A( x , ..., x n ) ∈ O ( R n ) , ∀ x i ∈ R . Proof : We first must state the general definition of O ( R n ) based on our generalOuroboros space definition for x = [ x ... x n ] T ∈ R n : O ( R n ) = { f : R n → B | f ( f ( x ) , ..., f ( x )) = f ( x ) , ∀ x ∈ R n , ∀ B ⊆ R } In the case of A( x , ..., x ), it is obvious that B = R ⊆ R , so this condition ismet. Now we simply need to show that A is a self-referential function. We beginas usual by choosing an arbitrary string of n values from the domain, given by c , ..., c n ∈ R . By the definition of our function, we know that:A( c , ..., c n ) = 1 n n X i =1 c i = 1 n ( c + ... + c n )We can now observe that the value of A(A( c , ..., c n ) , ..., A( c , ..., c n )) is givenby:A(A( c , ..., c n ) , ..., A( c , ..., c n )) = A( 1 n ( c + ... + c n ) , ..., n ( c + ... + c n )) =1 n n X i =1 x i , ∀ i = 1 , ..., n ∋ x i = 1 n ( c + ... + c n )Knowing this, we can deduce that:A(A( c , ..., c n ) , ..., A( c , ..., c n )) = 1 n ( 1 n ( c + ... + c n ) + ... + 1 n ( c + ... + c n )) =1 n ( nn ( c + ... + c n )) = 1 n ( c + ... + c n ) = 1 n n X i =1 c i = A( c , ..., c n )Therefore, since A : R n → R , R ⊆ R , and A(A( x , ..., x n ) , ..., A( x , ..., x n )) =A( x , ..., x n ), we conclude that A( x , ..., x n ) ∈ O ( R n ).5his is quite an idea, for it is difficult to think of functions that satisfy theOuroboros equation by simply brainstorming strange functions. This relation-ship seems vaguely familiar to a relationship in probability theory regardingexpected value. For a random variable X , it always holds that E [ E [ X ]] = E [ X ],and this property seems to echo the Ouroboros equation. Moreover, whetherwe are discussing A(A( x ) , ..., A( x )) or E [ E [ X ]], we are referring to the averageof averages, both of which yield a self-same average.We now recall the Strong Law of Large Numbers, which is adapted fromthe definition given by Evans [2]. Suppose we have independent, identicallydistributed, integrable random random variables X , ..., X n on the probabilityspace S = (Ω , F , P ), where Ω is the sample space, F is an appropriate σ -algebra,and P is a probability measure. Also, let i = 1 , ..., n ∈ N . Then: P lim n →∞ n n X i =1 X i = E [ X i ] ! = 1Again, we have adapted Evans’ definition with slight notational differences [2].Now, suppose Ω ⊂ Ω denotes the sample subspace of all events ω z such that P ( ω z ) = 0. Let Ω C = Ω / Ω contain all events whose probabilities are not zero.Suppose now that X , ..., X n are still defined on S C = (Ω C , F C , P ), where F C isan analogously appropriate σ -algebra for Ω C , such that every random variableis both F -measurable and F C -measurable. Then we can say:lim n →∞ n n X i =1 X i = E [ X i ] (a . s . )Assume in this case that for each X i where i = 1 , ..., n ∈ N , we can say that X i : Ω C → R , ∀ i = 1 , ..., n . Assume also that we have the following equality ofexpected values for any i = 1 , ..., n : Z Ω X i d P = E [ X i ] = Z Ω C X i d P If we use our previous definition of:A( x , ..., x n ) = 1 n n X i =1 x i Then we can go on to write:A( X , ..., X n ) = 1 n n X i =1 X i Furthermore, if we impose an infinite limit on this expression, we find that:lim n →∞ A( X , ..., X n ) = lim n →∞ n n X i =1 X i = E [ X i ] (a . s . )6rom our general definition, we can equivalently define: O ( R ∞ ) = { f : R ∞ → B | f ( f ( x ) , f ( x ) , ... ) = f ( x ) , ∀ x ∈ R ∞ , ∀ B ⊆ R } Here, we can think of x of being in the form: x = lim n →∞ [ x ... x n ] T Finally, from Theorem 1 and our assumption of equal expected values, we canremove the “almost surely” and make the ultimate conclusion that:lim n →∞ A( X , ..., X n ) = E [ X i ] ∈ O ( R ∞ )We can generalize our previous discussion into the form of a second theoremthat encompasses all of our assumptions and conclusions, summarized with aquick proof. Theorem 2 : Suppose X , ..., X n is a sequence of independent, identicallydistributed, integrable, F -measurable and F C -measurable random variables de-fined on the probability spaces S = (Ω , F , P ) and S C = (Ω C , F C , P ), whereΩ ⊂ Ω denotes the sample subspace of all events ω z such that P ( X = ω z ) = 0and Ω C = Ω / Ω , such that F and F C are the appropriate σ -algebras for Ω andΩ C , respectively. Moreover, assume that each X i ( ω C ) = X i takes the form X i : Ω C → R , ∀ ω C ∈ Ω C , ∀ i = 1 , ..., n ∈ N . Furthermore, let us assume ∀ i = 1 , ..., n that: Z Ω X i d P = E [ X i ] = Z Ω C X i d P Then, in accordance with the Strong Law of Large Numbers, for any i = 1 , ..., n ,it holds that: E [ X i ] ∈ O ( R ∞ ) Proof : Suppose that all of the conditions in Theorem 2 are met as needed.Let the general arithmetic average function (A( x , ..., x n )) be defined as wehave previously defined it. Observe that under the previously stated conditions,A( X , ..., X n ) : R n → R for any positive integer n . Naturally, as we said before:lim n →∞ A( X , ..., X n ) = lim n →∞ n n X i =1 X i Similarly, we see that when n → ∞ , A( X , ..., X n ) : R ∞ → R , which in turnmeans that: lim n →∞ A( X , ..., X n ) ∈ O ( R ∞ )By the Strong Law of Large Numbers, we can conclude that for i = 1 , ..., n ∈ N : E [ X i ] = lim n →∞ n n X i =1 X i = lim n →∞ A( X , ..., X n ) (a . s . )7his means that E [ X i ] ∈ O ( R ∞ ) almost surely . However, we have assumedthat our random variables are also defined on S C (as given in the theorem) andare F C -measurable, which removes all zero-probability events from the samplespace, while also assuming that the expected value in question remains un-changed, regardless of the two sample spaces. This removes the “almost surely”condition and allows us to conclude that E [ X i ] ∈ O ( R ∞ ).Breaking away from this fascinating result, we should now briefly discuss thecase of mixed domains, or rather, domains that can be expressed as Cartesianproducts of unequal sets. Suppose we have a domain given by ∆ = [ A × ... × A n ]where A k = A j , ∀ k = j . We denote the Ouroboros space for all i = 1 , ..., n ∈ N such that ∆ = [ A × ... × A n ] for this domain by the following definition: O (∆) = { f : ∆ → B | f ( f ( x ) , ..., f ( x )) = f ( x ) , ∀ x ∈ ∆ , ∀ B ⊆ A i } This slightly altered definition requires that B is a subset of each A i , and wewill now prove the following lemma stating that A i B , for all i . Lemma 3 : If ∆ = [ A × ... × A n ], f ∈ O (∆), and f : ∆ → B , then B ⊆ A i for all A i and i = 1 , ..., n . Proof : Assume f : ∆ → B , f ∈ O (∆) with ∆ = [ A × ... × A n ], and ∃ A i ⊂ B forsome i ∈ , ..., n ⊂ N , in which there is some x i ∈ A i such that f ( x i ) ∈ B/A i = C , which means that f ( x , ..., x i , ..., x n ) / ∈ A i because A i * C and C * A i .Therefore, f ( f ( x , ..., x i , ..., x n ) , ..., f ( x , ..., x i , ..., x n )) = f ( x , ..., x i , ..., x n ) be-cause f ( x , ..., x i , ..., x n ) is not contained in at least one part of the domain,leading to an undefined result. Thus, f cannot be an Ouroboros function,which contradicts our initial assumption, so Lemma 3 holds for every A i . Conclusion
We have constructed an important type of function space that consists of all ofthe functions that satisfy the enigmatic Ouroboros equation [1]. We observedthat this notion can be extended to functions of multiple variables, and gave sev-eral examples of higher dimensional Ouroboros functions. In this exploration,we discovered that we can write the arithmetic average as a function of n vari-ables, and, more interestingly, that this arithmetic average function is indeedan Ouroboros function. Finally, we made use of the Strong Law of Large Num-bers to prove that under certain conditions, the expected value of a randomvariable can be an Ouroboros function for an infinite domain. We also pointedout that there can be Ouroboros functions for mixed domains. The concept ofa self-referential function is imbued with immense mathematical, scientific, andphilosophical significance, and we have linked the concept of an Ouroboros spacein functional analysis to probability theory. Undoubtedly, further investigationinto these concepts will yield indescribably fruitful results going forward.8 eferenceseferences