Transparallel mind: classical computing with quantum power

Abstract

Inspired by the extraordinary computing power promised by quantum computers, the quantum mind hypothesis postulated that quantum mechanical phenomena are the source of neuronal synchronization, which, in turn, might underlie consciousness. Here, I present an alternative inspired by a classical computing method with quantum power. This method relies on special distributed representations called hyperstrings. Hyperstrings are superpositions of up to an exponential number of strings, which—by a single-processor classical computer—can be evaluated in a transparallel fashion, that is, simultaneously as if only one string were concerned. Building on a neurally plausible model of human visual perceptual organization, in which hyperstrings are formal counterparts of transient neural assemblies, I postulate that synchronization in such assemblies is a manifestation of transparallel information processing. This accounts for the high combinatorial capacity and speed of human visual perceptual organization and strengthens ideas that self-organizing cognitive architecture bridges the gap between neurons and consciousness.

Appendix

Structural information theory (SIT) adopts the simplicity principle, which holds that the simplest organization of a visual stimulus is the one most likely perceived by humans. To enable quantifiable predictions, SIT developed a formal coding model to determine simplest codes of symbol strings. This model provides coding rules for the extraction of regularities, and a metric of the complexity, or structural information load $I$, of codes.

The coding rules serve the extraction of the transparent holographic regularities repetition (or iteration I), symmetry (S), and alternation (A). They can be applied to any substring of an input string, and a code of the input string consists of a string of symbols and coded substrings, such that decoding the code returns the input string. Formally, SIT’s coding language and complexity metric are defined as follows.

Definition 1

A code $\overline{X}$ of a string $X$ is a string $t_1t_2 \ldots t_m$ of code terms $t_i$ such that $X = D(t_1)\ldots D(t_m)$, where the decoding function $D : t \rightarrow D(t)$ takes one of the following forms:

for strings $y$, $p$, and $x_i$ ($i = 1,2,\ldots ,n$). The code parts $(\overline{y})$, $(\overline{p})$, and $(\overline{x_i})$ are chunks. The chunk $(\overline{y})$ in an I-form or an A-form is a repeat, and the chunk $(\overline{p})$ in an S-form is a pivot which, as a limit case, may be empty. The chunk string $(\overline{x_1})(\overline{x_2})\ldots (\overline{x_n})$ in an S-form is an S-argument consisting of S-chunks $(\overline{x_i})$, and in an A-form, it is an A-argument consisting of A-chunks $(\overline{x_i})$.

Definition 2

Let $\overline{X}$ be a code of string $X=s_1 s_2\ldots s_N$. The complexity $I$ of $\overline{X}$ in structural information parameters (sip) is given by the sum of (a) the number of remaining symbols $s_i$ ($1 \le i \le N$) and (b) the number of chunks $(\overline{y})$ in which $y$ is neither a symbol nor an S-chunk.

The last part of Definition 2 may seem somewhat ad hoc, but has a solid theoretical basis in terms of degrees of freedom in the hierarchical organization described by a code. Furthermore, Definition 1 implies that a string may be encodable into many different codes. For instance, a code may involve not only recursive encodings of strings inside chunks—that is, from $(y)$ into $(\overline{y})$—but also hierarchically recursive encodings of S- or A-arguments $(\overline{x_1})(\overline{x_2})\ldots (\overline{x_n})$ into $\overline{(\overline{x_1})(\overline{x_2})\ldots (\overline{x_n})}$. The following sample of codes for one and the same string may give a gist of the abundance of coding possibilities:

Code 1 is a code with six code terms, namely, one S-form, two I-forms, and three symbols. Code 2 is an A-form with chunks containing strings that may be encoded as given in Code 3. Code 4 is an S-form with an empty pivot and illustrates that, in general, S-forms describe broken symmetry; mirror symmetry then is the limit case in which every S-chunk contains only one symbol. Code 5 gives a hierarchical recoding of the S-argument in Code 4. Code 6 is an I-form in which the repeat has been encoded into an A-form with an A-argument that has been recoded hierarchically into an S-form.

The computation of a simplest code for a string requires an exhaustive search for ISA-forms into which its substrings can be encoded, followed by the selection of a simplest code for the entire string. This also requires the hierarchical recoding of S- and A-arguments: A substring of length $k$ can be encoded into $O(2^k)$ S-forms and $O(k2^k)$ A-forms, and to pinpoint a simplest one, simplest codes of the arguments of these S- and A-forms have to be determined as well—and so on, with $O(\log N)$ recursion steps, because $k/2$ is the maximal length of the argument of an S- or A-form into which a substring of length $k$ can be encoded. Hence, recoding S- and A-arguments separately would require a superexponential $O(2^{N \log N})$ total amount of work.

This combinatorial explosion can be nipped in the bud by gathering the arguments of S- and A-forms in distributed representations. The next definitions and proofs show that S-arguments and A-arguments group naturally into distributed representations consisting of one or more independent hyperstrings—which enable the hierarchical recoding of up to an exponential number of S- or A-arguments in a transparellel fashion, that is, simultaneously as if only one argument were concerned. Here, only A-forms $\langle (y)\rangle /\langle (x_1)(x_2)...(x_n)\rangle$ with repeat $y$ consisting of one element are considered, but Definition 4 and Theorem 1 below hold mutatis mutandis for other A-forms as well.

• Graph-theoretical definition of hyperstrings:

Definition 3

A hyperstring is a simple semi-Hamiltonian directed acyclic graph $(V,E)$ with a labeling of the edges in $E$ such that, for all vertices $i,j,p,q \in V$:

$$\begin{aligned} \hbox {either }\pi (i,j) = \pi (p,q)\hbox { or }\pi (i,j) \cap \pi (p,q) = \emptyset \end{aligned}$$

where substring set $\pi (v_1,v_2)$ is the set of label strings represented by the paths between vertices $v_1$ and $v_2$; the subgraph on the vertices and edges in these paths is a hypersubstring.

• Definition of distributed representations called A-graphs, which represent all A-forms covering suffixes of strings, that is, all A-forms into which those suffixes can be encoded (see Fig. 9 for an example):

Definition 4

For a string $T = s_1s_2\ldots s_N$, the A-graph $\mathcal {A}(T)$ is a simple directed acyclic graph $(V,E)$ with $V = \lbrace 1,2,\ldots ,N+1\rbrace$ and, for all $1 \le i < j \le N$, edges $(i,j)$ and $(j,N+1)$ labeled with, respectively, the chunks $(s_i\ldots s_{j-1})$ and $(s_j\ldots s_N)$ if and only if $s_i = s_j$.

• Definition of diafixes, which are substrings centered around the midpoint of a string (this notion complements the known notions of prefixes and suffixes, and facilitates the explication of the subsequent definition of S-graphs):

Definition 5

A diafix of a string $T = s_1s_2\ldots s_N$ is a substring $s_{i+1}\ldots s_{N-i}$ ($0 \le i < N/2$).

• Definition of distributed representations called S-graphs, which represent all S-forms covering diafixes of strings (see Fig. 10 for an example):

Definition 6

For a string $T = s_1s_2\ldots s_N$, the S-graph $\mathcal {S}(T)$ is a simple directed acyclic graph $(V,E)$ with $V = \lbrace 1,2,\ldots ,\lfloor N/2 \rfloor + 2\rbrace$ and, for all $1 \le i < j < \lfloor N/2 \rfloor + 2$, edges $(i,j)$ and $(j,\lfloor N/2 \rfloor + 2)$ labeled with, respectively, the chunk $(s_i\ldots s_{j-1})$ and the possibly empty chunk $(s_j\ldots s_{N-j+1})$ if and only if $s_i\ldots s_{j-1} = s_{N-j+2}\ldots s_{N-i+1}$.

Fig. 9

The A-graph for string $T = akagakakag$, with three independent hyperstrings (connected only at vertex $11$) for the three sets of A-forms with repeats $a$, $k$, and $g$, respectively, which cover suffixes of $T$. An A-graph may contain so-called pseudo A-pair edges—like, here, edge $(10,11)$—which do not correspond to actual repeat plus A-chunk pairs; they cannot end up in codes but are needed to maintain the integrity of hyperstrings during recoding

Fig. 10

The S-graph for string $T=\textit{ababfdedgpfdedgbaba}$, with two independent hyperstrings given by the solid edges, which represent S-chunks in S-forms covering diafixes of $T$. The dashed edges represent the pivots, which come into play after hyperstring recoding

Theorem 1

The A-graph $\mathcal {A}(T)$ for a string $T = s_1s_2\ldots s_N$ consists of at most $N+1$ disconnected vertices and at most $\lfloor N/2 \rfloor$ independent subgraphs (i.e., subgraphs that share only the sink vertex $N+1$), each of which is a hyperstring.

Proof

• (1) By Definition 4, vertex $i$ ($i \le N$) in $\mathcal {A}(T)$ does not have incoming or outgoing edges if and only if $s_i$ is a unique element in $T$. Since $T$ contains at most $N$ unique elements, $\mathcal {A}(T)$ contains at most $N+1$ disconnected vertices, as required.

• (2) Let $s_{i_1},s_{i_2},\ldots ,s_{i_n}$ ($i_p < i_{p+1}$) be a complete set of identical elements in $T$. Then, by Definition 4, the vertices $i_1,i_2,\ldots ,i_n$ in $\mathcal {A}(T)$ are connected with each other and with vertex $N+1$ but not with any other vertex. Hence, the subgraph on the vertices $i_1,i_2,\ldots ,i_n,N+1$ forms an independent subgraph. For every complete set of identical elements in $T$, $n$ may be as small as $2$, so that $\mathcal {A}(T)$ contains at most $\lfloor N/2 \rfloor$ independent subgraphs, as required.

• (3) The independent subgraphs must be semi-Hamiltonian to be hyperstrings. Now, let $s_{i_1},s_{i_2},\ldots ,s_{i_n}$ ($i_p < i_{p+1}$) again be a complete set of identical elements in $T$. Then, by Definition 4, $\mathcal {A}(T)$ contains edges $(i_p,i_{p+1})$, $p = 1,2,\ldots ,n-1$, and it contains edge $(i_n,N+1)$. Together, these edges form a Hamiltonian path through the independent subgraph on the vertices $i_1,i_2,\ldots ,i_n,N+1$, as required.

• (4) The only thing left to prove is that the substring sets are pairwise either identical or disjoint. Now, for $i < j$ and $k \ge 1$, let substring sets $\pi (i,i+k)$ and $\pi (j,j+k)$ in $\mathcal {A}(T)$ be not disjoint, that is, let them share at least one chunk string. Then, the substrings $s_{i}\ldots s_{i+k-1}$ and $s_{j}\ldots s_{j+k-1}$ of $T$ are necessarily identical and, also necessarily, $s_{i} = s_{i+k}$ and either $s_{j} = s_{j+k}$ or $j+k = N+1$. Hence, by Definition 4, these identical substrings of $T$ yield, in $\mathcal {A}(T)$, edges $(i,i+k)$ and $(j,j+k)$ labeled with the identical chunks $(s_{i}\ldots s_{i+k-1})$ and $(s_{j}\ldots s_{j+k-1})$, respectively. Furthermore, obviously, these identical substrings of $T$ can be chunked into exactly the same strings of two or more identically beginning chunks. By Definition 4, all these chunks are represented in $\mathcal {A}(T)$, so that each of these chunkings is represented not only by a path $(i,\ldots ,i+k)$ but also by a path $(j,\ldots ,j+k)$. This implies that the substring sets $\pi (i,i+k)$ and $\pi (j,j+k)$ are identical. The foregoing holds not only for the entire A-graph but, because of their independence, also for every independent subgraph. Hence, in sum, every independent subgraph is a hyperstring, as required.$\square$

Lemma 1

Let the strings $c_1 = s_1s_2\ldots s_k$ and $c_2 = s_1s_2\ldots s_p$ ($k<p$) be such that $c_2$ can be written in the following two ways:

$$\begin{aligned} c_2 = c_1X \quad with \quad X = s_{k+1}\ldots s_p\\ c_2 = Yc_1 \quad with \quad Y = s_1\ldots s_{p-k} \end{aligned}$$

Then, $X=Y$ if $q=p/(p-k)$ is an integer; otherwise $Y=VW$ and $X=WV$, where $V = s_1\ldots s_r$ and $W = s_{r+1}\ldots s_{p-k}$, with $r= p- \lfloor q \rfloor (p-k)$.

Proof

• (1) If $1<q<2$, then $c_2=c_1Wc_1$, so that $Y=c_1W$ and $X=Wc_1$. Then, too, $r=k$, so that $c_1=V$. Hence, $Y=VW$ and $X=WV$, as required.

• (2) If $q=2$, then $c_2=c_1c_1$. Hence, $X=Y=c_1$, as required.

• (3) If $q>2$, then the two copies of $c_1$ in $c_2$ overlap each other as follows:

  $$\begin{aligned} \begin{array}{lccccccccccc} c_2 = c_1 X =s_1 &{} \ldots &{} s_{p-k} &{} s_{p-k+1} &{} \ldots &{} s_k &{} s_{k+1} &{} \ldots &{} s_p\\ c_2 = Y c_1 =&{} Y &{} &{} s_1 &{} \ldots &{} s_{2k-p} &{} s_{2k-p+1} &{} \ldots &{} s_k \end{array} \end{aligned}$$

  Hence, $s_i=s_{p-k+i}$ for $i=1,2,\ldots ,k$. That is, $c_2$ is a prefix of an infinite repetition of $Y$.

• (3a) If $q$ is an integer, then $c_2$ is a $q$-fold repetition of $Y$, that is, $c_2=YY\ldots Y$. This implies (because also $c_2=Yc_1$) that $c_1$ is a $(q-1)$-fold repetition of $Y$, so, $c_2$ can also be written as $c_2=c_1Y$. This implies $X=Y$, as required.

• (3b) If $q$ is not an integer, then $c_2$ is a $\lfloor q \rfloor$-fold repetition of $Y$ plus a residual prefix $V$ of $Y$, that is, $c_2=YY\ldots YV$. Now, $Y=VW$, so that $c_2$ can also be written as $c_2=VWVW\ldots VWV$. This implies (because also $c_2=Yc_1=VWc_1$) that $c_1=VW\ldots VWV$, that is, $c_1$ is a $(\lfloor q \rfloor - 1)$-fold repetition of $Y = VW$ plus a residual part $V$. This, in turn, implies that $c_2$ can also be written as $c_2=c_1WV$, so that $X=WV$, as required.$\square$

Lemma 2

Let $\mathcal {S}(T)$ be the S-graph for a string $T = s_1s_2\ldots s_N$. Then:

• (1) If $\mathcal {S}(T)$ contains edges $(i,i+k)$ and $(i,i+p)$, with $k < p < \lfloor N/2 \rfloor + 2 - i$, then it also contains a path $(i+k,\ldots ,i+p)$.

• (2) If $\mathcal {S}(T)$ contains edges $(i-k,i)$ and $(i-p,i)$, with $k<p$ and $i < \lfloor N/2 \rfloor + 2$, then it also contains a path $(i-p,\ldots ,i-k)$.

Proof

• (1) Edge $(i,i+k)$ represents S-chunk $(c_1) = (s_i\ldots s_{i+k-1})$, and edge $(i,i+p)$ represents S-chunk $(c_2) = (s_i\ldots s_{i+p-1})$. This implies that diafix $D = s_i\ldots s_{N-i+1}$ of $T$ can be written in two ways:

  $$\begin{aligned} \begin{array}{ccccc} D &{} = &{} c_2 &{} \ldots &{} c_2\\ D &{} = &{} c_1 &{} \ldots &{} c_1 \end{array} \end{aligned}$$

  This implies that $c_2$ (which is longer than $c_1$) can be written in two ways:

  $$\begin{aligned} c_2 = c_1X \quad \hbox { with }\quad X = s_{i+k}\ldots s_{i+p-1}\\ c_2 = Yc_1 \quad \hbox { with }\quad Y = s_i\ldots s_{i+p-k-1} \end{aligned}$$

  Hence, by Lemma 1, either $X = Y$ or $Y = VW$ and $X =WV$ for some $V$ and $W$. If $X = Y$, then $D = c_1Y\ldots Yc_1$ so that, by Definition 6, $Y$ is an S-chunk represented by an edge that yields a path $(i+k,\ldots ,i+p)$ as required. If $Y = VW$ and $X =WV$, then $D=c_1WV\ldots VWc_1$ so that, by Definition 6, $W$ and $V$ are S-chunks represented by subsequent edges that yield a path $(i+k,\ldots ,i+p)$ as required.

• (2) This time, edge $(i-k,i)$ represents S-chunk $(c_1) = (s_{i-k}\ldots s_{i-1})$, and edge $(i-p,i)$ represents S-chunk $(c_2) = (s_{i-p}\ldots s_{i-1})$. This implies that diafix $D = s_{i-p}\ldots s_{N-i+p+1}$ of $T$ can be written in two ways:

  $$\begin{aligned} \begin{array}{ccccc} D &{} = &{} c_2 &{} \ldots &{} c_2\\ D &{} = &{} Yc_1 &{} \ldots &{} c_1X \end{array} \end{aligned}$$

  with $X$=$s_{i-p+k}\ldots s_{i-1}$ and $Y$=$s_{i-p}\ldots s_{i-k-1}$. Hence, as before, $c_2 = c_1X$ and $c_2 = Yc_1$, so that, by Lemma 1, either $X=Y$ or $Y = VW$ and $X =WV$ for some $V$ and $W$. This implies either $D=Yc_1\ldots c_1Y$ or $D=VWc_1\ldots c_1WV$. Hence, this time, Definition 6 implies that both cases yield a path $(i-p,\ldots ,i-k)$, as required.$\square$

Lemma 3

In the S-graph $\mathcal {S}(T)$ for a string $T = s_1s_2\ldots s_N$, the substring sets $\pi (v_1,v_2)$ ($1 \le v_1 < v_2 < \lfloor N/2 \rfloor + 2$) are pairwise identical or disjoint.

Proof

Let, for $i < j$ and $k \ge 1$, substring sets $\pi (i,i+k)$ and $\pi (j,j+k)$ in $\mathcal {S}(T)$ be nondisjoint, that is, let them share at least one S-chunk string. Then, the substrings $s_{i}\ldots s_{i+k-1}$ and $s_{j}\ldots s_{j+k-1}$ in the left-hand half of $T$ are necessarily identical to each other. Furthermore, by Definition 6, the substring in each chunk of these S-chunk strings is identical to its symmetrically positioned counterpart in the right-hand half of $T$, so that also the substrings $s_{N-i-k+2}\ldots s_{N-i+1}$ and $s_{N-j-k+2}\ldots s_{N-j+1}$ in the right-hand half of $T$ are identical to each other. Hence, the diafixes $D_1=s_{i}\ldots s_{N-i+1}$ and $D_2=s_{j}\ldots s_{N-j+1}$ can be written as

$$\begin{aligned} D_1 = s_{i}\ldots s_{i+k-1}\;\;p_1\;\;s_{N-i-k+2}\ldots s_{N-i+1}\\ D_2 = s_{i}\ldots s_{i+k-1}\;\;p_2\;\;s_{N-i-k+2}\ldots s_{N-i+1} \end{aligned}$$

with $p_1=s_{i+k}\ldots s_{N-i-k+1}$ and $p_2=s_{j+k}\ldots s_{N-j-k+1}$. Now, by means of any S-chunk string $C$ in $\pi (i,i+k)$, diafix $D_1$ can be encoded into the covering S-form $S[C,(p_1)]$. If pivot $(p_1)$ is replaced by $(p_2)$, then one gets the covering S-form $S[C,(p_2)]$ for diafix $D_2$. This implies that any S-chunk string in $\pi (i,i+k)$ is in $\pi (j,j+k)$, and vice versa. Hence, nondisjoint substring sets $\pi (i,i+k)$ and $\pi (j,j+k)$ are identical, as required. $\square$

Theorem 2

The S-graph $\mathcal {S}(T)$ for a string $T = s_1s_2\ldots s_N$ consists of at most $\lfloor N/2 \rfloor + 2$ disconnected vertices and at most $\lfloor N/4 \rfloor$ independent subgraphs that, without the sink vertex $\lfloor N/2 \rfloor + 2$ and its incoming pivot edges, form one disconnected hyperstring each.

Proof

From Definition 6, it is obvious that there may be disconnected vertices and that their number is at most $\lfloor N/2 \rfloor + 2$, so let us turn to the more interesting part. If $\mathcal {S}(T)$ contains one or more paths $(i,\ldots ,j)$ ($i<j<\lfloor N/2 \rfloor +2$) then, by Lemma 2, one of these paths visits every vertex $v$ with $i<v<j$ and $v$ connected to $i$ or $j$. This implies that, without the pivot edges and apart from disconnected vertices, $\mathcal {S}(T)$ consists of disconnected semi-Hamiltonian subgraphs. Obviously, the number of such subgraphs is at most $\lfloor N/4 \rfloor$, and if these subgraphs are expanded to include the pivot edges, they form one independent subgraph each. More important, by Lemma 3, these disconnected semi-Hamiltonian subgraphs form one hyperstring each, as required. $\square$