A Tutorial in Data Science: Lecture 10 – The Fundamental Law of Data Science (Inter-Communication is Functional Chaos)

by | Jan 12, 2021 | Math Lecture

We will prove below that a system is inter-communicating if and only if it is chaotic. This is the fundamental law of data science, since it is both analytical, as coming from definitions, and synthetic, as explaining empirical observation, thus satisfying the Kantian transcendental deduction for laws of nature as analytical-synthetic.

According to Robert Devaney, the father of Chaos Theory, a function f is Chaotic if f satisfies the three criteria: (i) f has a sensitive dependence on the initial conditions, (ii) the periodic points of f are dense in X, and (iii) f is topologically transitive [1].

  1. \exists \epsilon >0 \ s.t. \ \forall x \in X \ \& \ \delta>0 \ \exists y \in X \ w/ \ d(x,y)<\delta \ \& \ n\in \mathbb{N} \ s.t. \ d(f^n(x),f^n(y)\geq \epsilon.
  2. \forall \delta>0 \ \forall x \in X \ \exists p' \in Per(f)={p: \exists n \in \mathbb{N} \ s.t. f^{n}(p)=p, \ p \in X } \ s.t \ d(x,p')<\delta
  3. \exists y \in X \ s.t. \ \forall \delta>0 \ \forall x \in X \ \exists y' \in \mathcal{O}(y)={f^n(x):n \in \mathbb{N}_0}\ s.t \ d(x,y')<\delta
    Stronger Criteria Statement: \forall \mathcal{U}, \mathcal{V} \subset X, \ \text{open and non-empty}, \ \exists m \in \mathbb{N} \ s.t. \ f^m(\mathcal{U}) \cap \mathcal{V} \neq \emptyset
    (Standard) Conditions of Equivalency: 2nd Category and no isolated points

The iterates of an underlying finite-difference function, f, with the outcomes landing in different intervals or partitions that are marked via sampling as states, give the codes that form the symbolic shift space of possibilities as its total messages. The set of all possible state-strings from an experiment with an origin and a b-sized state-space is the one-sided shift-space \Sigma^+_b, i.e. strings of numbers less than b. This has a natural operation, the shift-map, \sigma, which shifts items by their index over by 1 to the left, i.e. deleting the first item. Clearly, all code-string combinations are not possible for every experience, but the language \mathcal{L} of the experiment as the sampling of word-results from sets of iteration will be based upon the underlying functionality f at-play in the experiment, which as a stochastic process has a transition matrix P giving the probability that one state will occur after another. This may be presupposed and confirmed via experimentation, or it may be discovered through frequency estimation as approximators. If we look at only the communicativity of this transition matrix, as whether states communicate at all (\rho_{xy}>0 \implies 1) or not (\rho_{xy}>0 \implies 0) then we have the reduced form of the transition matrix A that sets up a \textbf{subshift of finite type}, \Sigma^+_A, which is a set of all possible code-strings from the experiment, i.e. a subset of \Sigma^+_b closed under \sigma. Thus, we consider matrix A, a {0, \cdots, d-1} \times {0, \cdots , d-1} matrix with entrees zero or one. A engenders the subshift of finite type, \Sigma_A^+={x=(x_k){k \in \mathbb{N}} \in \Sigma_d^+: a_{x_i, x_{i+1}}=1}. We let \sigma_A denote the shift map on \Sigma^+_A \subset \Sigma^+_b.

Within this experimental set-up, it follows that the shift map, \sigma_A is \textbf{transitive} if and only if the language of the subshift of finite type \textbf{intercommunicates}, for all i,j \in {0,, \cdots, d-1} \ \exists n=n(i,j) \in \mathbb{N} and a word w=w_1 \cdots w_n \in \mathcal{L}(\Sigma_A^+) \ s.t. \ w_i=i \ and \ w_n=j.

If \theta_A is transitive \exists x \in \Sigma^+A \ s.t. \ \mathcal{O}(x)={\sigma_A^k(x):k \in \mathbb{N}} \ \text{is dense in} \ \Sigma^+_A i.e.  \forall y \in \Sigma^+_A, \forall \delta>0 \ \exists m \in \mathbb{N} \ s.t. \ d(\sigma_A^m(x),y)<\delta. Thus, \forall w \in \mathcal{L}(\Sigma^+_b), \ \exists n \in \mathbb{N} s.t.  (x_k)_n^{n+|w|}=w \implies \exists W: \mathbb{N} \rightarrow \mathcal{L}(\Sigma_d^+)  s.t.  x=w(1)\cdots w(k) \cdots \ \forall v \in \mathcal{L}(\Sigma_d^+) \ \exists l \in \mathbb{N}  s.t. v=w(l) and specifically \exists m_k \ s.t. \sigma_A^{(m_k)}(x)=w(k)w(k+1) \ \cdots. \forall i \in {0,1, \cdots d-1}, i \in \mathcal{L}(\Sigma_A^+). Thus, \exists l_i \ s.t. \ w(l_i)=i. Let w(l_j)=j. If l_j>l_i, (x_k){m_{l_i}}^{m_{l_j}+1} =w=w(l_i)\cdots w(l_j)=w_1 \cdots w_n \in \mathcal{L}(\Sigma^+_A), \ w_1=i & w_n=j. If l_jl_i \ s.t. \ w(r_j)=j.

If \forall i,j \in {0,1, \cdots d-1} \ \exists n=n(i,j) \in \mathbb{N} and a word w=w_1\cdots w_n \in \mathcal{L}_n(\Sigma^+_A)  s.t.  w_1=i \ \& \ w_n=j. \ \forall w \in \mathcal{L}, \exists n s.t. w=w(n), \ x=w(1)\cdots w(n) \cdots.

Next we show that for every topologically transitive subshift of finite type, the periodic points are dense.

For p* \in Per(\sigma_A) with p=w(l) the repeating component of p, the periodic sequence of the word p, it is also a subword to the transitive concatenated word x. Since \forall y \in \Sigma^+_A \forall \delta> 0 \ \exists l,n \ s.t. \sigma_A^n(x)=w(l)\cdots \ d(\sigma_A^n(x),y)<\delta, \ d(w(l),y)<\delta, where w(l)*\in Per(\sigma_A) is the repetition of w(l).

It is also true that every shiftmap has a sensitive dependence upon initial conditions.

Thus, it follows that intercommunicating subshifts of finite type are chaotic and chaotic systems intercommunicate in the language subshifts of finite type.

References

  1. Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd Ed. Addison-Wesley Publishing Company, Inc, The Advanced Book Program, 1948. p.148-152.

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